Subgroups — C₃₁×#62193 #21518952

Order 7936.

1682 conjugacy classes of subgroups.

The subgroup lattice has 1090 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×#3522	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3387	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3375	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3365	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3356	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3353	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3343	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3338	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3333	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3327	C₂	1	2	yes	maximal, complemented
13	C₃₁×#3313	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3304	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#3258	C₂	1	2	yes	maximal, complemented
16	C₃₁×#3254	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#3243	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×#3104	C₂	1	1	yes	maximal, characteristic, complemented
21	C₃₁×#3083	C₂	1	2	yes	maximal, complemented
22	C₃₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
23	C₃₁×#2993	C₂	1	1	yes	maximal, characteristic, complemented
24	C₃₁×#563	—	2	2	no	—
25	C₃₁×#559	—	2	1	no	—
26	C₃₁×#559	C₂²	1	1	yes	characteristic
27	Q₈²×C₃₁	C₂²	1	1	yes	characteristic
28	Q₈²×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#557	—	2	1	no	—
30	C₃₁×#557	C₂²	1	1	yes	characteristic
31	C₃₁×#556	—	2	1	no	—
32	C₃₁×#556	C₂²	1	2	yes	—
33	C₃₁×#556	C₂²	1	1	yes	characteristic
34	C₃₁×#554	—	2	1	no	—
35	C₃₁×#554	C₂²	1	1	yes	characteristic
36	C₃₁×#554	C₂²	1	1	yes	characteristic
37	C₃₁×#552	C₂²	1	2	yes	—
38	C₃₁×#550	C₂²	1	1	yes	characteristic
39	C₃₁×#548	C₂²	1	2	yes	—
40	C₃₁×#547	—	2	2	no	—
41	C₃₁×#547	C₂²	1	2	yes	—
42	C₃₁×#544	C₂²	1	2	yes	—
43	C₃₁×#542	—	2	2	no	—
44	C₃₁×#542	—	2	1	no	—
45	C₃₁×#542	C₂²	1	2	yes	—
46	C₃₁×#541	—	2	2	no	—
47	C₃₁×#541	—	2	1	no	—
48	C₃₁×#541	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	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
57	C₃₁×#501	C₂²	1	2	yes	—
58	C₃₁×#500	C₂²	1	1	yes	characteristic
59	C₃₁×#500	C₂²	1	2	yes	—
60	C₃₁×#498	C₂²	1	1	yes	characteristic
61	C₃₁×#489	C₂²	1	2	yes	—
62	C₃₁×#488	C₂²	1	2	yes	—
63	C₃₁×#486	C₂²	1	1	yes	characteristic
64	C₃₁×#486	C₂²	1	1	yes	characteristic
65	C₃₁×#485	C₂²	1	2	yes	—
66	C₃₁×#484	C₂²	1	2	yes	—
67	C₃₁×#484	C₂²	1	2	yes	—
68	C₃₁×#484	C₂²	1	1	yes	characteristic
69	C₃₁×#484	C₂²	1	1	yes	characteristic
70	C₃₁×#483	C₂²	1	2	yes	—
71	C₃₁×#482	C₂²	1	2	yes	—
72	C₃₁×#482	C₂²	1	2	yes	—
73	C₃₁×#482	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×#478	C₂²	1	2	yes	—
79	C₃₁×#478	C₂²	1	2	yes	—
80	C₃₁×#478	C₂²	1	2	yes	—
81	C₃₁×#477	C₂²	1	1	yes	characteristic
82	C₃₁×#477	C₂²	1	1	yes	characteristic
83	C₃₁×#477	C₂²	1	1	yes	characteristic
84	C₃₁×#477	C₂²	1	1	yes	characteristic
85	C₃₁×#477	C₂²	1	2	yes	—
86	C₃₁×#477	C₂²	1	2	yes	—
87	C₃₁×#477	C₂²	1	1	yes	characteristic
88	C₃₁×#477	C₂²	1	1	yes	characteristic
89	C₃₁×#475	C₂²	1	1	yes	characteristic
90	C₃₁×#475	C₂²	1	1	yes	characteristic
91	C₃₁×#475	C₂²	1	2	yes	—
92	C₃₁×#474	C₂²	1	2	yes	—
93	C₃₁×#474	C₂²	1	2	yes	—
94	C₃₁×#470	C₂²	1	2	yes	—
95	C₃₁×#468	C₂²	1	2	yes	—
96	C₃₁×#468	C₂²	1	2	yes	—
97	C₃₁×#468	C₂²	1	2	yes	—
98	C₃₁×#467	C₂²	1	2	yes	—
99	C₃₁×#465	C₂²	1	2	yes	—
100	C₃₁×#464	C₂²	1	2	yes	—
101	C₃₁×#464	C₂²	1	2	yes	—
102	C₃₁×#462	C₂²	1	1	yes	characteristic
103	C₃₁×#462	C₂²	1	1	yes	characteristic
104	C₃₁×#461	C₂²	1	2	yes	—
105	C₃₁×#460	C₂²	1	1	yes	characteristic
106	C₃₁×#460	C₂²	1	1	yes	characteristic
107	C₃₁×#452	C₂²	1	2	yes	—
108	C₃₁×#452	C₂²	1	2	yes	—
109	C₃₁×#451	C₂²	1	1	yes	characteristic
110	C₃₁×#451	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
118	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
119	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
120	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
121	C₃₁×#436	C₂²	1	1	yes	characteristic
122	C₃₁×#436	C₂²	1	2	yes	—
123	C₃₁×#435	C₂²	1	2	yes	—
124	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
125	C₃₁×#433	C₂²	1	2	yes	—
126	C₃₁×#431	C₂²	1	1	yes	characteristic
127	C₃₁×#428	C₂²	1	2	yes	—
128	C₃₁×#428	C₂²	1	2	yes	—
129	C₂×C₃₁×#107	C₂²	1	2	yes	—
130	C₃₁×#424	C₂²	1	2	yes	—
131	C₃₁×#423	C₂²	1	1	yes	characteristic
132	C₃₁×#419	C₂²	1	2	yes	—
133	C₃₁×#417	C₂²	1	2	yes	—
134	C₃₁×#416	C₂²	1	2	yes	—
135	C₂×#41×C₃₁	—	2	2	no	—
136	C₂×#41×C₃₁	—	2	1	no	—
137	C₂×#41×C₃₁	C₂³	1	2	yes	—
138	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
140	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
141	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
142	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
143	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
144	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
145	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
146	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
147	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
148	C₃₁×C₄⋊Q₈	—	2	1	no	—
149	C₃₁×C₄⋊Q₈	—	2	1	no	—
150	C₃₁×C₄⋊Q₈	—	2	2	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	2	yes	—
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×C₄⋊Q₈	C₂³	1	1	yes	characteristic
161	C₃₁×#127	—	2	1	no	—
162	C₃₁×#127	C₂³	1	1	yes	characteristic
163	C₃₁×#126	—	2	2	no	—
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	D₄	1	2	yes	—
171	C₃₁×#125	—	2	1	no	—
172	C₃₁×#125	—	2	2	no	—
173	C₃₁×#125	—	2	2	no	—
174	C₃₁×#125	—	2	2	no	—
175	C₃₁×#125	—	2	1	no	—
176	C₃₁×#125	—	2	2	no	—
177	C₃₁×#125	—	2	2	no	—
178	C₃₁×#125	—	2	1	no	—
179	C₃₁×#125	D₄	1	1	yes	characteristic
180	C₃₁×#125	C₂³	1	2	yes	—
181	C₃₁×#125	C₂³	1	2	yes	—
182	C₃₁×#124	—	2	1	no	—
183	C₃₁×#124	—	2	1	no	—
184	C₃₁×#124	—	2	2	no	—
185	C₃₁×#124	—	2	2	no	—
186	C₃₁×#124	—	2	2	no	—
187	C₃₁×#124	—	2	1	no	—
188	C₃₁×#124	D₄	1	1	yes	characteristic
189	C₃₁×#124	C₂³	1	2	yes	—
190	C₃₁×#123	—	2	2	no	—
191	C₃₁×#123	—	2	2	no	—
192	C₃₁×#123	—	2	2	no	—
193	C₃₁×#123	—	2	2	no	—
194	C₃₁×#123	—	2	2	no	—
195	C₃₁×#123	D₄	1	2	yes	—
196	C₃₁×C₂²⋊Q₈	—	2	1	no	—
197	C₃₁×C₂²⋊Q₈	—	2	1	no	—
198	C₃₁×C₂²⋊Q₈	—	2	1	no	—
199	C₃₁×C₂²⋊Q₈	—	2	2	no	—
200	C₃₁×C₂²⋊Q₈	—	2	2	no	—
201	C₃₁×C₂²⋊Q₈	—	2	2	no	—
202	C₃₁×C₂²⋊Q₈	—	2	2	no	—
203	C₃₁×C₂²⋊Q₈	—	2	2	no	—
204	C₃₁×C₂²⋊Q₈	—	2	2	no	—
205	C₃₁×C₂²⋊Q₈	—	2	2	no	—
206	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
207	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
208	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
209	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
210	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
211	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
212	C₃₁×#121	—	2	2	no	—
213	C₃₁×#121	—	2	1	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	C₄×Q₈×C₃₁	—	2	1	no	—
219	C₄×Q₈×C₃₁	—	2	1	no	—
220	C₄×Q₈×C₃₁	—	2	2	no	—
221	C₄×Q₈×C₃₁	—	2	1	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	—	2	1	no	—
228	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
229	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
230	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
231	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
232	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
233	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
234	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
235	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
236	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
237	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
238	C₄×D₄×C₃₁	—	2	1	no	—
239	C₄×D₄×C₃₁	—	2	1	no	—
240	C₄×D₄×C₃₁	—	2	1	no	—
241	C₄×D₄×C₃₁	—	2	1	no	—
242	C₄×D₄×C₃₁	—	2	2	no	—
243	C₄×D₄×C₃₁	—	2	2	no	—
244	C₄×D₄×C₃₁	—	2	2	no	—
245	C₄×D₄×C₃₁	—	2	1	no	—
246	C₄×D₄×C₃₁	C₂³	1	2	yes	—
247	C₄×D₄×C₃₁	C₂³	1	2	yes	—
248	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
249	C₄×D₄×C₃₁	C₂³	1	2	yes	—
250	C₃₁×#117	—	2	1	no	—
251	C₃₁×#117	—	2	2	no	—
252	C₃₁×#117	—	2	2	no	—
253	C₃₁×#117	C₂³	1	2	yes	—
254	C₃₁×#117	C₂³	1	2	yes	—
255	C₃₁×#117	C₂³	1	2	yes	—
256	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
257	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
258	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
259	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
260	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
261	C₂×C₄²×C₃₁	—	2	1	no	—
262	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
263	C₃₁×#107	C₂³	1	1	yes	characteristic
264	C₃₁×#107	C₂³	1	2	yes	—
265	C₃₁×#107	C₂³	1	2	yes	—
266	C₃₁×#107	C₂³	1	2	yes	—
267	C₃₁×#107	C₂³	1	1	yes	characteristic
268	C₃₁×#107	C₂³	1	2	yes	—
269	C₃₁×#107	C₂³	1	2	yes	—
270	C₃₁×#107	C₂³	1	1	yes	characteristic
271	C₃₁×#107	C₂³	1	1	yes	characteristic
272	C₃₁×#107	C₂³	1	2	yes	—
273	C₃₁×#106	C₂³	1	2	yes	—
274	C₃₁×#106	C₂³	1	2	yes	—
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	1	yes	characteristic
287	C₃₁×#103	C₂³	1	2	yes	—
288	C₃₁×#103	C₂³	1	2	yes	—
289	C₃₁×#103	C₂³	1	2	yes	—
290	C₃₁×#103	C₂³	1	1	yes	characteristic
291	C₃₁×#103	C₂³	1	1	yes	characteristic
292	C₃₁×#103	C₂³	1	1	yes	characteristic
293	C₃₁×#103	C₂³	1	1	yes	characteristic
294	C₃₁×#103	C₂³	1	2	yes	—
295	C₃₁×#103	C₂³	1	1	yes	characteristic
296	C₃₁×#103	C₂³	1	1	yes	characteristic
297	C₃₁×#103	C₂³	1	2	yes	—
298	C₃₁×#102	C₂³	1	2	yes	—
299	C₃₁×#102	C₂³	1	2	yes	—
300	C₃₁×#102	C₂³	1	2	yes	—
301	C₃₁×#102	C₂³	1	2	yes	—
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	4	no	—
313	#41×C₃₁	—	4	2	no	—
314	#41×C₃₁	—	8	2	no	—
315	#41×C₃₁	—	8	4	no	—
316	C₂×Q₈×C₃₁	—	2	1	no	—
317	C₂×Q₈×C₃₁	—	2	1	no	—
318	C₂×Q₈×C₃₁	—	2	1	no	—
319	C₂×Q₈×C₃₁	—	2	1	no	—
320	C₂×Q₈×C₃₁	—	2	2	no	—
321	C₂×Q₈×C₃₁	—	2	2	no	—
322	C₂×Q₈×C₃₁	—	2	1	no	—
323	C₂×Q₈×C₃₁	—	2	2	no	—
324	C₂×Q₈×C₃₁	—	4	1	no	—
325	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
326	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
327	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
328	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
329	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
336	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
337	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
338	C₂²×C₄×C₃₁	—	2	1	no	—
339	C₂²×C₄×C₃₁	—	2	1	no	—
340	C₂²×C₄×C₃₁	—	2	2	no	—
341	C₂²×C₄×C₃₁	—	2	2	no	—
342	C₂²×C₄×C₃₁	—	2	2	no	—
343	C₂²×C₄×C₃₁	—	2	2	no	—
344	C₂²×C₄×C₃₁	—	2	2	no	—
345	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
346	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
347	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
348	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
349	Q₁₆×C₃₁	—	2	2	no	—
350	Q₁₆×C₃₁	—	2	2	no	—
351	Q₁₆×C₃₁	—	4	1	no	—
352	Q₁₆×C₃₁	—	4	1	no	—
353	SD₁₆×C₃₁	—	4	2	no	—
354	SD₁₆×C₃₁	—	4	2	no	—
355	M₁₆×C₃₁	—	4	2	no	—
356	M₁₆×C₃₁	—	4	2	no	—
357	C₂×C₈×C₃₁	—	4	2	no	—
358	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
359	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
360	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
361	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
362	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
363	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
364	C₄⋊C₄×C₃₁	—	2	1	no	—
365	C₄⋊C₄×C₃₁	—	2	2	no	—
366	C₄⋊C₄×C₃₁	—	2	2	no	—
367	C₄⋊C₄×C₃₁	—	2	1	no	—
368	C₄⋊C₄×C₃₁	—	2	1	no	—
369	C₄⋊C₄×C₃₁	—	2	1	no	—
370	C₄⋊C₄×C₃₁	—	2	1	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	2	no	—
375	C₄⋊C₄×C₃₁	—	2	1	no	—
376	C₄⋊C₄×C₃₁	—	2	1	no	—
377	C₄⋊C₄×C₃₁	—	2	1	no	—
378	C₄⋊C₄×C₃₁	—	2	2	no	—
379	C₄⋊C₄×C₃₁	—	2	1	no	—
380	C₄⋊C₄×C₃₁	—	2	2	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	1	no	—
385	C₄⋊C₄×C₃₁	—	2	1	no	—
386	C₄⋊C₄×C₃₁	—	2	2	no	—
387	C₄⋊C₄×C₃₁	—	2	2	no	—
388	C₄⋊C₄×C₃₁	—	2	2	no	—
389	C₄⋊C₄×C₃₁	—	2	1	no	—
390	C₄⋊C₄×C₃₁	—	2	1	no	—
391	C₄⋊C₄×C₃₁	—	2	2	no	—
392	C₄⋊C₄×C₃₁	—	2	2	no	—
393	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
394	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
395	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
399	C₄⋊C₄×C₃₁	C₂⁴	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	1	yes	characteristic
404	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
405	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
406	C₂²⋊C₄×C₃₁	—	2	1	no	—
407	C₂²⋊C₄×C₃₁	—	2	1	no	—
408	C₂²⋊C₄×C₃₁	—	2	2	no	—
409	C₂²⋊C₄×C₃₁	—	2	2	no	—
410	C₂²⋊C₄×C₃₁	—	2	2	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	2	no	—
416	C₂²⋊C₄×C₃₁	—	2	2	no	—
417	C₂²⋊C₄×C₃₁	—	2	2	no	—
418	C₂²⋊C₄×C₃₁	—	2	2	no	—
419	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
420	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
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	C₄²×C₃₁	—	2	1	no	—
425	C₄²×C₃₁	—	2	1	no	—
426	C₄²×C₃₁	—	2	1	no	—
427	C₄²×C₃₁	—	2	1	no	—
428	C₄²×C₃₁	—	2	2	no	—
429	C₄²×C₃₁	—	2	2	no	—
430	C₄²×C₃₁	—	2	1	no	—
431	C₄²×C₃₁	—	2	1	no	—
432	C₄²×C₃₁	—	2	2	no	—
433	C₄²×C₃₁	—	2	2	no	—
434	C₄²×C₃₁	—	2	2	no	—
435	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
436	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
437	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
438	C₄²×C₃₁	C₂⁴	1	2	yes	—
439	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
440	#62193	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
441	C₂³×C₃₁	—	2	2	no	—
442	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
443	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
444	Q₈×C₃₁	—	2	2	no	—
445	Q₈×C₃₁	—	2	1	no	—
446	Q₈×C₃₁	—	2	1	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	—	8	1	no	—
458	Q₈×C₃₁	—	8	2	no	—
459	Q₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
460	D₄×C₃₁	—	2	2	no	—
461	D₄×C₃₁	—	2	2	no	—
462	D₄×C₃₁	—	2	2	no	—
463	D₄×C₃₁	—	4	2	no	—
464	D₄×C₃₁	—	4	4	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	C₂×C₄×C₃₁	—	2	1	no	—
470	C₂×C₄×C₃₁	—	2	2	no	—
471	C₂×C₄×C₃₁	—	2	1	no	—
472	C₂×C₄×C₃₁	—	2	2	no	—
473	C₂×C₄×C₃₁	—	2	2	no	—
474	C₂×C₄×C₃₁	—	2	1	no	—
475	C₂×C₄×C₃₁	—	2	1	no	—
476	C₂×C₄×C₃₁	—	2	1	no	—
477	C₂×C₄×C₃₁	—	2	2	no	—
478	C₂×C₄×C₃₁	—	2	1	no	—
479	C₂×C₄×C₃₁	—	2	1	no	—
480	C₂×C₄×C₃₁	—	2	2	no	—
481	C₂×C₄×C₃₁	—	2	1	no	—
482	C₂×C₄×C₃₁	—	2	1	no	—
483	C₂×C₄×C₃₁	—	2	2	no	—
484	C₂×C₄×C₃₁	—	2	1	no	—
485	C₂×C₄×C₃₁	—	4	1	no	—
486	C₂×C₄×C₃₁	—	4	4	no	—
487	C₂×C₄×C₃₁	—	4	2	no	—
488	C₂×C₄×C₃₁	—	4	2	no	—
489	C₂×C₄×C₃₁	—	8	2	no	—
490	C₂×C₄×C₃₁	—	8	2	no	—
491	C₂×C₄×C₃₁	—	8	2	no	—
492	C₂×C₄×C₃₁	—	8	2	no	—
493	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
494	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
495	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
496	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
497	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
498	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
499	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
500	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
501	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
502	C₈×C₃₁	—	2	1	no	—
503	C₈×C₃₁	—	2	1	no	—
504	C₈×C₃₁	—	2	1	no	—
505	C₈×C₃₁	—	2	2	no	—
506	C₈×C₃₁	—	2	2	no	—
507	C₈×C₃₁	—	2	1	no	—
508	#3555	C₂×C₃₁	1	1	yes	characteristic, complemented
509	#3522	C₂×C₃₁	1	1	yes	characteristic
510	#3387	C₂×C₃₁	1	1	yes	characteristic, complemented
511	Q₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
512	#3375	C₂×C₃₁	1	1	yes	characteristic, complemented
513	#3365	C₂×C₃₁	1	1	yes	characteristic, complemented
514	#3356	C₂×C₃₁	1	1	yes	characteristic, complemented
515	#3353	C₂×C₃₁	1	2	yes	complemented
516	#3343	C₂×C₃₁	1	2	yes	complemented
517	#3338	C₂×C₃₁	1	1	yes	characteristic, complemented
518	#3333	C₂×C₃₁	1	1	yes	characteristic, complemented
519	#3327	C₂×C₃₁	1	2	yes	complemented
520	#3313	C₂×C₃₁	1	2	yes	complemented
521	#3304	C₂×C₃₁	1	1	yes	characteristic, complemented
522	#3258	C₂×C₃₁	1	2	yes	complemented
523	#3254	C₂×C₃₁	1	1	yes	characteristic, complemented
524	#3243	C₂×C₃₁	1	2	yes	complemented
525	#3228	C₂×C₃₁	1	1	yes	characteristic, complemented
526	#3120	C₂×C₃₁	1	2	yes	complemented
527	#3104	C₂×C₃₁	1	1	yes	characteristic, complemented
528	#3083	C₂×C₃₁	1	2	yes	complemented
529	#3078	C₂×C₃₁	1	1	yes	characteristic, complemented
530	#2993	C₂×C₃₁	1	1	yes	characteristic, complemented
531	C₂²×C₃₁	—	2	1	no	—
532	C₂²×C₃₁	—	2	2	no	—
533	C₂²×C₃₁	—	2	1	no	—
534	C₂²×C₃₁	—	2	2	no	—
535	C₂²×C₃₁	—	2	1	no	—
536	C₂²×C₃₁	—	2	2	no	—
537	C₂²×C₃₁	—	4	2	no	—
538	C₂²×C₃₁	—	4	2	no	—
539	C₂²×C₃₁	—	4	2	no	—
540	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
541	C₄×C₃₁	—	2	1	no	—
542	C₄×C₃₁	—	2	1	no	—
543	C₄×C₃₁	—	2	2	no	—
544	C₄×C₃₁	—	2	2	no	—
545	C₄×C₃₁	—	2	1	no	—
546	C₄×C₃₁	—	2	2	no	—
547	C₄×C₃₁	—	2	1	no	—
548	C₄×C₃₁	—	2	2	no	—
549	C₄×C₃₁	—	4	1	no	—
550	C₄×C₃₁	—	4	1	no	—
551	C₄×C₃₁	—	4	2	no	—
552	C₄×C₃₁	—	4	1	no	—
553	C₄×C₃₁	—	4	2	no	—
554	C₄×C₃₁	—	4	1	no	—
555	C₄×C₃₁	—	4	1	no	—
556	C₄×C₃₁	—	4	1	no	—
557	C₄×C₃₁	—	4	1	no	—
558	C₄×C₃₁	—	4	2	no	—
559	C₄×C₃₁	#578	1	2	yes	—
560	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
561	C₄×C₃₁	#585	1	1	yes	characteristic
562	#563	—	2	2	no	—
563	#559	—	2	1	no	—
564	#559	C₂²×C₃₁	1	1	yes	characteristic
565	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
566	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
567	#557	—	2	1	no	—
568	#557	C₂²×C₃₁	1	1	yes	characteristic
569	#556	—	2	1	no	—
570	#556	C₂²×C₃₁	1	2	yes	—
571	#556	C₂²×C₃₁	1	1	yes	characteristic
572	#554	—	2	1	no	—
573	#554	C₂²×C₃₁	1	1	yes	characteristic
574	#554	C₂²×C₃₁	1	1	yes	characteristic
575	#552	C₂²×C₃₁	1	2	yes	—
576	#550	C₂²×C₃₁	1	1	yes	characteristic
577	#548	C₂²×C₃₁	1	2	yes	—
578	#547	—	2	2	no	—
579	#547	C₂²×C₃₁	1	2	yes	—
580	#544	C₂²×C₃₁	1	2	yes	—
581	#542	—	2	2	no	—
582	#542	—	2	1	no	—
583	#542	C₂²×C₃₁	1	2	yes	—
584	#541	—	2	2	no	—
585	#541	—	2	1	no	—
586	#541	C₂²×C₃₁	1	2	yes	—
587	#533	C₂²×C₃₁	1	1	yes	characteristic
588	#533	C₂²×C₃₁	1	1	yes	characteristic
589	#532	—	2	1	no	—
590	#532	C₂²×C₃₁	1	1	yes	characteristic
591	#529	—	2	1	no	—
592	#529	C₂²×C₃₁	1	1	yes	characteristic
593	#520	C₂²×C₃₁	1	2	yes	—
594	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
595	#501	C₂²×C₃₁	1	2	yes	—
596	#500	C₂²×C₃₁	1	1	yes	characteristic
597	#500	C₂²×C₃₁	1	2	yes	—
598	#498	C₂²×C₃₁	1	1	yes	characteristic
599	#489	C₂²×C₃₁	1	2	yes	—
600	#488	C₂²×C₃₁	1	2	yes	—
601	#486	C₂²×C₃₁	1	1	yes	characteristic
602	#486	C₂²×C₃₁	1	1	yes	characteristic
603	#485	C₂²×C₃₁	1	2	yes	—
604	#484	C₂²×C₃₁	1	2	yes	—
605	#484	C₂²×C₃₁	1	2	yes	—
606	#484	C₂²×C₃₁	1	1	yes	characteristic
607	#484	C₂²×C₃₁	1	1	yes	characteristic
608	#483	C₂²×C₃₁	1	2	yes	—
609	#482	C₂²×C₃₁	1	2	yes	—
610	#482	C₂²×C₃₁	1	2	yes	—
611	#482	C₂²×C₃₁	1	2	yes	—
612	#480	C₂²×C₃₁	1	2	yes	—
613	#479	C₂²×C₃₁	1	2	yes	—
614	#479	C₂²×C₃₁	1	2	yes	—
615	#479	C₂²×C₃₁	1	2	yes	—
616	#478	C₂²×C₃₁	1	2	yes	—
617	#478	C₂²×C₃₁	1	2	yes	—
618	#478	C₂²×C₃₁	1	2	yes	—
619	#477	C₂²×C₃₁	1	1	yes	characteristic
620	#477	C₂²×C₃₁	1	1	yes	characteristic
621	#477	C₂²×C₃₁	1	1	yes	characteristic
622	#477	C₂²×C₃₁	1	1	yes	characteristic
623	#477	C₂²×C₃₁	1	2	yes	—
624	#477	C₂²×C₃₁	1	2	yes	—
625	#477	C₂²×C₃₁	1	1	yes	characteristic
626	#477	C₂²×C₃₁	1	1	yes	characteristic
627	#475	C₂²×C₃₁	1	1	yes	characteristic
628	#475	C₂²×C₃₁	1	1	yes	characteristic
629	#475	C₂²×C₃₁	1	2	yes	—
630	#474	C₂²×C₃₁	1	2	yes	—
631	#474	C₂²×C₃₁	1	2	yes	—
632	#470	C₂²×C₃₁	1	2	yes	—
633	#468	C₂²×C₃₁	1	2	yes	—
634	#468	C₂²×C₃₁	1	2	yes	—
635	#468	C₂²×C₃₁	1	2	yes	—
636	#467	C₂²×C₃₁	1	2	yes	—
637	#465	C₂²×C₃₁	1	2	yes	—
638	#464	C₂²×C₃₁	1	2	yes	—
639	#464	C₂²×C₃₁	1	2	yes	—
640	#462	C₂²×C₃₁	1	1	yes	characteristic
641	#462	C₂²×C₃₁	1	1	yes	characteristic
642	#461	C₂²×C₃₁	1	2	yes	—
643	#460	C₂²×C₃₁	1	1	yes	characteristic
644	#460	C₂²×C₃₁	1	1	yes	characteristic
645	#452	C₂²×C₃₁	1	2	yes	—
646	#452	C₂²×C₃₁	1	2	yes	—
647	#451	C₂²×C₃₁	1	1	yes	characteristic
648	#451	C₂²×C₃₁	1	1	yes	characteristic
649	#446	C₂²×C₃₁	1	1	yes	characteristic
650	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
651	#441	C₂²×C₃₁	1	2	yes	—
652	#440	C₂²×C₃₁	1	2	yes	—
653	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
654	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
655	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
656	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
657	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
658	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
659	#436	C₂²×C₃₁	1	1	yes	characteristic
660	#436	C₂²×C₃₁	1	2	yes	—
661	#435	C₂²×C₃₁	1	2	yes	—
662	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
663	#433	C₂²×C₃₁	1	2	yes	—
664	#431	C₂²×C₃₁	1	1	yes	characteristic
665	#428	C₂²×C₃₁	1	2	yes	—
666	#428	C₂²×C₃₁	1	2	yes	—
667	C₂×#107	C₂²×C₃₁	1	2	yes	—
668	#424	C₂²×C₃₁	1	2	yes	—
669	#423	C₂²×C₃₁	1	1	yes	characteristic
670	#419	C₂²×C₃₁	1	2	yes	—
671	#417	C₂²×C₃₁	1	2	yes	—
672	#416	C₂²×C₃₁	1	2	yes	—
673	C₂×C₃₁	—	4	1	no	—
674	C₂×C₃₁	—	4	2	no	—
675	C₂×C₃₁	—	8	2	no	—
676	C₂×C₃₁	#3489	1	1	yes	characteristic
677	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
678	C₂×C₃₁	#3585	1	1	yes	characteristic
679	C₂×#41	—	2	2	no	—
680	C₂×#41	—	2	1	no	—
681	C₂×#41	C₂³×C₃₁	1	2	yes	—
682	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
683	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
684	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
685	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
686	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
687	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
688	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
689	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
690	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
691	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
692	C₄⋊Q₈	—	2	1	no	—
693	C₄⋊Q₈	—	2	1	no	—
694	C₄⋊Q₈	—	2	2	no	—
695	C₄⋊Q₈	—	2	2	no	—
696	C₄⋊Q₈	—	2	1	no	—
697	C₄⋊Q₈	—	2	2	no	—
698	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
699	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
700	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
701	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
702	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
703	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
704	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
705	#127	—	2	1	no	—
706	#127	C₂³×C₃₁	1	1	yes	characteristic
707	#126	—	2	2	no	—
708	#126	—	2	2	no	—
709	#126	—	2	2	no	—
710	#126	—	2	2	no	—
711	#126	—	2	2	no	—
712	#126	—	2	2	no	—
713	#126	—	2	2	no	—
714	#126	D₄×C₃₁	1	2	yes	—
715	#125	—	2	1	no	—
716	#125	—	2	2	no	—
717	#125	—	2	2	no	—
718	#125	—	2	2	no	—
719	#125	—	2	1	no	—
720	#125	—	2	2	no	—
721	#125	—	2	2	no	—
722	#125	—	2	1	no	—
723	#125	D₄×C₃₁	1	1	yes	characteristic
724	#125	C₂³×C₃₁	1	2	yes	—
725	#125	C₂³×C₃₁	1	2	yes	—
726	#124	—	2	1	no	—
727	#124	—	2	1	no	—
728	#124	—	2	2	no	—
729	#124	—	2	2	no	—
730	#124	—	2	2	no	—
731	#124	—	2	1	no	—
732	#124	D₄×C₃₁	1	1	yes	characteristic
733	#124	C₂³×C₃₁	1	2	yes	—
734	#123	—	2	2	no	—
735	#123	—	2	2	no	—
736	#123	—	2	2	no	—
737	#123	—	2	2	no	—
738	#123	—	2	2	no	—
739	#123	D₄×C₃₁	1	2	yes	—
740	C₂²⋊Q₈	—	2	1	no	—
741	C₂²⋊Q₈	—	2	1	no	—
742	C₂²⋊Q₈	—	2	1	no	—
743	C₂²⋊Q₈	—	2	2	no	—
744	C₂²⋊Q₈	—	2	2	no	—
745	C₂²⋊Q₈	—	2	2	no	—
746	C₂²⋊Q₈	—	2	2	no	—
747	C₂²⋊Q₈	—	2	2	no	—
748	C₂²⋊Q₈	—	2	2	no	—
749	C₂²⋊Q₈	—	2	2	no	—
750	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
751	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
752	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
753	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
754	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
755	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
756	#121	—	2	2	no	—
757	#121	—	2	1	no	—
758	#121	—	2	2	no	—
759	#121	D₄×C₃₁	1	1	yes	characteristic
760	#121	C₂³×C₃₁	1	2	yes	—
761	#121	C₂³×C₃₁	1	2	yes	—
762	C₄×Q₈	—	2	1	no	—
763	C₄×Q₈	—	2	1	no	—
764	C₄×Q₈	—	2	2	no	—
765	C₄×Q₈	—	2	1	no	—
766	C₄×Q₈	—	2	1	no	—
767	C₄×Q₈	—	2	1	no	—
768	C₄×Q₈	—	2	1	no	—
769	C₄×Q₈	—	2	1	no	—
770	C₄×Q₈	—	2	1	no	—
771	C₄×Q₈	—	2	1	no	—
772	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
773	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
774	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
775	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
776	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
777	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
778	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
779	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
780	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
781	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
782	C₄×D₄	—	2	1	no	—
783	C₄×D₄	—	2	1	no	—
784	C₄×D₄	—	2	1	no	—
785	C₄×D₄	—	2	1	no	—
786	C₄×D₄	—	2	2	no	—
787	C₄×D₄	—	2	2	no	—
788	C₄×D₄	—	2	2	no	—
789	C₄×D₄	—	2	1	no	—
790	C₄×D₄	C₂³×C₃₁	1	2	yes	—
791	C₄×D₄	C₂³×C₃₁	1	2	yes	—
792	C₄×D₄	C₂³×C₃₁	1	2	yes	—
793	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
794	#117	—	2	1	no	—
795	#117	—	2	2	no	—
796	#117	—	2	2	no	—
797	#117	C₂³×C₃₁	1	2	yes	—
798	#117	C₂³×C₃₁	1	2	yes	—
799	#117	C₂³×C₃₁	1	2	yes	—
800	C₂×C₄⋊C₄	—	2	1	no	—
801	C₂×C₄⋊C₄	—	2	2	no	—
802	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
803	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
804	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
805	C₂×C₄²	—	2	1	no	—
806	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
807	#107	C₂³×C₃₁	1	1	yes	characteristic
808	#107	C₂³×C₃₁	1	2	yes	—
809	#107	C₂³×C₃₁	1	2	yes	—
810	#107	C₂³×C₃₁	1	2	yes	—
811	#107	C₂³×C₃₁	1	1	yes	characteristic
812	#107	C₂³×C₃₁	1	2	yes	—
813	#107	C₂³×C₃₁	1	2	yes	—
814	#107	C₂³×C₃₁	1	1	yes	characteristic
815	#107	C₂³×C₃₁	1	1	yes	characteristic
816	#107	C₂³×C₃₁	1	2	yes	—
817	#106	C₂³×C₃₁	1	2	yes	—
818	#106	C₂³×C₃₁	1	2	yes	—
819	#106	C₂³×C₃₁	1	2	yes	—
820	#106	C₂³×C₃₁	1	2	yes	—
821	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
822	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
823	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
824	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
825	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
826	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
827	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
828	#103	C₂³×C₃₁	1	1	yes	characteristic
829	#103	C₂³×C₃₁	1	2	yes	—
830	#103	C₂³×C₃₁	1	1	yes	characteristic
831	#103	C₂³×C₃₁	1	2	yes	—
832	#103	C₂³×C₃₁	1	2	yes	—
833	#103	C₂³×C₃₁	1	2	yes	—
834	#103	C₂³×C₃₁	1	1	yes	characteristic
835	#103	C₂³×C₃₁	1	1	yes	characteristic
836	#103	C₂³×C₃₁	1	1	yes	characteristic
837	#103	C₂³×C₃₁	1	1	yes	characteristic
838	#103	C₂³×C₃₁	1	2	yes	—
839	#103	C₂³×C₃₁	1	1	yes	characteristic
840	#103	C₂³×C₃₁	1	1	yes	characteristic
841	#103	C₂³×C₃₁	1	2	yes	—
842	#102	C₂³×C₃₁	1	2	yes	—
843	#102	C₂³×C₃₁	1	2	yes	—
844	#102	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	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
849	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
850	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
851	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
852	#97	C₂³×C₃₁	1	2	yes	—
853	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
854	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
855	C₄×C₈	C₂³×C₃₁	1	2	yes	—
856	C₃₁	#62193	1	1	yes	characteristic, direct product factor, Sylow
857	#41	—	2	4	no	—
858	#41	—	4	2	no	—
859	#41	—	8	2	no	—
860	#41	—	8	4	no	—
861	C₂×Q₈	—	2	1	no	—
862	C₂×Q₈	—	2	1	no	—
863	C₂×Q₈	—	2	1	no	—
864	C₂×Q₈	—	2	1	no	—
865	C₂×Q₈	—	2	2	no	—
866	C₂×Q₈	—	2	2	no	—
867	C₂×Q₈	—	2	1	no	—
868	C₂×Q₈	—	2	2	no	—
869	C₂×Q₈	—	4	1	no	—
870	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
871	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
872	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
873	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
874	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
875	C₂×D₄	—	2	2	no	—
876	C₂×D₄	—	2	2	no	—
877	C₂×D₄	—	2	2	no	—
878	C₂×D₄	—	2	1	no	—
879	C₂×D₄	—	2	1	no	—
880	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
881	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
882	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
883	C₂²×C₄	—	2	1	no	—
884	C₂²×C₄	—	2	1	no	—
885	C₂²×C₄	—	2	2	no	—
886	C₂²×C₄	—	2	2	no	—
887	C₂²×C₄	—	2	2	no	—
888	C₂²×C₄	—	2	2	no	—
889	C₂²×C₄	—	2	2	no	—
890	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
891	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
892	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
893	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
894	Q₁₆	—	2	2	no	—
895	Q₁₆	—	2	2	no	—
896	Q₁₆	—	4	1	no	—
897	Q₁₆	—	4	1	no	—
898	SD₁₆	—	4	2	no	—
899	SD₁₆	—	4	2	no	—
900	M₁₆	—	4	2	no	—
901	M₁₆	—	4	2	no	—
902	C₂×C₈	—	4	2	no	—
903	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
904	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
905	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
906	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
907	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
908	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
909	C₄⋊C₄	—	2	1	no	—
910	C₄⋊C₄	—	2	2	no	—
911	C₄⋊C₄	—	2	2	no	—
912	C₄⋊C₄	—	2	1	no	—
913	C₄⋊C₄	—	2	1	no	—
914	C₄⋊C₄	—	2	1	no	—
915	C₄⋊C₄	—	2	1	no	—
916	C₄⋊C₄	—	2	2	no	—
917	C₄⋊C₄	—	2	1	no	—
918	C₄⋊C₄	—	2	1	no	—
919	C₄⋊C₄	—	2	2	no	—
920	C₄⋊C₄	—	2	1	no	—
921	C₄⋊C₄	—	2	1	no	—
922	C₄⋊C₄	—	2	1	no	—
923	C₄⋊C₄	—	2	2	no	—
924	C₄⋊C₄	—	2	1	no	—
925	C₄⋊C₄	—	2	2	no	—
926	C₄⋊C₄	—	2	2	no	—
927	C₄⋊C₄	—	2	2	no	—
928	C₄⋊C₄	—	2	2	no	—
929	C₄⋊C₄	—	2	1	no	—
930	C₄⋊C₄	—	2	1	no	—
931	C₄⋊C₄	—	2	2	no	—
932	C₄⋊C₄	—	2	2	no	—
933	C₄⋊C₄	—	2	2	no	—
934	C₄⋊C₄	—	2	1	no	—
935	C₄⋊C₄	—	2	1	no	—
936	C₄⋊C₄	—	2	2	no	—
937	C₄⋊C₄	—	2	2	no	—
938	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
939	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
940	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
941	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
942	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
943	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
944	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
945	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
946	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
947	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
948	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
949	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
950	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
951	C₂²⋊C₄	—	2	1	no	—
952	C₂²⋊C₄	—	2	1	no	—
953	C₂²⋊C₄	—	2	2	no	—
954	C₂²⋊C₄	—	2	2	no	—
955	C₂²⋊C₄	—	2	2	no	—
956	C₂²⋊C₄	—	2	2	no	—
957	C₂²⋊C₄	—	2	2	no	—
958	C₂²⋊C₄	—	2	2	no	—
959	C₂²⋊C₄	—	2	2	no	—
960	C₂²⋊C₄	—	2	2	no	—
961	C₂²⋊C₄	—	2	2	no	—
962	C₂²⋊C₄	—	2	2	no	—
963	C₂²⋊C₄	—	2	2	no	—
964	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
965	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
966	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
967	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
968	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
969	C₄²	—	2	1	no	—
970	C₄²	—	2	1	no	—
971	C₄²	—	2	1	no	—
972	C₄²	—	2	1	no	—
973	C₄²	—	2	2	no	—
974	C₄²	—	2	2	no	—
975	C₄²	—	2	1	no	—
976	C₄²	—	2	1	no	—
977	C₄²	—	2	2	no	—
978	C₄²	—	2	2	no	—
979	C₄²	—	2	2	no	—
980	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
981	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
982	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
983	C₄²	C₂⁴×C₃₁	1	2	yes	—
984	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
985	C₂³	—	2	2	no	—
986	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
987	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
988	Q₈	—	2	2	no	—
989	Q₈	—	2	1	no	—
990	Q₈	—	2	1	no	—
991	Q₈	—	2	1	no	—
992	Q₈	—	2	1	no	—
993	Q₈	—	2	1	no	—
994	Q₈	—	2	1	no	—
995	Q₈	—	2	1	no	—
996	Q₈	—	4	2	no	—
997	Q₈	—	4	2	no	—
998	Q₈	—	8	1	no	—
999	Q₈	—	8	1	no	—
1000	Q₈	—	8	2	no	—
1001	Q₈	—	8	1	no	—
1002	Q₈	—	8	2	no	—
1003	Q₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
1004	D₄	—	2	2	no	—
1005	D₄	—	2	2	no	—
1006	D₄	—	2	2	no	—
1007	D₄	—	4	2	no	—
1008	D₄	—	4	4	no	—
1009	D₄	—	4	4	no	—
1010	D₄	—	4	2	no	—
1011	D₄	—	8	1	no	—
1012	D₄	—	8	2	no	—
1013	C₂×C₄	—	2	1	no	—
1014	C₂×C₄	—	2	2	no	—
1015	C₂×C₄	—	2	1	no	—
1016	C₂×C₄	—	2	2	no	—
1017	C₂×C₄	—	2	2	no	—
1018	C₂×C₄	—	2	1	no	—
1019	C₂×C₄	—	2	1	no	—
1020	C₂×C₄	—	2	1	no	—
1021	C₂×C₄	—	2	2	no	—
1022	C₂×C₄	—	2	1	no	—
1023	C₂×C₄	—	2	1	no	—
1024	C₂×C₄	—	2	2	no	—
1025	C₂×C₄	—	2	1	no	—
1026	C₂×C₄	—	2	1	no	—
1027	C₂×C₄	—	2	2	no	—
1028	C₂×C₄	—	2	1	no	—
1029	C₂×C₄	—	4	1	no	—
1030	C₂×C₄	—	4	4	no	—
1031	C₂×C₄	—	4	2	no	—
1032	C₂×C₄	—	4	2	no	—
1033	C₂×C₄	—	8	2	no	—
1034	C₂×C₄	—	8	2	no	—
1035	C₂×C₄	—	8	2	no	—
1036	C₂×C₄	—	8	2	no	—
1037	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1038	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1039	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1040	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1041	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1042	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1043	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1044	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1045	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
1046	C₈	—	2	1	no	—
1047	C₈	—	2	1	no	—
1048	C₈	—	2	1	no	—
1049	C₈	—	2	2	no	—
1050	C₈	—	2	2	no	—
1051	C₈	—	2	1	no	—
1052	C₂²	—	2	1	no	—
1053	C₂²	—	2	2	no	—
1054	C₂²	—	2	1	no	—
1055	C₂²	—	2	2	no	—
1056	C₂²	—	2	1	no	—
1057	C₂²	—	2	2	no	—
1058	C₂²	—	4	2	no	—
1059	C₂²	—	4	2	no	—
1060	C₂²	—	4	2	no	—
1061	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
1062	C₄	—	2	2	no	—
1063	C₄	—	2	1	no	—
1064	C₄	—	2	2	no	—
1065	C₄	—	2	1	no	—
1066	C₄	—	2	2	no	—
1067	C₄	—	2	1	no	—
1068	C₄	—	2	1	no	—
1069	C₄	—	2	2	no	—
1070	C₄	—	4	1	no	—
1071	C₄	—	4	1	no	—
1072	C₄	—	4	1	no	—
1073	C₄	—	4	2	no	—
1074	C₄	—	4	1	no	—
1075	C₄	—	4	1	no	—
1076	C₄	—	4	2	no	—
1077	C₄	—	4	1	no	—
1078	C₄	—	4	2	no	—
1079	C₄	—	4	1	no	—
1080	C₄	C₃₁×#578	1	2	yes	—
1081	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
1082	C₄	C₃₁×#585	1	1	yes	characteristic
1083	C₂	—	4	1	no	—
1084	C₂	—	4	2	no	—
1085	C₂	—	8	2	no	—
1086	C₂	C₃₁×#3489	1	1	yes	characteristic
1087	C₂	C₂×C₃₁×#578	1	1	yes	characteristic
1088	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3522	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. Q₈×Q₁₆×C₃₁	C₂	semidirect
5. C₃₁×#3375	C₂	semidirect
6. C₃₁×#3365	C₂	semidirect
7. C₃₁×#3356	C₂	semidirect
8. C₃₁×#3353	C₂	semidirect
9. C₃₁×#3343	C₂	semidirect
10. C₃₁×#3338	C₂	semidirect
11. C₃₁×#3333	C₂	semidirect
12. C₃₁×#3327	C₂	semidirect
13. C₃₁×#3313	C₂	semidirect
14. C₃₁×#3304	C₂	semidirect
15. C₃₁×#3258	C₂	semidirect
16. C₃₁×#3254	C₂	semidirect
17. C₃₁×#3243	C₂	semidirect
18. C₃₁×#3228	C₂	semidirect
19. C₃₁×#3120	C₂	semidirect
20. C₃₁×#3104	C₂	semidirect
21. C₃₁×#3083	C₂	semidirect
22. C₃₁×#3078	C₂	semidirect
23. C₃₁×#2993	C₂	semidirect
440. #62193	C₃₁	direct (N × H)
508. #3555	C₂×C₃₁	semidirect
510. #3387	C₂×C₃₁	semidirect
511. Q₈×Q₁₆	C₂×C₃₁	semidirect
512. #3375	C₂×C₃₁	semidirect
513. #3365	C₂×C₃₁	semidirect
514. #3356	C₂×C₃₁	semidirect
515. #3353	C₂×C₃₁	semidirect
516. #3343	C₂×C₃₁	semidirect
517. #3338	C₂×C₃₁	semidirect
518. #3333	C₂×C₃₁	semidirect
519. #3327	C₂×C₃₁	semidirect
520. #3313	C₂×C₃₁	semidirect
521. #3304	C₂×C₃₁	semidirect
522. #3258	C₂×C₃₁	semidirect
523. #3254	C₂×C₃₁	semidirect
524. #3243	C₂×C₃₁	semidirect
525. #3228	C₂×C₃₁	semidirect
526. #3120	C₂×C₃₁	semidirect
527. #3104	C₂×C₃₁	semidirect
528. #3083	C₂×C₃₁	semidirect
529. #3078	C₂×C₃₁	semidirect
530. #2993	C₂×C₃₁	semidirect
856. C₃₁	#62193	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×#3522	2
C₃₁×#3387	2
Q₈×Q₁₆×C₃₁	2
C₃₁×#3375	2
C₃₁×#3365	2
C₃₁×#3356	2
C₃₁×#3353	2
C₃₁×#3343	2
C₃₁×#3338	2
C₃₁×#3333	2
C₃₁×#3327	2
C₃₁×#3313	2
C₃₁×#3304	2
C₃₁×#3258	2
C₃₁×#3254	2
C₃₁×#3243	2
C₃₁×#3228	2
C₃₁×#3120	2
C₃₁×#3104	2
C₃₁×#3083	2
C₃₁×#3078	2
C₃₁×#2993	2
#62193	31

Subgroups — C31×#62193 #21518952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62193 #21518952