Subgroups — C₃₁×#62221 #21518980

Order 7936.

1696 conjugacy classes of subgroups.

The subgroup lattice has 1104 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3517	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3387	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3362	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3353	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3350	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3337	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3335	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3333	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3317	C₂	1	2	yes	maximal, complemented
13	C₃₁×#3304	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3304	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#3260	C₂	1	2	yes	maximal, complemented
16	C₃₁×#3246	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#3241	C₂	1	2	yes	maximal, complemented
18	C₃₁×#3228	C₂	1	1	yes	maximal, characteristic, complemented
19	C₃₁×#3120	C₂	1	2	yes	maximal, complemented
20	C₃₁×#3107	C₂	1	1	yes	maximal, characteristic, complemented
21	C₃₁×#3084	C₂	1	2	yes	maximal, complemented
22	C₃₁×#3076	C₂	1	1	yes	maximal, characteristic, complemented
23	C₃₁×#2994	C₂	1	1	yes	maximal, characteristic, complemented
24	C₃₁×#563	—	2	2	no	—
25	Q₈²×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#557	—	2	1	no	—
27	C₃₁×#557	C₂²	1	1	yes	characteristic
28	C₃₁×#557	C₂²	1	1	yes	characteristic
29	C₃₁×#556	—	2	1	no	—
30	C₃₁×#556	C₂²	1	2	yes	—
31	C₃₁×#556	C₂²	1	1	yes	characteristic
32	C₃₁×#554	C₂²	1	1	yes	characteristic
33	C₃₁×#553	—	2	2	no	—
34	C₃₁×#553	C₂²	1	2	yes	—
35	C₃₁×#552	C₂²	1	2	yes	—
36	C₃₁×#550	—	2	1	no	—
37	C₃₁×#550	C₂²	1	1	yes	characteristic
38	C₃₁×#550	C₂²	1	2	yes	—
39	C₃₁×#550	C₂²	1	1	yes	characteristic
40	C₃₁×#547	—	2	1	no	—
41	C₃₁×#547	C₂²	1	1	yes	characteristic
42	C₃₁×#544	C₂²	1	2	yes	—
43	C₃₁×#542	—	2	1	no	—
44	C₃₁×#541	—	2	2	no	—
45	C₃₁×#541	—	2	1	no	—
46	C₃₁×#541	C₂²	1	2	yes	—
47	C₃₁×#540	—	2	2	no	—
48	C₃₁×#540	C₂²	1	2	yes	—
49	C₃₁×#533	C₂²	1	1	yes	characteristic
50	C₃₁×#533	C₂²	1	1	yes	characteristic
51	C₃₁×#532	—	2	1	no	—
52	C₃₁×#532	C₂²	1	1	yes	characteristic
53	C₃₁×#529	—	2	1	no	—
54	C₃₁×#529	C₂²	1	1	yes	characteristic
55	C₃₁×#520	C₂²	1	2	yes	—
56	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
57	C₃₁×#501	C₂²	1	2	yes	—
58	C₃₁×#500	C₂²	1	1	yes	characteristic
59	C₃₁×#499	C₂²	1	2	yes	—
60	C₃₁×#498	C₂²	1	1	yes	characteristic
61	C₃₁×#490	C₂²	1	2	yes	—
62	C₃₁×#486	C₂²	1	1	yes	characteristic
63	C₃₁×#486	C₂²	1	2	yes	—
64	C₃₁×#486	C₂²	1	1	yes	characteristic
65	C₃₁×#485	C₂²	1	1	yes	characteristic
66	C₃₁×#485	C₂²	1	1	yes	characteristic
67	C₃₁×#484	C₂²	1	2	yes	—
68	C₃₁×#483	C₂²	1	2	yes	—
69	C₃₁×#482	C₂²	1	2	yes	—
70	C₃₁×#482	C₂²	1	2	yes	—
71	C₃₁×#482	C₂²	1	2	yes	—
72	C₃₁×#482	C₂²	1	2	yes	—
73	C₃₁×#481	C₂²	1	2	yes	—
74	C₃₁×#480	C₂²	1	2	yes	—
75	C₃₁×#479	C₂²	1	2	yes	—
76	C₃₁×#479	C₂²	1	2	yes	—
77	C₃₁×#479	C₂²	1	2	yes	—
78	C₃₁×#479	C₂²	1	2	yes	—
79	C₃₁×#478	C₂²	1	2	yes	—
80	C₃₁×#478	C₂²	1	2	yes	—
81	C₃₁×#478	C₂²	1	2	yes	—
82	C₃₁×#478	C₂²	1	2	yes	—
83	C₃₁×#477	C₂²	1	1	yes	characteristic
84	C₃₁×#476	C₂²	1	1	yes	characteristic
85	C₃₁×#476	C₂²	1	2	yes	—
86	C₃₁×#476	C₂²	1	2	yes	—
87	C₃₁×#475	C₂²	1	1	yes	characteristic
88	C₃₁×#475	C₂²	1	1	yes	characteristic
89	C₃₁×#475	C₂²	1	1	yes	characteristic
90	C₃₁×#475	C₂²	1	2	yes	—
91	C₃₁×#475	C₂²	1	1	yes	characteristic
92	C₃₁×#474	C₂²	1	1	yes	characteristic
93	C₃₁×#474	C₂²	1	1	yes	characteristic
94	C₃₁×#468	C₂²	1	2	yes	—
95	C₃₁×#468	C₂²	1	2	yes	—
96	C₃₁×#465	C₂²	1	2	yes	—
97	C₃₁×#465	C₂²	1	2	yes	—
98	C₃₁×#465	C₂²	1	2	yes	—
99	C₃₁×#465	C₂²	1	2	yes	—
100	C₃₁×#462	C₂²	1	1	yes	characteristic
101	C₃₁×#461	C₂²	1	1	yes	characteristic
102	C₃₁×#461	C₂²	1	2	yes	—
103	C₃₁×#460	C₂²	1	1	yes	characteristic
104	C₃₁×#460	C₂²	1	1	yes	characteristic
105	C₃₁×#460	C₂²	1	2	yes	—
106	C₃₁×#460	C₂²	1	2	yes	—
107	C₃₁×#452	C₂²	1	2	yes	—
108	C₃₁×#452	C₂²	1	1	yes	characteristic
109	C₃₁×#452	C₂²	1	1	yes	characteristic
110	C₃₁×#452	C₂²	1	2	yes	—
111	C₃₁×#446	C₂²	1	1	yes	characteristic
112	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
113	C₃₁×#441	C₂²	1	2	yes	—
114	C₃₁×#440	C₂²	1	2	yes	—
115	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
116	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
117	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
118	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
119	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
120	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
121	C₃₁×#436	C₂²	1	1	yes	characteristic
122	C₃₁×#435	C₂²	1	2	yes	—
123	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
124	C₈×D₄×C₃₁	C₂²	1	2	yes	—
125	C₃₁×#433	C₂²	1	2	yes	—
126	C₃₁×#431	C₂²	1	1	yes	characteristic
127	C₃₁×#428	C₂²	1	2	yes	—
128	C₃₁×#427	C₂²	1	2	yes	—
129	C₂×C₃₁×#106	C₂²	1	2	yes	—
130	C₃₁×#424	C₂²	1	2	yes	—
131	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
132	C₃₁×#417	C₂²	1	2	yes	—
133	C₃₁×#416	C₂²	1	2	yes	—
134	C₃₁×#416	C₂²	1	2	yes	—
135	C₂×#41×C₃₁	—	2	1	no	—
136	C₂×#41×C₃₁	—	2	1	no	—
137	C₂×#41×C₃₁	—	2	1	no	—
138	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
139	C₂×#41×C₃₁	C₂³	1	2	yes	—
140	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
141	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
142	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
143	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
144	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
145	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
146	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
147	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
148	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
149	C₃₁×C₄⋊Q₈	—	2	1	no	—
150	C₃₁×C₄⋊Q₈	—	2	1	no	—
151	C₃₁×C₄⋊Q₈	—	2	2	no	—
152	C₃₁×C₄⋊Q₈	—	2	1	no	—
153	C₃₁×C₄⋊Q₈	—	2	2	no	—
154	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
155	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
156	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
157	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
158	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
159	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
160	C₃₁×#127	—	2	1	no	—
161	C₃₁×#127	—	2	2	no	—
162	C₃₁×#127	C₂³	1	1	yes	characteristic
163	C₃₁×#127	C₂³	1	2	yes	—
164	C₃₁×#126	—	2	2	no	—
165	C₃₁×#126	—	2	2	no	—
166	C₃₁×#126	—	2	2	no	—
167	C₃₁×#126	—	2	2	no	—
168	C₃₁×#126	—	2	2	no	—
169	C₃₁×#126	—	2	2	no	—
170	C₃₁×#126	—	2	2	no	—
171	C₃₁×#126	D₄	1	2	yes	—
172	C₃₁×#125	—	2	2	no	—
173	C₃₁×#125	—	2	2	no	—
174	C₃₁×#125	—	2	2	no	—
175	C₃₁×#125	—	2	2	no	—
176	C₃₁×#125	—	2	1	no	—
177	C₃₁×#125	—	2	2	no	—
178	C₃₁×#125	—	2	2	no	—
179	C₃₁×#125	—	2	1	no	—
180	C₃₁×#125	D₄	1	1	yes	characteristic
181	C₃₁×#125	C₂³	1	2	yes	—
182	C₃₁×#125	C₂³	1	2	yes	—
183	C₃₁×#125	C₂³	1	2	yes	—
184	C₃₁×#124	—	2	1	no	—
185	C₃₁×#124	—	2	2	no	—
186	C₃₁×#124	—	2	1	no	—
187	C₃₁×#124	D₄	1	1	yes	characteristic
188	C₃₁×#123	—	2	2	no	—
189	C₃₁×#123	—	2	2	no	—
190	C₃₁×#123	—	2	2	no	—
191	C₃₁×#123	—	2	2	no	—
192	C₃₁×#123	—	2	2	no	—
193	C₃₁×#123	D₄	1	2	yes	—
194	C₃₁×C₂²⋊Q₈	—	2	1	no	—
195	C₃₁×C₂²⋊Q₈	—	2	1	no	—
196	C₃₁×C₂²⋊Q₈	—	2	1	no	—
197	C₃₁×C₂²⋊Q₈	—	2	1	no	—
198	C₃₁×C₂²⋊Q₈	—	2	2	no	—
199	C₃₁×C₂²⋊Q₈	—	2	2	no	—
200	C₃₁×C₂²⋊Q₈	—	2	2	no	—
201	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
202	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
203	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
204	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
205	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
206	C₃₁×#121	—	2	2	no	—
207	C₃₁×#121	—	2	1	no	—
208	C₃₁×#121	—	2	1	no	—
209	C₃₁×#121	—	2	2	no	—
210	C₃₁×#121	—	2	2	no	—
211	C₃₁×#121	—	2	2	no	—
212	C₃₁×#121	—	2	2	no	—
213	C₃₁×#121	—	2	2	no	—
214	C₃₁×#121	—	2	2	no	—
215	C₃₁×#121	D₄	1	1	yes	characteristic
216	C₃₁×#121	C₂³	1	2	yes	—
217	C₃₁×#121	C₂³	1	2	yes	—
218	C₃₁×#121	C₂³	1	2	yes	—
219	C₄×Q₈×C₃₁	—	2	1	no	—
220	C₄×Q₈×C₃₁	—	2	1	no	—
221	C₄×Q₈×C₃₁	—	2	2	no	—
222	C₄×Q₈×C₃₁	—	2	1	no	—
223	C₄×Q₈×C₃₁	—	2	1	no	—
224	C₄×Q₈×C₃₁	—	2	1	no	—
225	C₄×Q₈×C₃₁	—	2	1	no	—
226	C₄×Q₈×C₃₁	—	2	1	no	—
227	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
228	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
229	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
230	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
231	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
232	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
233	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
234	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
235	C₄×D₄×C₃₁	—	2	1	no	—
236	C₄×D₄×C₃₁	—	2	1	no	—
237	C₄×D₄×C₃₁	—	2	1	no	—
238	C₄×D₄×C₃₁	—	2	1	no	—
239	C₄×D₄×C₃₁	—	2	2	no	—
240	C₄×D₄×C₃₁	—	2	2	no	—
241	C₄×D₄×C₃₁	—	2	2	no	—
242	C₄×D₄×C₃₁	—	2	2	no	—
243	C₄×D₄×C₃₁	—	2	1	no	—
244	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
245	C₄×D₄×C₃₁	C₂³	1	2	yes	—
246	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
247	C₄×D₄×C₃₁	C₂³	1	2	yes	—
248	C₄×D₄×C₃₁	C₂³	1	2	yes	—
249	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
250	C₃₁×#117	—	2	1	no	—
251	C₃₁×#117	—	2	2	no	—
252	C₃₁×#117	—	2	2	no	—
253	C₃₁×#117	—	2	1	no	—
254	C₃₁×#117	C₂³	1	2	yes	—
255	C₃₁×#117	C₂³	1	2	yes	—
256	C₃₁×#117	C₂³	1	1	yes	characteristic
257	C₃₁×#117	C₂³	1	2	yes	—
258	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
259	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
260	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
261	C₂×C₄²×C₃₁	—	2	1	no	—
262	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
263	C₃₁×#107	C₂³	1	2	yes	—
264	C₃₁×#107	C₂³	1	2	yes	—
265	C₃₁×#107	C₂³	1	2	yes	—
266	C₃₁×#107	C₂³	1	1	yes	characteristic
267	C₃₁×#107	C₂³	1	1	yes	characteristic
268	C₃₁×#107	C₂³	1	2	yes	—
269	C₃₁×#106	C₂³	1	1	yes	characteristic
270	C₃₁×#106	C₂³	1	2	yes	—
271	C₃₁×#106	C₂³	1	2	yes	—
272	C₃₁×#106	C₂³	1	2	yes	—
273	C₃₁×#106	C₂³	1	2	yes	—
274	C₃₁×#106	C₂³	1	1	yes	characteristic
275	C₃₁×#106	C₂³	1	2	yes	—
276	C₃₁×#106	C₂³	1	2	yes	—
277	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
278	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
279	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
280	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
281	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
282	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
283	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
284	C₃₁×#103	C₂³	1	1	yes	characteristic
285	C₃₁×#103	C₂³	1	2	yes	—
286	C₃₁×#103	C₂³	1	2	yes	—
287	C₃₁×#103	C₂³	1	1	yes	characteristic
288	C₃₁×#103	C₂³	1	1	yes	characteristic
289	C₃₁×#103	C₂³	1	1	yes	characteristic
290	C₃₁×#103	C₂³	1	2	yes	—
291	C₃₁×#103	C₂³	1	1	yes	characteristic
292	C₃₁×#103	C₂³	1	1	yes	characteristic
293	C₃₁×#103	C₂³	1	2	yes	—
294	C₃₁×#102	C₂³	1	2	yes	—
295	C₃₁×#102	C₂³	1	2	yes	—
296	C₃₁×#102	C₂³	1	2	yes	—
297	C₃₁×#102	C₂³	1	2	yes	—
298	C₃₁×#102	C₂³	1	2	yes	—
299	C₃₁×#102	C₂³	1	2	yes	—
300	C₃₁×#102	C₂³	1	1	yes	characteristic
301	C₃₁×#102	C₂³	1	1	yes	characteristic
302	C₃₁×#102	C₂³	1	2	yes	—
303	C₃₁×#102	C₂³	1	2	yes	—
304	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
305	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
306	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
307	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
308	C₃₁×#97	C₂³	1	2	yes	—
309	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
310	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
311	C₄×C₈×C₃₁	C₂³	1	2	yes	—
312	#41×C₃₁	—	2	2	no	—
313	#41×C₃₁	—	2	4	no	—
314	#41×C₃₁	—	4	1	no	—
315	#41×C₃₁	—	4	2	no	—
316	#41×C₃₁	—	8	2	no	—
317	#41×C₃₁	—	8	2	no	—
318	#41×C₃₁	—	8	2	no	—
319	C₂×Q₈×C₃₁	—	2	1	no	—
320	C₂×Q₈×C₃₁	—	2	1	no	—
321	C₂×Q₈×C₃₁	—	2	1	no	—
322	C₂×Q₈×C₃₁	—	2	1	no	—
323	C₂×Q₈×C₃₁	—	2	2	no	—
324	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
325	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
326	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
327	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
328	C₂×D₄×C₃₁	—	2	1	no	—
329	C₂×D₄×C₃₁	—	2	2	no	—
330	C₂×D₄×C₃₁	—	2	2	no	—
331	C₂×D₄×C₃₁	—	2	2	no	—
332	C₂×D₄×C₃₁	—	2	2	no	—
333	C₂×D₄×C₃₁	—	2	1	no	—
334	C₂×D₄×C₃₁	—	2	1	no	—
335	C₂×D₄×C₃₁	—	2	2	no	—
336	C₂×D₄×C₃₁	—	2	2	no	—
337	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
338	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
339	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
340	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
341	C₂²×C₄×C₃₁	—	2	2	no	—
342	C₂²×C₄×C₃₁	—	2	2	no	—
343	C₂²×C₄×C₃₁	—	2	1	no	—
344	C₂²×C₄×C₃₁	—	2	1	no	—
345	C₂²×C₄×C₃₁	—	2	2	no	—
346	C₂²×C₄×C₃₁	—	2	1	no	—
347	C₂²×C₄×C₃₁	—	2	2	no	—
348	C₂²×C₄×C₃₁	—	2	2	no	—
349	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
350	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
351	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
352	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
353	Q₁₆×C₃₁	—	2	2	no	—
354	Q₁₆×C₃₁	—	2	2	no	—
355	SD₁₆×C₃₁	—	4	1	no	—
356	SD₁₆×C₃₁	—	4	2	no	—
357	SD₁₆×C₃₁	—	4	2	no	—
358	SD₁₆×C₃₁	—	4	1	no	—
359	M₁₆×C₃₁	—	4	2	no	—
360	C₂×C₈×C₃₁	—	4	2	no	—
361	C₂×C₈×C₃₁	—	4	2	no	—
362	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
363	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
364	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
365	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
366	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
367	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
368	C₄⋊C₄×C₃₁	—	2	1	no	—
369	C₄⋊C₄×C₃₁	—	2	2	no	—
370	C₄⋊C₄×C₃₁	—	2	2	no	—
371	C₄⋊C₄×C₃₁	—	2	2	no	—
372	C₄⋊C₄×C₃₁	—	2	1	no	—
373	C₄⋊C₄×C₃₁	—	2	1	no	—
374	C₄⋊C₄×C₃₁	—	2	1	no	—
375	C₄⋊C₄×C₃₁	—	2	2	no	—
376	C₄⋊C₄×C₃₁	—	2	1	no	—
377	C₄⋊C₄×C₃₁	—	2	1	no	—
378	C₄⋊C₄×C₃₁	—	2	1	no	—
379	C₄⋊C₄×C₃₁	—	2	2	no	—
380	C₄⋊C₄×C₃₁	—	2	1	no	—
381	C₄⋊C₄×C₃₁	—	2	2	no	—
382	C₄⋊C₄×C₃₁	—	2	2	no	—
383	C₄⋊C₄×C₃₁	—	2	2	no	—
384	C₄⋊C₄×C₃₁	—	2	2	no	—
385	C₄⋊C₄×C₃₁	—	2	2	no	—
386	C₄⋊C₄×C₃₁	—	2	1	no	—
387	C₄⋊C₄×C₃₁	—	2	1	no	—
388	C₄⋊C₄×C₃₁	—	2	2	no	—
389	C₄⋊C₄×C₃₁	—	2	2	no	—
390	C₄⋊C₄×C₃₁	—	2	1	no	—
391	C₄⋊C₄×C₃₁	—	2	1	no	—
392	C₄⋊C₄×C₃₁	—	2	2	no	—
393	C₄⋊C₄×C₃₁	—	2	2	no	—
394	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
395	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
396	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
397	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
398	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
399	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
400	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
401	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
402	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
403	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
404	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
405	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
406	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
407	C₂²⋊C₄×C₃₁	—	2	1	no	—
408	C₂²⋊C₄×C₃₁	—	2	1	no	—
409	C₂²⋊C₄×C₃₁	—	2	2	no	—
410	C₂²⋊C₄×C₃₁	—	2	1	no	—
411	C₂²⋊C₄×C₃₁	—	2	2	no	—
412	C₂²⋊C₄×C₃₁	—	2	2	no	—
413	C₂²⋊C₄×C₃₁	—	2	2	no	—
414	C₂²⋊C₄×C₃₁	—	2	2	no	—
415	C₂²⋊C₄×C₃₁	—	2	1	no	—
416	C₂²⋊C₄×C₃₁	—	2	1	no	—
417	C₂²⋊C₄×C₃₁	—	2	2	no	—
418	C₂²⋊C₄×C₃₁	—	2	2	no	—
419	C₂²⋊C₄×C₃₁	—	2	2	no	—
420	C₂²⋊C₄×C₃₁	—	2	2	no	—
421	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
422	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
423	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
424	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
425	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
426	C₄²×C₃₁	—	2	1	no	—
427	C₄²×C₃₁	—	2	1	no	—
428	C₄²×C₃₁	—	2	1	no	—
429	C₄²×C₃₁	—	2	2	no	—
430	C₄²×C₃₁	—	2	2	no	—
431	C₄²×C₃₁	—	2	1	no	—
432	C₄²×C₃₁	—	2	1	no	—
433	C₄²×C₃₁	—	2	2	no	—
434	C₄²×C₃₁	—	2	2	no	—
435	C₄²×C₃₁	—	2	2	no	—
436	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
437	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
438	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
439	C₄²×C₃₁	C₂⁴	1	2	yes	—
440	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
441	#62221	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
442	C₂³×C₃₁	—	2	1	no	—
443	C₂³×C₃₁	—	2	2	no	—
444	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
445	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
446	Q₈×C₃₁	—	2	2	no	—
447	Q₈×C₃₁	—	2	1	no	—
448	Q₈×C₃₁	—	2	1	no	—
449	Q₈×C₃₁	—	2	1	no	—
450	Q₈×C₃₁	—	2	1	no	—
451	Q₈×C₃₁	—	2	1	no	—
452	Q₈×C₃₁	—	4	2	no	—
453	Q₈×C₃₁	—	4	2	no	—
454	Q₈×C₃₁	—	8	1	no	—
455	Q₈×C₃₁	—	8	1	no	—
456	Q₈×C₃₁	—	8	2	no	—
457	Q₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
458	D₄×C₃₁	—	2	2	no	—
459	D₄×C₃₁	—	2	2	no	—
460	D₄×C₃₁	—	2	1	no	—
461	D₄×C₃₁	—	2	2	no	—
462	D₄×C₃₁	—	2	1	no	—
463	D₄×C₃₁	—	4	2	no	—
464	D₄×C₃₁	—	4	2	no	—
465	D₄×C₃₁	—	4	4	no	—
466	D₄×C₃₁	—	4	2	no	—
467	D₄×C₃₁	—	8	1	no	—
468	D₄×C₃₁	—	8	2	no	—
469	D₄×C₃₁	—	8	2	no	—
470	D₄×C₃₁	—	8	2	no	—
471	D₄×C₃₁	—	8	2	no	—
472	D₄×C₃₁	—	8	2	no	—
473	C₂×C₄×C₃₁	—	2	1	no	—
474	C₂×C₄×C₃₁	—	2	2	no	—
475	C₂×C₄×C₃₁	—	2	1	no	—
476	C₂×C₄×C₃₁	—	2	2	no	—
477	C₂×C₄×C₃₁	—	2	1	no	—
478	C₂×C₄×C₃₁	—	2	2	no	—
479	C₂×C₄×C₃₁	—	2	1	no	—
480	C₂×C₄×C₃₁	—	2	1	no	—
481	C₂×C₄×C₃₁	—	2	2	no	—
482	C₂×C₄×C₃₁	—	2	1	no	—
483	C₂×C₄×C₃₁	—	2	1	no	—
484	C₂×C₄×C₃₁	—	2	1	no	—
485	C₂×C₄×C₃₁	—	2	2	no	—
486	C₂×C₄×C₃₁	—	2	2	no	—
487	C₂×C₄×C₃₁	—	2	1	no	—
488	C₂×C₄×C₃₁	—	4	1	no	—
489	C₂×C₄×C₃₁	—	4	2	no	—
490	C₂×C₄×C₃₁	—	4	4	no	—
491	C₂×C₄×C₃₁	—	4	2	no	—
492	C₂×C₄×C₃₁	—	8	1	no	—
493	C₂×C₄×C₃₁	—	8	2	no	—
494	C₂×C₄×C₃₁	—	8	2	no	—
495	C₂×C₄×C₃₁	—	8	2	no	—
496	C₂×C₄×C₃₁	—	8	2	no	—
497	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
498	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
499	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
500	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
501	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
502	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
503	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
504	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
505	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
506	C₈×C₃₁	—	2	1	no	—
507	C₈×C₃₁	—	2	1	no	—
508	C₈×C₃₁	—	2	1	no	—
509	C₈×C₃₁	—	2	2	no	—
510	C₈×C₃₁	—	2	2	no	—
511	C₈×C₃₁	—	2	1	no	—
512	#3555	C₂×C₃₁	1	1	yes	characteristic, complemented
513	#3517	C₂×C₃₁	1	1	yes	characteristic
514	#3387	C₂×C₃₁	1	1	yes	characteristic, complemented
515	#3383	C₂×C₃₁	1	1	yes	characteristic, complemented
516	#3368	C₂×C₃₁	1	1	yes	characteristic, complemented
517	#3362	C₂×C₃₁	1	1	yes	characteristic, complemented
518	#3353	C₂×C₃₁	1	1	yes	characteristic, complemented
519	#3350	C₂×C₃₁	1	2	yes	complemented
520	#3337	C₂×C₃₁	1	2	yes	complemented
521	#3335	C₂×C₃₁	1	1	yes	characteristic, complemented
522	#3333	C₂×C₃₁	1	1	yes	characteristic, complemented
523	#3317	C₂×C₃₁	1	2	yes	complemented
524	#3304	C₂×C₃₁	1	1	yes	characteristic, complemented
525	#3304	C₂×C₃₁	1	2	yes	complemented
526	#3260	C₂×C₃₁	1	2	yes	complemented
527	#3246	C₂×C₃₁	1	1	yes	characteristic, complemented
528	#3241	C₂×C₃₁	1	2	yes	complemented
529	#3228	C₂×C₃₁	1	1	yes	characteristic, complemented
530	#3120	C₂×C₃₁	1	2	yes	complemented
531	#3107	C₂×C₃₁	1	1	yes	characteristic, complemented
532	#3084	C₂×C₃₁	1	2	yes	complemented
533	#3076	C₂×C₃₁	1	1	yes	characteristic, complemented
534	#2994	C₂×C₃₁	1	1	yes	characteristic, complemented
535	C₂²×C₃₁	—	2	1	no	—
536	C₂²×C₃₁	—	2	2	no	—
537	C₂²×C₃₁	—	2	1	no	—
538	C₂²×C₃₁	—	2	2	no	—
539	C₂²×C₃₁	—	2	1	no	—
540	C₂²×C₃₁	—	2	2	no	—
541	C₂²×C₃₁	—	4	1	no	—
542	C₂²×C₃₁	—	4	2	no	—
543	C₂²×C₃₁	—	4	1	no	—
544	C₂²×C₃₁	—	4	2	no	—
545	C₂²×C₃₁	—	4	1	no	—
546	C₂²×C₃₁	—	4	2	no	—
547	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
548	C₄×C₃₁	—	2	1	no	—
549	C₄×C₃₁	—	2	1	no	—
550	C₄×C₃₁	—	2	2	no	—
551	C₄×C₃₁	—	2	1	no	—
552	C₄×C₃₁	—	2	2	no	—
553	C₄×C₃₁	—	2	2	no	—
554	C₄×C₃₁	—	2	1	no	—
555	C₄×C₃₁	—	2	2	no	—
556	C₄×C₃₁	—	4	1	no	—
557	C₄×C₃₁	—	4	1	no	—
558	C₄×C₃₁	—	4	2	no	—
559	C₄×C₃₁	—	4	1	no	—
560	C₄×C₃₁	—	4	1	no	—
561	C₄×C₃₁	—	4	1	no	—
562	C₄×C₃₁	—	4	1	no	—
563	C₄×C₃₁	—	4	2	no	—
564	C₄×C₃₁	—	4	2	no	—
565	C₄×C₃₁	#577	1	2	yes	—
566	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
567	C₄×C₃₁	#585	1	1	yes	characteristic
568	#563	—	2	2	no	—
569	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
570	#557	—	2	1	no	—
571	#557	C₂²×C₃₁	1	1	yes	characteristic
572	#557	C₂²×C₃₁	1	1	yes	characteristic
573	#556	—	2	1	no	—
574	#556	C₂²×C₃₁	1	2	yes	—
575	#556	C₂²×C₃₁	1	1	yes	characteristic
576	#554	C₂²×C₃₁	1	1	yes	characteristic
577	#553	—	2	2	no	—
578	#553	C₂²×C₃₁	1	2	yes	—
579	#552	C₂²×C₃₁	1	2	yes	—
580	#550	—	2	1	no	—
581	#550	C₂²×C₃₁	1	1	yes	characteristic
582	#550	C₂²×C₃₁	1	2	yes	—
583	#550	C₂²×C₃₁	1	1	yes	characteristic
584	#547	—	2	1	no	—
585	#547	C₂²×C₃₁	1	1	yes	characteristic
586	#544	C₂²×C₃₁	1	2	yes	—
587	#542	—	2	1	no	—
588	#541	—	2	2	no	—
589	#541	—	2	1	no	—
590	#541	C₂²×C₃₁	1	2	yes	—
591	#540	—	2	2	no	—
592	#540	C₂²×C₃₁	1	2	yes	—
593	#533	C₂²×C₃₁	1	1	yes	characteristic
594	#533	C₂²×C₃₁	1	1	yes	characteristic
595	#532	—	2	1	no	—
596	#532	C₂²×C₃₁	1	1	yes	characteristic
597	#529	—	2	1	no	—
598	#529	C₂²×C₃₁	1	1	yes	characteristic
599	#520	C₂²×C₃₁	1	2	yes	—
600	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
601	#501	C₂²×C₃₁	1	2	yes	—
602	#500	C₂²×C₃₁	1	1	yes	characteristic
603	#499	C₂²×C₃₁	1	2	yes	—
604	#498	C₂²×C₃₁	1	1	yes	characteristic
605	#490	C₂²×C₃₁	1	2	yes	—
606	#486	C₂²×C₃₁	1	1	yes	characteristic
607	#486	C₂²×C₃₁	1	2	yes	—
608	#486	C₂²×C₃₁	1	1	yes	characteristic
609	#485	C₂²×C₃₁	1	1	yes	characteristic
610	#485	C₂²×C₃₁	1	1	yes	characteristic
611	#484	C₂²×C₃₁	1	2	yes	—
612	#483	C₂²×C₃₁	1	2	yes	—
613	#482	C₂²×C₃₁	1	2	yes	—
614	#482	C₂²×C₃₁	1	2	yes	—
615	#482	C₂²×C₃₁	1	2	yes	—
616	#482	C₂²×C₃₁	1	2	yes	—
617	#481	C₂²×C₃₁	1	2	yes	—
618	#480	C₂²×C₃₁	1	2	yes	—
619	#479	C₂²×C₃₁	1	2	yes	—
620	#479	C₂²×C₃₁	1	2	yes	—
621	#479	C₂²×C₃₁	1	2	yes	—
622	#479	C₂²×C₃₁	1	2	yes	—
623	#478	C₂²×C₃₁	1	2	yes	—
624	#478	C₂²×C₃₁	1	2	yes	—
625	#478	C₂²×C₃₁	1	2	yes	—
626	#478	C₂²×C₃₁	1	2	yes	—
627	#477	C₂²×C₃₁	1	1	yes	characteristic
628	#476	C₂²×C₃₁	1	1	yes	characteristic
629	#476	C₂²×C₃₁	1	2	yes	—
630	#476	C₂²×C₃₁	1	2	yes	—
631	#475	C₂²×C₃₁	1	1	yes	characteristic
632	#475	C₂²×C₃₁	1	1	yes	characteristic
633	#475	C₂²×C₃₁	1	1	yes	characteristic
634	#475	C₂²×C₃₁	1	2	yes	—
635	#475	C₂²×C₃₁	1	1	yes	characteristic
636	#474	C₂²×C₃₁	1	1	yes	characteristic
637	#474	C₂²×C₃₁	1	1	yes	characteristic
638	#468	C₂²×C₃₁	1	2	yes	—
639	#468	C₂²×C₃₁	1	2	yes	—
640	#465	C₂²×C₃₁	1	2	yes	—
641	#465	C₂²×C₃₁	1	2	yes	—
642	#465	C₂²×C₃₁	1	2	yes	—
643	#465	C₂²×C₃₁	1	2	yes	—
644	#462	C₂²×C₃₁	1	1	yes	characteristic
645	#461	C₂²×C₃₁	1	1	yes	characteristic
646	#461	C₂²×C₃₁	1	2	yes	—
647	#460	C₂²×C₃₁	1	1	yes	characteristic
648	#460	C₂²×C₃₁	1	1	yes	characteristic
649	#460	C₂²×C₃₁	1	2	yes	—
650	#460	C₂²×C₃₁	1	2	yes	—
651	#452	C₂²×C₃₁	1	2	yes	—
652	#452	C₂²×C₃₁	1	1	yes	characteristic
653	#452	C₂²×C₃₁	1	1	yes	characteristic
654	#452	C₂²×C₃₁	1	2	yes	—
655	#446	C₂²×C₃₁	1	1	yes	characteristic
656	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
657	#441	C₂²×C₃₁	1	2	yes	—
658	#440	C₂²×C₃₁	1	2	yes	—
659	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
660	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
661	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
662	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
663	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
664	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
665	#436	C₂²×C₃₁	1	1	yes	characteristic
666	#435	C₂²×C₃₁	1	2	yes	—
667	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
668	C₈×D₄	C₂²×C₃₁	1	2	yes	—
669	#433	C₂²×C₃₁	1	2	yes	—
670	#431	C₂²×C₃₁	1	1	yes	characteristic
671	#428	C₂²×C₃₁	1	2	yes	—
672	#427	C₂²×C₃₁	1	2	yes	—
673	C₂×#106	C₂²×C₃₁	1	2	yes	—
674	#424	C₂²×C₃₁	1	2	yes	—
675	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
676	#417	C₂²×C₃₁	1	2	yes	—
677	#416	C₂²×C₃₁	1	2	yes	—
678	#416	C₂²×C₃₁	1	2	yes	—
679	C₂×C₃₁	—	4	1	no	—
680	C₂×C₃₁	—	4	2	no	—
681	C₂×C₃₁	—	8	1	no	—
682	C₂×C₃₁	—	8	2	no	—
683	C₂×C₃₁	#3489	1	1	yes	characteristic
684	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
685	C₂×C₃₁	#3584	1	1	yes	characteristic
686	C₂×#41	—	2	1	no	—
687	C₂×#41	—	2	1	no	—
688	C₂×#41	—	2	1	no	—
689	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
690	C₂×#41	C₂³×C₃₁	1	2	yes	—
691	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
692	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
693	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
694	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
695	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
696	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
697	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
698	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
699	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
700	C₄⋊Q₈	—	2	1	no	—
701	C₄⋊Q₈	—	2	1	no	—
702	C₄⋊Q₈	—	2	2	no	—
703	C₄⋊Q₈	—	2	1	no	—
704	C₄⋊Q₈	—	2	2	no	—
705	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
706	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
707	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
708	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
709	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
710	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
711	#127	—	2	1	no	—
712	#127	—	2	2	no	—
713	#127	C₂³×C₃₁	1	1	yes	characteristic
714	#127	C₂³×C₃₁	1	2	yes	—
715	#126	—	2	2	no	—
716	#126	—	2	2	no	—
717	#126	—	2	2	no	—
718	#126	—	2	2	no	—
719	#126	—	2	2	no	—
720	#126	—	2	2	no	—
721	#126	—	2	2	no	—
722	#126	D₄×C₃₁	1	2	yes	—
723	#125	—	2	2	no	—
724	#125	—	2	2	no	—
725	#125	—	2	2	no	—
726	#125	—	2	2	no	—
727	#125	—	2	1	no	—
728	#125	—	2	2	no	—
729	#125	—	2	2	no	—
730	#125	—	2	1	no	—
731	#125	D₄×C₃₁	1	1	yes	characteristic
732	#125	C₂³×C₃₁	1	2	yes	—
733	#125	C₂³×C₃₁	1	2	yes	—
734	#125	C₂³×C₃₁	1	2	yes	—
735	#124	—	2	1	no	—
736	#124	—	2	2	no	—
737	#124	—	2	1	no	—
738	#124	D₄×C₃₁	1	1	yes	characteristic
739	#123	—	2	2	no	—
740	#123	—	2	2	no	—
741	#123	—	2	2	no	—
742	#123	—	2	2	no	—
743	#123	—	2	2	no	—
744	#123	D₄×C₃₁	1	2	yes	—
745	C₂²⋊Q₈	—	2	1	no	—
746	C₂²⋊Q₈	—	2	1	no	—
747	C₂²⋊Q₈	—	2	1	no	—
748	C₂²⋊Q₈	—	2	1	no	—
749	C₂²⋊Q₈	—	2	2	no	—
750	C₂²⋊Q₈	—	2	2	no	—
751	C₂²⋊Q₈	—	2	2	no	—
752	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
753	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
754	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
755	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
756	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
757	#121	—	2	2	no	—
758	#121	—	2	1	no	—
759	#121	—	2	1	no	—
760	#121	—	2	2	no	—
761	#121	—	2	2	no	—
762	#121	—	2	2	no	—
763	#121	—	2	2	no	—
764	#121	—	2	2	no	—
765	#121	—	2	2	no	—
766	#121	D₄×C₃₁	1	1	yes	characteristic
767	#121	C₂³×C₃₁	1	2	yes	—
768	#121	C₂³×C₃₁	1	2	yes	—
769	#121	C₂³×C₃₁	1	2	yes	—
770	C₄×Q₈	—	2	1	no	—
771	C₄×Q₈	—	2	1	no	—
772	C₄×Q₈	—	2	2	no	—
773	C₄×Q₈	—	2	1	no	—
774	C₄×Q₈	—	2	1	no	—
775	C₄×Q₈	—	2	1	no	—
776	C₄×Q₈	—	2	1	no	—
777	C₄×Q₈	—	2	1	no	—
778	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
779	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
780	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
781	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
782	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
783	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
784	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
785	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
786	C₄×D₄	—	2	1	no	—
787	C₄×D₄	—	2	1	no	—
788	C₄×D₄	—	2	1	no	—
789	C₄×D₄	—	2	1	no	—
790	C₄×D₄	—	2	2	no	—
791	C₄×D₄	—	2	2	no	—
792	C₄×D₄	—	2	2	no	—
793	C₄×D₄	—	2	2	no	—
794	C₄×D₄	—	2	1	no	—
795	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
796	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
797	C₄×D₄	C₂³×C₃₁	1	2	yes	—
798	C₄×D₄	C₂³×C₃₁	1	2	yes	—
799	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
800	C₄×D₄	C₂³×C₃₁	1	2	yes	—
801	#117	—	2	1	no	—
802	#117	—	2	2	no	—
803	#117	—	2	2	no	—
804	#117	—	2	1	no	—
805	#117	C₂³×C₃₁	1	1	yes	characteristic
806	#117	C₂³×C₃₁	1	2	yes	—
807	#117	C₂³×C₃₁	1	2	yes	—
808	#117	C₂³×C₃₁	1	2	yes	—
809	C₂×C₄⋊C₄	—	2	2	no	—
810	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
811	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
812	C₂×C₄²	—	2	1	no	—
813	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
814	#107	C₂³×C₃₁	1	2	yes	—
815	#107	C₂³×C₃₁	1	2	yes	—
816	#107	C₂³×C₃₁	1	2	yes	—
817	#107	C₂³×C₃₁	1	1	yes	characteristic
818	#107	C₂³×C₃₁	1	1	yes	characteristic
819	#107	C₂³×C₃₁	1	2	yes	—
820	#106	C₂³×C₃₁	1	1	yes	characteristic
821	#106	C₂³×C₃₁	1	2	yes	—
822	#106	C₂³×C₃₁	1	2	yes	—
823	#106	C₂³×C₃₁	1	2	yes	—
824	#106	C₂³×C₃₁	1	2	yes	—
825	#106	C₂³×C₃₁	1	1	yes	characteristic
826	#106	C₂³×C₃₁	1	2	yes	—
827	#106	C₂³×C₃₁	1	2	yes	—
828	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
829	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
830	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
831	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
832	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
833	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
834	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
835	#103	C₂³×C₃₁	1	1	yes	characteristic
836	#103	C₂³×C₃₁	1	2	yes	—
837	#103	C₂³×C₃₁	1	2	yes	—
838	#103	C₂³×C₃₁	1	1	yes	characteristic
839	#103	C₂³×C₃₁	1	1	yes	characteristic
840	#103	C₂³×C₃₁	1	1	yes	characteristic
841	#103	C₂³×C₃₁	1	2	yes	—
842	#103	C₂³×C₃₁	1	1	yes	characteristic
843	#103	C₂³×C₃₁	1	1	yes	characteristic
844	#103	C₂³×C₃₁	1	2	yes	—
845	#102	C₂³×C₃₁	1	2	yes	—
846	#102	C₂³×C₃₁	1	2	yes	—
847	#102	C₂³×C₃₁	1	2	yes	—
848	#102	C₂³×C₃₁	1	2	yes	—
849	#102	C₂³×C₃₁	1	2	yes	—
850	#102	C₂³×C₃₁	1	2	yes	—
851	#102	C₂³×C₃₁	1	1	yes	characteristic
852	#102	C₂³×C₃₁	1	1	yes	characteristic
853	#102	C₂³×C₃₁	1	2	yes	—
854	#102	C₂³×C₃₁	1	2	yes	—
855	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
856	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
857	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
858	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
859	#97	C₂³×C₃₁	1	2	yes	—
860	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
861	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
862	C₄×C₈	C₂³×C₃₁	1	2	yes	—
863	C₃₁	#62221	1	1	yes	characteristic, direct product factor, Sylow
864	#41	—	2	2	no	—
865	#41	—	2	4	no	—
866	#41	—	4	1	no	—
867	#41	—	4	2	no	—
868	#41	—	8	2	no	—
869	#41	—	8	2	no	—
870	#41	—	8	2	no	—
871	C₂×Q₈	—	2	1	no	—
872	C₂×Q₈	—	2	1	no	—
873	C₂×Q₈	—	2	1	no	—
874	C₂×Q₈	—	2	1	no	—
875	C₂×Q₈	—	2	2	no	—
876	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
877	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
878	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
879	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
880	C₂×D₄	—	2	1	no	—
881	C₂×D₄	—	2	2	no	—
882	C₂×D₄	—	2	2	no	—
883	C₂×D₄	—	2	2	no	—
884	C₂×D₄	—	2	2	no	—
885	C₂×D₄	—	2	1	no	—
886	C₂×D₄	—	2	1	no	—
887	C₂×D₄	—	2	2	no	—
888	C₂×D₄	—	2	2	no	—
889	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
890	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
891	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
892	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
893	C₂²×C₄	—	2	2	no	—
894	C₂²×C₄	—	2	2	no	—
895	C₂²×C₄	—	2	1	no	—
896	C₂²×C₄	—	2	1	no	—
897	C₂²×C₄	—	2	2	no	—
898	C₂²×C₄	—	2	1	no	—
899	C₂²×C₄	—	2	2	no	—
900	C₂²×C₄	—	2	2	no	—
901	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
902	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
903	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
904	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
905	Q₁₆	—	2	2	no	—
906	Q₁₆	—	2	2	no	—
907	SD₁₆	—	4	1	no	—
908	SD₁₆	—	4	2	no	—
909	SD₁₆	—	4	2	no	—
910	SD₁₆	—	4	1	no	—
911	M₁₆	—	4	2	no	—
912	C₂×C₈	—	4	2	no	—
913	C₂×C₈	—	4	2	no	—
914	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
915	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
916	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
917	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
918	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
919	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
920	C₄⋊C₄	—	2	1	no	—
921	C₄⋊C₄	—	2	2	no	—
922	C₄⋊C₄	—	2	2	no	—
923	C₄⋊C₄	—	2	2	no	—
924	C₄⋊C₄	—	2	1	no	—
925	C₄⋊C₄	—	2	1	no	—
926	C₄⋊C₄	—	2	1	no	—
927	C₄⋊C₄	—	2	2	no	—
928	C₄⋊C₄	—	2	1	no	—
929	C₄⋊C₄	—	2	1	no	—
930	C₄⋊C₄	—	2	1	no	—
931	C₄⋊C₄	—	2	2	no	—
932	C₄⋊C₄	—	2	1	no	—
933	C₄⋊C₄	—	2	2	no	—
934	C₄⋊C₄	—	2	2	no	—
935	C₄⋊C₄	—	2	2	no	—
936	C₄⋊C₄	—	2	2	no	—
937	C₄⋊C₄	—	2	2	no	—
938	C₄⋊C₄	—	2	1	no	—
939	C₄⋊C₄	—	2	1	no	—
940	C₄⋊C₄	—	2	2	no	—
941	C₄⋊C₄	—	2	2	no	—
942	C₄⋊C₄	—	2	1	no	—
943	C₄⋊C₄	—	2	1	no	—
944	C₄⋊C₄	—	2	2	no	—
945	C₄⋊C₄	—	2	2	no	—
946	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
947	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
948	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
949	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
950	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
951	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
952	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
953	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
954	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
955	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
956	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
957	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
958	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
959	C₂²⋊C₄	—	2	1	no	—
960	C₂²⋊C₄	—	2	1	no	—
961	C₂²⋊C₄	—	2	2	no	—
962	C₂²⋊C₄	—	2	1	no	—
963	C₂²⋊C₄	—	2	2	no	—
964	C₂²⋊C₄	—	2	2	no	—
965	C₂²⋊C₄	—	2	2	no	—
966	C₂²⋊C₄	—	2	2	no	—
967	C₂²⋊C₄	—	2	1	no	—
968	C₂²⋊C₄	—	2	1	no	—
969	C₂²⋊C₄	—	2	2	no	—
970	C₂²⋊C₄	—	2	2	no	—
971	C₂²⋊C₄	—	2	2	no	—
972	C₂²⋊C₄	—	2	2	no	—
973	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
974	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
975	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
976	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
977	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
978	C₄²	—	2	1	no	—
979	C₄²	—	2	1	no	—
980	C₄²	—	2	1	no	—
981	C₄²	—	2	2	no	—
982	C₄²	—	2	2	no	—
983	C₄²	—	2	1	no	—
984	C₄²	—	2	1	no	—
985	C₄²	—	2	2	no	—
986	C₄²	—	2	2	no	—
987	C₄²	—	2	2	no	—
988	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
989	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
990	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
991	C₄²	C₂⁴×C₃₁	1	2	yes	—
992	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
993	C₂³	—	2	1	no	—
994	C₂³	—	2	2	no	—
995	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
996	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
997	Q₈	—	2	2	no	—
998	Q₈	—	2	1	no	—
999	Q₈	—	2	1	no	—
1000	Q₈	—	2	1	no	—
1001	Q₈	—	2	1	no	—
1002	Q₈	—	2	1	no	—
1003	Q₈	—	4	2	no	—
1004	Q₈	—	4	2	no	—
1005	Q₈	—	8	1	no	—
1006	Q₈	—	8	1	no	—
1007	Q₈	—	8	2	no	—
1008	Q₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
1009	D₄	—	2	2	no	—
1010	D₄	—	2	2	no	—
1011	D₄	—	2	1	no	—
1012	D₄	—	2	2	no	—
1013	D₄	—	2	1	no	—
1014	D₄	—	4	2	no	—
1015	D₄	—	4	2	no	—
1016	D₄	—	4	4	no	—
1017	D₄	—	4	2	no	—
1018	D₄	—	8	1	no	—
1019	D₄	—	8	2	no	—
1020	D₄	—	8	2	no	—
1021	D₄	—	8	2	no	—
1022	D₄	—	8	2	no	—
1023	D₄	—	8	2	no	—
1024	C₂×C₄	—	2	1	no	—
1025	C₂×C₄	—	2	2	no	—
1026	C₂×C₄	—	2	1	no	—
1027	C₂×C₄	—	2	2	no	—
1028	C₂×C₄	—	2	1	no	—
1029	C₂×C₄	—	2	2	no	—
1030	C₂×C₄	—	2	1	no	—
1031	C₂×C₄	—	2	1	no	—
1032	C₂×C₄	—	2	2	no	—
1033	C₂×C₄	—	2	1	no	—
1034	C₂×C₄	—	2	1	no	—
1035	C₂×C₄	—	2	1	no	—
1036	C₂×C₄	—	2	2	no	—
1037	C₂×C₄	—	2	2	no	—
1038	C₂×C₄	—	2	1	no	—
1039	C₂×C₄	—	4	1	no	—
1040	C₂×C₄	—	4	2	no	—
1041	C₂×C₄	—	4	4	no	—
1042	C₂×C₄	—	4	2	no	—
1043	C₂×C₄	—	8	1	no	—
1044	C₂×C₄	—	8	2	no	—
1045	C₂×C₄	—	8	2	no	—
1046	C₂×C₄	—	8	2	no	—
1047	C₂×C₄	—	8	2	no	—
1048	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1049	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1050	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1051	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1052	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1053	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1054	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1055	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1056	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
1057	C₈	—	2	1	no	—
1058	C₈	—	2	1	no	—
1059	C₈	—	2	1	no	—
1060	C₈	—	2	2	no	—
1061	C₈	—	2	2	no	—
1062	C₈	—	2	1	no	—
1063	C₂²	—	2	1	no	—
1064	C₂²	—	2	2	no	—
1065	C₂²	—	2	1	no	—
1066	C₂²	—	2	2	no	—
1067	C₂²	—	2	1	no	—
1068	C₂²	—	2	2	no	—
1069	C₂²	—	4	1	no	—
1070	C₂²	—	4	2	no	—
1071	C₂²	—	4	1	no	—
1072	C₂²	—	4	2	no	—
1073	C₂²	—	4	1	no	—
1074	C₂²	—	4	2	no	—
1075	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
1076	C₄	—	2	1	no	—
1077	C₄	—	2	2	no	—
1078	C₄	—	2	2	no	—
1079	C₄	—	2	1	no	—
1080	C₄	—	2	2	no	—
1081	C₄	—	2	1	no	—
1082	C₄	—	2	1	no	—
1083	C₄	—	2	2	no	—
1084	C₄	—	4	1	no	—
1085	C₄	—	4	1	no	—
1086	C₄	—	4	1	no	—
1087	C₄	—	4	2	no	—
1088	C₄	—	4	2	no	—
1089	C₄	—	4	1	no	—
1090	C₄	—	4	1	no	—
1091	C₄	—	4	2	no	—
1092	C₄	—	4	1	no	—
1093	C₄	C₃₁×#577	1	2	yes	—
1094	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
1095	C₄	C₃₁×#585	1	1	yes	characteristic
1096	C₂	—	4	1	no	—
1097	C₂	—	4	2	no	—
1098	C₂	—	8	1	no	—
1099	C₂	—	8	2	no	—
1100	C₂	C₃₁×#3489	1	1	yes	characteristic
1101	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
1102	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3517	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#529 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#3555	C₂	semidirect
3. C₃₁×#3387	C₂	semidirect
4. C₃₁×#3383	C₂	semidirect
5. C₃₁×#3368	C₂	semidirect
6. C₃₁×#3362	C₂	semidirect
7. C₃₁×#3353	C₂	semidirect
8. C₃₁×#3350	C₂	semidirect
9. C₃₁×#3337	C₂	semidirect
10. C₃₁×#3335	C₂	semidirect
11. C₃₁×#3333	C₂	semidirect
12. C₃₁×#3317	C₂	semidirect
13. C₃₁×#3304	C₂	semidirect
14. C₃₁×#3304	C₂	semidirect
15. C₃₁×#3260	C₂	semidirect
16. C₃₁×#3246	C₂	semidirect
17. C₃₁×#3241	C₂	semidirect
18. C₃₁×#3228	C₂	semidirect
19. C₃₁×#3120	C₂	semidirect
20. C₃₁×#3107	C₂	semidirect
21. C₃₁×#3084	C₂	semidirect
22. C₃₁×#3076	C₂	semidirect
23. C₃₁×#2994	C₂	semidirect
441. #62221	C₃₁	direct (N × H)
512. #3555	C₂×C₃₁	semidirect
514. #3387	C₂×C₃₁	semidirect
515. #3383	C₂×C₃₁	semidirect
516. #3368	C₂×C₃₁	semidirect
517. #3362	C₂×C₃₁	semidirect
518. #3353	C₂×C₃₁	semidirect
519. #3350	C₂×C₃₁	semidirect
520. #3337	C₂×C₃₁	semidirect
521. #3335	C₂×C₃₁	semidirect
522. #3333	C₂×C₃₁	semidirect
523. #3317	C₂×C₃₁	semidirect
524. #3304	C₂×C₃₁	semidirect
525. #3304	C₂×C₃₁	semidirect
526. #3260	C₂×C₃₁	semidirect
527. #3246	C₂×C₃₁	semidirect
528. #3241	C₂×C₃₁	semidirect
529. #3228	C₂×C₃₁	semidirect
530. #3120	C₂×C₃₁	semidirect
531. #3107	C₂×C₃₁	semidirect
532. #3084	C₂×C₃₁	semidirect
533. #3076	C₂×C₃₁	semidirect
534. #2994	C₂×C₃₁	semidirect
863. C₃₁	#62221	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3555	2
C₃₁×#3517	2
C₃₁×#3387	2
C₃₁×#3383	2
C₃₁×#3368	2
C₃₁×#3362	2
C₃₁×#3353	2
C₃₁×#3350	2
C₃₁×#3337	2
C₃₁×#3335	2
C₃₁×#3333	2
C₃₁×#3317	2
C₃₁×#3304	2
C₃₁×#3260	2
C₃₁×#3246	2
C₃₁×#3241	2
C₃₁×#3228	2
C₃₁×#3120	2
C₃₁×#3107	2
C₃₁×#3084	2
C₃₁×#3076	2
C₃₁×#2994	2
#62221	31

Subgroups — C31×#62221 #21518980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62221 #21518980