Subgroups — C₃₁×#62220 #21518979

Order 7936.

1640 conjugacy classes of subgroups.

The subgroup lattice has 1076 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×#3518	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×#3369	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3349	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3342	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3340	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3328	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3313	C₂	1	2	yes	maximal, complemented
13	C₃₁×#3300	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3299	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#3258	C₂	1	2	yes	maximal, complemented
16	C₃₁×#3249	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#3243	C₂	1	2	yes	maximal, complemented
18	C₃₁×#3231	C₂	1	1	yes	maximal, characteristic, complemented
19	C₃₁×#3127	C₂	1	2	yes	maximal, complemented
20	C₃₁×#3107	C₂	1	1	yes	maximal, characteristic, complemented
21	C₃₁×#3082	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	Q₈²×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#557	—	2	1	no	—
28	C₃₁×#557	C₂²	1	1	yes	characteristic
29	C₃₁×#556	—	2	2	no	—
30	C₃₁×#556	—	2	1	no	—
31	C₃₁×#556	—	2	1	no	—
32	C₃₁×#556	C₂²	1	1	yes	characteristic
33	C₃₁×#556	C₂²	1	1	yes	characteristic
34	C₃₁×#556	C₂²	1	2	yes	—
35	C₃₁×#556	C₂²	1	2	yes	—
36	C₃₁×#555	—	2	1	no	—
37	C₃₁×#555	C₂²	1	1	yes	characteristic
38	C₃₁×#554	C₂²	1	1	yes	characteristic
39	C₃₁×#552	C₂²	1	2	yes	—
40	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#548	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	—	2	1	no	—
50	C₃₁×#533	C₂²	1	1	yes	characteristic
51	C₃₁×#533	C₂²	1	1	yes	characteristic
52	C₃₁×#533	C₂²	1	1	yes	characteristic
53	C₃₁×#529	—	2	1	no	—
54	C₃₁×#529	C₂²	1	1	yes	characteristic
55	C₃₁×#519	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	2	yes	—
59	C₃₁×#500	C₂²	1	1	yes	characteristic
60	C₃₁×#498	C₂²	1	1	yes	characteristic
61	C₃₁×#489	C₂²	1	2	yes	—
62	C₃₁×#488	C₂²	1	2	yes	—
63	C₃₁×#487	C₂²	1	1	yes	characteristic
64	C₃₁×#487	C₂²	1	1	yes	characteristic
65	C₃₁×#485	C₂²	1	2	yes	—
66	C₃₁×#485	C₂²	1	1	yes	characteristic
67	C₃₁×#485	C₂²	1	2	yes	—
68	C₃₁×#485	C₂²	1	2	yes	—
69	C₃₁×#485	C₂²	1	2	yes	—
70	C₃₁×#485	C₂²	1	1	yes	characteristic
71	C₃₁×#485	C₂²	1	2	yes	—
72	C₃₁×#484	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×#479	C₂²	1	2	yes	—
79	C₃₁×#479	C₂²	1	2	yes	—
80	C₃₁×#478	C₂²	1	2	yes	—
81	C₃₁×#477	C₂²	1	1	yes	characteristic
82	C₃₁×#477	C₂²	1	2	yes	—
83	C₃₁×#477	C₂²	1	2	yes	—
84	C₃₁×#477	C₂²	1	1	yes	characteristic
85	C₃₁×#477	C₂²	1	2	yes	—
86	C₃₁×#477	C₂²	1	1	yes	characteristic
87	C₃₁×#476	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	2	yes	—
92	C₃₁×#474	C₂²	1	1	yes	characteristic
93	C₃₁×#474	C₂²	1	1	yes	characteristic
94	C₃₁×#470	C₂²	1	2	yes	—
95	C₃₁×#468	C₂²	1	2	yes	—
96	C₃₁×#467	C₂²	1	2	yes	—
97	C₃₁×#467	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×#462	C₂²	1	1	yes	characteristic
105	C₃₁×#462	C₂²	1	2	yes	—
106	C₃₁×#461	C₂²	1	1	yes	characteristic
107	C₃₁×#452	C₂²	1	1	yes	characteristic
108	C₃₁×#452	C₂²	1	1	yes	characteristic
109	C₃₁×#451	C₂²	1	2	yes	—
110	C₃₁×#451	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×#441	C₂²	1	2	yes	—
115	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
116	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
117	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
118	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
119	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
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	C₃₁×#427	C₂²	1	2	yes	—
130	C₃₁×#424	C₂²	1	2	yes	—
131	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
132	C₃₁×#419	C₂²	1	2	yes	—
133	C₃₁×#416	C₂²	1	2	yes	—
134	C₂×C₃₁×#103	C₂²	1	2	yes	—
135	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
136	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
137	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
138	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
139	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
140	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
141	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
142	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
143	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
144	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
145	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
146	C₃₁×C₄⋊Q₈	—	2	1	no	—
147	C₃₁×C₄⋊Q₈	—	2	1	no	—
148	C₃₁×C₄⋊Q₈	—	2	1	no	—
149	C₃₁×C₄⋊Q₈	—	2	2	no	—
150	C₃₁×C₄⋊Q₈	—	2	2	no	—
151	C₃₁×C₄⋊Q₈	—	2	1	no	—
152	C₃₁×C₄⋊Q₈	—	2	2	no	—
153	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
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	2	yes	—
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×#126	—	2	2	no	—
162	C₃₁×#126	—	2	2	no	—
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	—	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	1	no	—
176	C₃₁×#125	—	2	2	no	—
177	C₃₁×#125	—	2	2	no	—
178	C₃₁×#125	—	2	1	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×#124	—	2	2	no	—
184	C₃₁×#124	—	2	1	no	—
185	C₃₁×#124	—	2	2	no	—
186	C₃₁×#124	—	2	2	no	—
187	C₃₁×#124	—	2	2	no	—
188	C₃₁×#124	—	2	2	no	—
189	C₃₁×#124	D₄	1	1	yes	characteristic
190	C₃₁×#124	C₂³	1	2	yes	—
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	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	D₄	1	1	yes	characteristic
206	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
207	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
208	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
209	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
210	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
211	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
212	C₃₁×#121	—	2	1	no	—
213	C₃₁×#121	—	2	2	no	—
214	C₃₁×#121	D₄	1	1	yes	characteristic
215	C₃₁×#121	C₂³	1	2	yes	—
216	C₄×Q₈×C₃₁	—	2	1	no	—
217	C₄×Q₈×C₃₁	—	2	1	no	—
218	C₄×Q₈×C₃₁	—	2	2	no	—
219	C₄×Q₈×C₃₁	—	2	2	no	—
220	C₄×Q₈×C₃₁	—	2	2	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	—	2	1	no	—
228	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
229	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
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	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	2	yes	—
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	C₂³	1	1	yes	characteristic
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	C₃₁×#117	—	2	1	no	—
248	C₃₁×#117	—	2	1	no	—
249	C₃₁×#117	—	2	1	no	—
250	C₃₁×#117	—	2	1	no	—
251	C₃₁×#117	—	2	2	no	—
252	C₃₁×#117	—	2	2	no	—
253	C₃₁×#117	—	2	2	no	—
254	C₃₁×#117	—	2	1	no	—
255	C₃₁×#117	C₂³	1	2	yes	—
256	C₃₁×#117	C₂³	1	2	yes	—
257	C₃₁×#117	C₂³	1	1	yes	characteristic
258	C₃₁×#117	C₂³	1	2	yes	—
259	C₃₁×#117	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	2	yes	—
267	C₃₁×#107	C₂³	1	2	yes	—
268	C₃₁×#107	C₂³	1	1	yes	characteristic
269	C₃₁×#107	C₂³	1	1	yes	characteristic
270	C₃₁×#107	C₂³	1	2	yes	—
271	C₃₁×#106	C₂³	1	1	yes	characteristic
272	C₃₁×#106	C₂³	1	2	yes	—
273	C₃₁×#106	C₂³	1	2	yes	—
274	C₃₁×#106	C₂³	1	2	yes	—
275	C₃₁×#106	C₂³	1	1	yes	characteristic
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	2	yes	—
288	C₃₁×#103	C₂³	1	2	yes	—
289	C₃₁×#103	C₂³	1	1	yes	characteristic
290	C₃₁×#103	C₂³	1	2	yes	—
291	C₃₁×#103	C₂³	1	2	yes	—
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×#103	C₂³	1	2	yes	—
299	C₃₁×#103	C₂³	1	2	yes	—
300	C₃₁×#102	C₂³	1	2	yes	—
301	C₃₁×#102	C₂³	1	2	yes	—
302	C₃₁×#102	C₂³	1	1	yes	characteristic
303	C₃₁×#102	C₂³	1	1	yes	characteristic
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	—	4	1	no	—
314	C₂×Q₈×C₃₁	—	2	1	no	—
315	C₂×Q₈×C₃₁	—	2	2	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	2	no	—
320	C₂×Q₈×C₃₁	—	2	2	no	—
321	C₂×Q₈×C₃₁	—	2	4	no	—
322	C₂×Q₈×C₃₁	—	4	2	no	—
323	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
324	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
331	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
332	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
333	C₂²×C₄×C₃₁	—	2	2	no	—
334	C₂²×C₄×C₃₁	—	2	2	no	—
335	C₂²×C₄×C₃₁	—	2	1	no	—
336	C₂²×C₄×C₃₁	—	2	1	no	—
337	C₂²×C₄×C₃₁	—	2	1	no	—
338	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
339	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
340	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
341	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
342	Q₁₆×C₃₁	—	2	2	no	—
343	Q₁₆×C₃₁	—	2	2	no	—
344	Q₁₆×C₃₁	—	4	2	no	—
345	Q₁₆×C₃₁	—	4	2	no	—
346	SD₁₆×C₃₁	—	4	1	no	—
347	SD₁₆×C₃₁	—	4	1	no	—
348	M₁₆×C₃₁	—	4	2	no	—
349	C₂×C₈×C₃₁	—	4	2	no	—
350	C₂×C₈×C₃₁	—	4	2	no	—
351	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
352	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
353	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
354	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
355	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
356	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
357	C₄⋊C₄×C₃₁	—	2	2	no	—
358	C₄⋊C₄×C₃₁	—	2	1	no	—
359	C₄⋊C₄×C₃₁	—	2	1	no	—
360	C₄⋊C₄×C₃₁	—	2	1	no	—
361	C₄⋊C₄×C₃₁	—	2	1	no	—
362	C₄⋊C₄×C₃₁	—	2	1	no	—
363	C₄⋊C₄×C₃₁	—	2	2	no	—
364	C₄⋊C₄×C₃₁	—	2	2	no	—
365	C₄⋊C₄×C₃₁	—	2	2	no	—
366	C₄⋊C₄×C₃₁	—	2	1	no	—
367	C₄⋊C₄×C₃₁	—	2	2	no	—
368	C₄⋊C₄×C₃₁	—	2	1	no	—
369	C₄⋊C₄×C₃₁	—	2	2	no	—
370	C₄⋊C₄×C₃₁	—	2	1	no	—
371	C₄⋊C₄×C₃₁	—	2	1	no	—
372	C₄⋊C₄×C₃₁	—	2	1	no	—
373	C₄⋊C₄×C₃₁	—	2	2	no	—
374	C₄⋊C₄×C₃₁	—	2	2	no	—
375	C₄⋊C₄×C₃₁	—	2	2	no	—
376	C₄⋊C₄×C₃₁	—	2	2	no	—
377	C₄⋊C₄×C₃₁	—	2	2	no	—
378	C₄⋊C₄×C₃₁	—	2	1	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	1	no	—
384	C₄⋊C₄×C₃₁	—	2	1	no	—
385	C₄⋊C₄×C₃₁	—	2	2	no	—
386	C₄⋊C₄×C₃₁	—	2	2	no	—
387	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
388	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
389	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
390	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
391	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
392	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
393	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
394	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
395	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
396	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
397	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
398	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
399	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
400	C₂²⋊C₄×C₃₁	—	2	2	no	—
401	C₂²⋊C₄×C₃₁	—	2	2	no	—
402	C₂²⋊C₄×C₃₁	—	2	1	no	—
403	C₂²⋊C₄×C₃₁	—	2	1	no	—
404	C₂²⋊C₄×C₃₁	—	2	2	no	—
405	C₂²⋊C₄×C₃₁	—	2	1	no	—
406	C₂²⋊C₄×C₃₁	—	2	2	no	—
407	C₂²⋊C₄×C₃₁	—	2	2	no	—
408	C₂²⋊C₄×C₃₁	—	2	2	no	—
409	C₂²⋊C₄×C₃₁	—	2	2	no	—
410	C₂²⋊C₄×C₃₁	—	2	1	no	—
411	C₂²⋊C₄×C₃₁	—	2	1	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	C₂×D₄	1	2	yes	—
416	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
417	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
418	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
419	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
420	C₄²×C₃₁	—	2	1	no	—
421	C₄²×C₃₁	—	2	1	no	—
422	C₄²×C₃₁	—	2	1	no	—
423	C₄²×C₃₁	—	2	2	no	—
424	C₄²×C₃₁	—	2	2	no	—
425	C₄²×C₃₁	—	2	2	no	—
426	C₄²×C₃₁	—	2	2	no	—
427	C₄²×C₃₁	—	2	2	no	—
428	C₄²×C₃₁	—	2	1	no	—
429	C₄²×C₃₁	—	2	1	no	—
430	C₄²×C₃₁	—	2	2	no	—
431	C₄²×C₃₁	—	2	2	no	—
432	C₄²×C₃₁	—	2	2	no	—
433	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
434	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
435	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
436	C₄²×C₃₁	C₂⁴	1	2	yes	—
437	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
438	#62220	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
439	C₂³×C₃₁	—	2	1	no	—
440	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
441	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
442	Q₈×C₃₁	—	2	2	no	—
443	Q₈×C₃₁	—	2	1	no	—
444	Q₈×C₃₁	—	2	1	no	—
445	Q₈×C₃₁	—	2	1	no	—
446	Q₈×C₃₁	—	2	2	no	—
447	Q₈×C₃₁	—	2	1	no	—
448	Q₈×C₃₁	—	2	1	no	—
449	Q₈×C₃₁	—	2	2	no	—
450	Q₈×C₃₁	—	4	4	no	—
451	Q₈×C₃₁	—	4	2	no	—
452	Q₈×C₃₁	—	8	1	no	—
453	Q₈×C₃₁	—	8	1	no	—
454	Q₈×C₃₁	—	8	1	no	—
455	Q₈×C₃₁	—	8	2	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	1	no	—
460	D₄×C₃₁	—	2	1	no	—
461	D₄×C₃₁	—	4	2	no	—
462	D₄×C₃₁	—	4	2	no	—
463	D₄×C₃₁	—	8	2	no	—
464	C₂×C₄×C₃₁	—	2	1	no	—
465	C₂×C₄×C₃₁	—	2	2	no	—
466	C₂×C₄×C₃₁	—	2	2	no	—
467	C₂×C₄×C₃₁	—	2	1	no	—
468	C₂×C₄×C₃₁	—	2	1	no	—
469	C₂×C₄×C₃₁	—	2	2	no	—
470	C₂×C₄×C₃₁	—	2	1	no	—
471	C₂×C₄×C₃₁	—	2	1	no	—
472	C₂×C₄×C₃₁	—	2	1	no	—
473	C₂×C₄×C₃₁	—	2	2	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	—	4	1	no	—
481	C₂×C₄×C₃₁	—	4	2	no	—
482	C₂×C₄×C₃₁	—	4	4	no	—
483	C₂×C₄×C₃₁	—	4	2	no	—
484	C₂×C₄×C₃₁	—	8	1	no	—
485	C₂×C₄×C₃₁	—	8	2	no	—
486	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
487	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
488	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
489	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
490	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
491	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
492	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
493	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
494	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
495	C₈×C₃₁	—	2	1	no	—
496	C₈×C₃₁	—	2	1	no	—
497	C₈×C₃₁	—	2	1	no	—
498	C₈×C₃₁	—	2	2	no	—
499	C₈×C₃₁	—	2	2	no	—
500	C₈×C₃₁	—	2	1	no	—
501	#3555	C₂×C₃₁	1	1	yes	characteristic, complemented
502	#3518	C₂×C₃₁	1	1	yes	characteristic
503	#3387	C₂×C₃₁	1	1	yes	characteristic, complemented
504	Q₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
505	#3369	C₂×C₃₁	1	1	yes	characteristic, complemented
506	#3364	C₂×C₃₁	1	1	yes	characteristic, complemented
507	#3357	C₂×C₃₁	1	1	yes	characteristic, complemented
508	#3349	C₂×C₃₁	1	2	yes	complemented
509	#3342	C₂×C₃₁	1	2	yes	complemented
510	#3340	C₂×C₃₁	1	1	yes	characteristic, complemented
511	#3328	C₂×C₃₁	1	1	yes	characteristic, complemented
512	#3313	C₂×C₃₁	1	2	yes	complemented
513	#3300	C₂×C₃₁	1	2	yes	complemented
514	#3299	C₂×C₃₁	1	1	yes	characteristic, complemented
515	#3258	C₂×C₃₁	1	2	yes	complemented
516	#3249	C₂×C₃₁	1	1	yes	characteristic, complemented
517	#3243	C₂×C₃₁	1	2	yes	complemented
518	#3231	C₂×C₃₁	1	1	yes	characteristic, complemented
519	#3127	C₂×C₃₁	1	2	yes	complemented
520	#3107	C₂×C₃₁	1	1	yes	characteristic, complemented
521	#3082	C₂×C₃₁	1	2	yes	complemented
522	#3076	C₂×C₃₁	1	1	yes	characteristic, complemented
523	#2994	C₂×C₃₁	1	1	yes	characteristic, complemented
524	C₂²×C₃₁	—	2	1	no	—
525	C₂²×C₃₁	—	2	2	no	—
526	C₂²×C₃₁	—	2	2	no	—
527	C₂²×C₃₁	—	2	1	no	—
528	C₂²×C₃₁	—	2	1	no	—
529	C₂²×C₃₁	—	2	2	no	—
530	C₂²×C₃₁	—	4	1	no	—
531	C₂²×C₃₁	—	4	1	no	—
532	C₂²×C₃₁	—	4	1	no	—
533	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
534	C₄×C₃₁	—	2	1	no	—
535	C₄×C₃₁	—	2	1	no	—
536	C₄×C₃₁	—	2	2	no	—
537	C₄×C₃₁	—	2	1	no	—
538	C₄×C₃₁	—	2	2	no	—
539	C₄×C₃₁	—	2	2	no	—
540	C₄×C₃₁	—	2	1	no	—
541	C₄×C₃₁	—	2	2	no	—
542	C₄×C₃₁	—	4	1	no	—
543	C₄×C₃₁	—	4	1	no	—
544	C₄×C₃₁	—	4	2	no	—
545	C₄×C₃₁	—	4	2	no	—
546	C₄×C₃₁	—	4	1	no	—
547	C₄×C₃₁	—	4	2	no	—
548	C₄×C₃₁	—	4	1	no	—
549	C₄×C₃₁	—	4	1	no	—
550	C₄×C₃₁	—	4	1	no	—
551	C₄×C₃₁	—	4	2	no	—
552	C₄×C₃₁	#578	1	2	yes	—
553	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
554	C₄×C₃₁	#585	1	1	yes	characteristic
555	#563	—	2	2	no	—
556	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
557	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
558	#557	—	2	1	no	—
559	#557	C₂²×C₃₁	1	1	yes	characteristic
560	#556	—	2	2	no	—
561	#556	—	2	1	no	—
562	#556	—	2	1	no	—
563	#556	C₂²×C₃₁	1	1	yes	characteristic
564	#556	C₂²×C₃₁	1	1	yes	characteristic
565	#556	C₂²×C₃₁	1	2	yes	—
566	#556	C₂²×C₃₁	1	2	yes	—
567	#555	—	2	1	no	—
568	#555	C₂²×C₃₁	1	1	yes	characteristic
569	#554	C₂²×C₃₁	1	1	yes	characteristic
570	#552	C₂²×C₃₁	1	2	yes	—
571	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
572	#548	C₂²×C₃₁	1	2	yes	—
573	#544	C₂²×C₃₁	1	2	yes	—
574	#542	—	2	2	no	—
575	#542	—	2	1	no	—
576	#542	C₂²×C₃₁	1	2	yes	—
577	#541	—	2	2	no	—
578	#541	—	2	1	no	—
579	#541	C₂²×C₃₁	1	2	yes	—
580	#533	—	2	1	no	—
581	#533	C₂²×C₃₁	1	1	yes	characteristic
582	#533	C₂²×C₃₁	1	1	yes	characteristic
583	#533	C₂²×C₃₁	1	1	yes	characteristic
584	#529	—	2	1	no	—
585	#529	C₂²×C₃₁	1	1	yes	characteristic
586	#519	C₂²×C₃₁	1	2	yes	—
587	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
588	#501	C₂²×C₃₁	1	2	yes	—
589	#500	C₂²×C₃₁	1	2	yes	—
590	#500	C₂²×C₃₁	1	1	yes	characteristic
591	#498	C₂²×C₃₁	1	1	yes	characteristic
592	#489	C₂²×C₃₁	1	2	yes	—
593	#488	C₂²×C₃₁	1	2	yes	—
594	#487	C₂²×C₃₁	1	1	yes	characteristic
595	#487	C₂²×C₃₁	1	1	yes	characteristic
596	#485	C₂²×C₃₁	1	2	yes	—
597	#485	C₂²×C₃₁	1	2	yes	—
598	#485	C₂²×C₃₁	1	1	yes	characteristic
599	#485	C₂²×C₃₁	1	2	yes	—
600	#485	C₂²×C₃₁	1	2	yes	—
601	#485	C₂²×C₃₁	1	1	yes	characteristic
602	#485	C₂²×C₃₁	1	2	yes	—
603	#484	C₂²×C₃₁	1	2	yes	—
604	#482	C₂²×C₃₁	1	2	yes	—
605	#480	C₂²×C₃₁	1	2	yes	—
606	#479	C₂²×C₃₁	1	2	yes	—
607	#479	C₂²×C₃₁	1	2	yes	—
608	#479	C₂²×C₃₁	1	2	yes	—
609	#479	C₂²×C₃₁	1	2	yes	—
610	#479	C₂²×C₃₁	1	2	yes	—
611	#478	C₂²×C₃₁	1	2	yes	—
612	#477	C₂²×C₃₁	1	1	yes	characteristic
613	#477	C₂²×C₃₁	1	2	yes	—
614	#477	C₂²×C₃₁	1	2	yes	—
615	#477	C₂²×C₃₁	1	1	yes	characteristic
616	#477	C₂²×C₃₁	1	2	yes	—
617	#477	C₂²×C₃₁	1	1	yes	characteristic
618	#476	C₂²×C₃₁	1	1	yes	characteristic
619	#475	C₂²×C₃₁	1	1	yes	characteristic
620	#475	C₂²×C₃₁	1	1	yes	characteristic
621	#475	C₂²×C₃₁	1	2	yes	—
622	#475	C₂²×C₃₁	1	2	yes	—
623	#474	C₂²×C₃₁	1	1	yes	characteristic
624	#474	C₂²×C₃₁	1	1	yes	characteristic
625	#470	C₂²×C₃₁	1	2	yes	—
626	#468	C₂²×C₃₁	1	2	yes	—
627	#467	C₂²×C₃₁	1	2	yes	—
628	#467	C₂²×C₃₁	1	2	yes	—
629	#467	C₂²×C₃₁	1	2	yes	—
630	#465	C₂²×C₃₁	1	2	yes	—
631	#464	C₂²×C₃₁	1	2	yes	—
632	#464	C₂²×C₃₁	1	2	yes	—
633	#462	C₂²×C₃₁	1	1	yes	characteristic
634	#462	C₂²×C₃₁	1	1	yes	characteristic
635	#462	C₂²×C₃₁	1	1	yes	characteristic
636	#462	C₂²×C₃₁	1	2	yes	—
637	#461	C₂²×C₃₁	1	1	yes	characteristic
638	#452	C₂²×C₃₁	1	1	yes	characteristic
639	#452	C₂²×C₃₁	1	1	yes	characteristic
640	#451	C₂²×C₃₁	1	2	yes	—
641	#451	C₂²×C₃₁	1	2	yes	—
642	#446	C₂²×C₃₁	1	1	yes	characteristic
643	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
644	#441	C₂²×C₃₁	1	2	yes	—
645	#441	C₂²×C₃₁	1	2	yes	—
646	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
647	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
648	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
649	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
650	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
651	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
652	#436	C₂²×C₃₁	1	1	yes	characteristic
653	#435	C₂²×C₃₁	1	2	yes	—
654	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
655	C₈×D₄	C₂²×C₃₁	1	2	yes	—
656	#433	C₂²×C₃₁	1	2	yes	—
657	#431	C₂²×C₃₁	1	1	yes	characteristic
658	#428	C₂²×C₃₁	1	2	yes	—
659	#427	C₂²×C₃₁	1	2	yes	—
660	#427	C₂²×C₃₁	1	2	yes	—
661	#424	C₂²×C₃₁	1	2	yes	—
662	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
663	#419	C₂²×C₃₁	1	2	yes	—
664	#416	C₂²×C₃₁	1	2	yes	—
665	C₂×#103	C₂²×C₃₁	1	2	yes	—
666	C₂×C₃₁	—	4	1	no	—
667	C₂×C₃₁	—	4	2	no	—
668	C₂×C₃₁	—	8	1	no	—
669	C₂×C₃₁	#3489	1	1	yes	characteristic
670	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
671	C₂×C₃₁	#3584	1	1	yes	characteristic
672	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
673	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
674	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
675	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
676	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
677	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
678	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
679	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
680	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
681	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
682	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
683	C₄⋊Q₈	—	2	1	no	—
684	C₄⋊Q₈	—	2	1	no	—
685	C₄⋊Q₈	—	2	1	no	—
686	C₄⋊Q₈	—	2	2	no	—
687	C₄⋊Q₈	—	2	2	no	—
688	C₄⋊Q₈	—	2	1	no	—
689	C₄⋊Q₈	—	2	2	no	—
690	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
691	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
692	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
693	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
694	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
695	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
696	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
697	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
698	#126	—	2	2	no	—
699	#126	—	2	2	no	—
700	#126	—	2	2	no	—
701	#126	—	2	2	no	—
702	#126	—	2	2	no	—
703	#126	—	2	2	no	—
704	#126	—	2	2	no	—
705	#126	—	2	2	no	—
706	#126	—	2	2	no	—
707	#126	—	2	2	no	—
708	#126	D₄×C₃₁	1	2	yes	—
709	#125	—	2	2	no	—
710	#125	—	2	2	no	—
711	#125	—	2	2	no	—
712	#125	—	2	1	no	—
713	#125	—	2	2	no	—
714	#125	—	2	2	no	—
715	#125	—	2	1	no	—
716	#125	—	2	1	no	—
717	#125	D₄×C₃₁	1	1	yes	characteristic
718	#125	C₂³×C₃₁	1	2	yes	—
719	#125	C₂³×C₃₁	1	2	yes	—
720	#124	—	2	2	no	—
721	#124	—	2	1	no	—
722	#124	—	2	2	no	—
723	#124	—	2	2	no	—
724	#124	—	2	2	no	—
725	#124	—	2	2	no	—
726	#124	D₄×C₃₁	1	1	yes	characteristic
727	#124	C₂³×C₃₁	1	2	yes	—
728	#123	—	2	2	no	—
729	#123	—	2	2	no	—
730	#123	D₄×C₃₁	1	2	yes	—
731	C₂²⋊Q₈	—	2	1	no	—
732	C₂²⋊Q₈	—	2	1	no	—
733	C₂²⋊Q₈	—	2	1	no	—
734	C₂²⋊Q₈	—	2	1	no	—
735	C₂²⋊Q₈	—	2	1	no	—
736	C₂²⋊Q₈	—	2	2	no	—
737	C₂²⋊Q₈	—	2	2	no	—
738	C₂²⋊Q₈	—	2	2	no	—
739	C₂²⋊Q₈	—	2	2	no	—
740	C₂²⋊Q₈	—	2	2	no	—
741	C₂²⋊Q₈	—	2	2	no	—
742	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
743	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
744	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
745	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
746	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
747	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
748	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
749	#121	—	2	1	no	—
750	#121	—	2	2	no	—
751	#121	D₄×C₃₁	1	1	yes	characteristic
752	#121	C₂³×C₃₁	1	2	yes	—
753	C₄×Q₈	—	2	1	no	—
754	C₄×Q₈	—	2	1	no	—
755	C₄×Q₈	—	2	2	no	—
756	C₄×Q₈	—	2	2	no	—
757	C₄×Q₈	—	2	2	no	—
758	C₄×Q₈	—	2	2	no	—
759	C₄×Q₈	—	2	1	no	—
760	C₄×Q₈	—	2	1	no	—
761	C₄×Q₈	—	2	1	no	—
762	C₄×Q₈	—	2	1	no	—
763	C₄×Q₈	—	2	1	no	—
764	C₄×Q₈	—	2	1	no	—
765	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
766	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
767	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
768	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
769	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
770	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
771	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
772	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
773	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
774	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
775	C₄×D₄	—	2	1	no	—
776	C₄×D₄	—	2	1	no	—
777	C₄×D₄	—	2	1	no	—
778	C₄×D₄	—	2	1	no	—
779	C₄×D₄	—	2	2	no	—
780	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
781	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
782	C₄×D₄	C₂³×C₃₁	1	2	yes	—
783	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
784	#117	—	2	1	no	—
785	#117	—	2	1	no	—
786	#117	—	2	1	no	—
787	#117	—	2	1	no	—
788	#117	—	2	2	no	—
789	#117	—	2	2	no	—
790	#117	—	2	2	no	—
791	#117	—	2	1	no	—
792	#117	C₂³×C₃₁	1	1	yes	characteristic
793	#117	C₂³×C₃₁	1	2	yes	—
794	#117	C₂³×C₃₁	1	2	yes	—
795	#117	C₂³×C₃₁	1	2	yes	—
796	#117	C₂³×C₃₁	1	2	yes	—
797	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
798	C₂×C₄²	—	2	1	no	—
799	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
800	#107	C₂³×C₃₁	1	2	yes	—
801	#107	C₂³×C₃₁	1	2	yes	—
802	#107	C₂³×C₃₁	1	2	yes	—
803	#107	C₂³×C₃₁	1	2	yes	—
804	#107	C₂³×C₃₁	1	2	yes	—
805	#107	C₂³×C₃₁	1	1	yes	characteristic
806	#107	C₂³×C₃₁	1	1	yes	characteristic
807	#107	C₂³×C₃₁	1	2	yes	—
808	#106	C₂³×C₃₁	1	1	yes	characteristic
809	#106	C₂³×C₃₁	1	2	yes	—
810	#106	C₂³×C₃₁	1	2	yes	—
811	#106	C₂³×C₃₁	1	2	yes	—
812	#106	C₂³×C₃₁	1	1	yes	characteristic
813	#106	C₂³×C₃₁	1	2	yes	—
814	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
815	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
816	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
817	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
818	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
819	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
820	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
821	#103	C₂³×C₃₁	1	1	yes	characteristic
822	#103	C₂³×C₃₁	1	2	yes	—
823	#103	C₂³×C₃₁	1	2	yes	—
824	#103	C₂³×C₃₁	1	2	yes	—
825	#103	C₂³×C₃₁	1	2	yes	—
826	#103	C₂³×C₃₁	1	1	yes	characteristic
827	#103	C₂³×C₃₁	1	2	yes	—
828	#103	C₂³×C₃₁	1	2	yes	—
829	#103	C₂³×C₃₁	1	1	yes	characteristic
830	#103	C₂³×C₃₁	1	1	yes	characteristic
831	#103	C₂³×C₃₁	1	2	yes	—
832	#103	C₂³×C₃₁	1	1	yes	characteristic
833	#103	C₂³×C₃₁	1	1	yes	characteristic
834	#103	C₂³×C₃₁	1	2	yes	—
835	#103	C₂³×C₃₁	1	2	yes	—
836	#103	C₂³×C₃₁	1	2	yes	—
837	#102	C₂³×C₃₁	1	2	yes	—
838	#102	C₂³×C₃₁	1	2	yes	—
839	#102	C₂³×C₃₁	1	1	yes	characteristic
840	#102	C₂³×C₃₁	1	1	yes	characteristic
841	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
842	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
843	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
844	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
845	#97	C₂³×C₃₁	1	2	yes	—
846	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
847	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
848	C₄×C₈	C₂³×C₃₁	1	2	yes	—
849	C₃₁	#62220	1	1	yes	characteristic, direct product factor, Sylow
850	#41	—	2	2	no	—
851	#41	—	4	1	no	—
852	C₂×Q₈	—	2	1	no	—
853	C₂×Q₈	—	2	2	no	—
854	C₂×Q₈	—	2	1	no	—
855	C₂×Q₈	—	2	1	no	—
856	C₂×Q₈	—	2	1	no	—
857	C₂×Q₈	—	2	2	no	—
858	C₂×Q₈	—	2	2	no	—
859	C₂×Q₈	—	2	4	no	—
860	C₂×Q₈	—	4	2	no	—
861	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
862	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
863	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
864	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
865	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
866	C₂×D₄	—	2	1	no	—
867	C₂×D₄	—	2	2	no	—
868	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
869	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
870	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
871	C₂²×C₄	—	2	2	no	—
872	C₂²×C₄	—	2	2	no	—
873	C₂²×C₄	—	2	1	no	—
874	C₂²×C₄	—	2	1	no	—
875	C₂²×C₄	—	2	1	no	—
876	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
877	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
878	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
879	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
880	Q₁₆	—	2	2	no	—
881	Q₁₆	—	2	2	no	—
882	Q₁₆	—	4	2	no	—
883	Q₁₆	—	4	2	no	—
884	SD₁₆	—	4	1	no	—
885	SD₁₆	—	4	1	no	—
886	M₁₆	—	4	2	no	—
887	C₂×C₈	—	4	2	no	—
888	C₂×C₈	—	4	2	no	—
889	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
890	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
891	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
892	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
893	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
894	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
895	C₄⋊C₄	—	2	2	no	—
896	C₄⋊C₄	—	2	1	no	—
897	C₄⋊C₄	—	2	1	no	—
898	C₄⋊C₄	—	2	1	no	—
899	C₄⋊C₄	—	2	1	no	—
900	C₄⋊C₄	—	2	1	no	—
901	C₄⋊C₄	—	2	2	no	—
902	C₄⋊C₄	—	2	2	no	—
903	C₄⋊C₄	—	2	2	no	—
904	C₄⋊C₄	—	2	1	no	—
905	C₄⋊C₄	—	2	2	no	—
906	C₄⋊C₄	—	2	1	no	—
907	C₄⋊C₄	—	2	2	no	—
908	C₄⋊C₄	—	2	1	no	—
909	C₄⋊C₄	—	2	1	no	—
910	C₄⋊C₄	—	2	1	no	—
911	C₄⋊C₄	—	2	2	no	—
912	C₄⋊C₄	—	2	2	no	—
913	C₄⋊C₄	—	2	2	no	—
914	C₄⋊C₄	—	2	2	no	—
915	C₄⋊C₄	—	2	2	no	—
916	C₄⋊C₄	—	2	1	no	—
917	C₄⋊C₄	—	2	1	no	—
918	C₄⋊C₄	—	2	2	no	—
919	C₄⋊C₄	—	2	2	no	—
920	C₄⋊C₄	—	2	2	no	—
921	C₄⋊C₄	—	2	1	no	—
922	C₄⋊C₄	—	2	1	no	—
923	C₄⋊C₄	—	2	2	no	—
924	C₄⋊C₄	—	2	2	no	—
925	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
926	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
927	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
928	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
929	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
930	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
931	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
932	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
933	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
934	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
935	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
936	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
937	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
938	C₂²⋊C₄	—	2	2	no	—
939	C₂²⋊C₄	—	2	2	no	—
940	C₂²⋊C₄	—	2	1	no	—
941	C₂²⋊C₄	—	2	1	no	—
942	C₂²⋊C₄	—	2	2	no	—
943	C₂²⋊C₄	—	2	1	no	—
944	C₂²⋊C₄	—	2	2	no	—
945	C₂²⋊C₄	—	2	2	no	—
946	C₂²⋊C₄	—	2	2	no	—
947	C₂²⋊C₄	—	2	2	no	—
948	C₂²⋊C₄	—	2	1	no	—
949	C₂²⋊C₄	—	2	1	no	—
950	C₂²⋊C₄	—	2	2	no	—
951	C₂²⋊C₄	—	2	2	no	—
952	C₂²⋊C₄	—	2	2	no	—
953	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
954	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
955	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
956	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
957	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
958	C₄²	—	2	1	no	—
959	C₄²	—	2	1	no	—
960	C₄²	—	2	1	no	—
961	C₄²	—	2	2	no	—
962	C₄²	—	2	2	no	—
963	C₄²	—	2	2	no	—
964	C₄²	—	2	2	no	—
965	C₄²	—	2	2	no	—
966	C₄²	—	2	1	no	—
967	C₄²	—	2	1	no	—
968	C₄²	—	2	2	no	—
969	C₄²	—	2	2	no	—
970	C₄²	—	2	2	no	—
971	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
972	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
973	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
974	C₄²	C₂⁴×C₃₁	1	2	yes	—
975	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
976	C₂³	—	2	1	no	—
977	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
978	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
979	Q₈	—	2	2	no	—
980	Q₈	—	2	1	no	—
981	Q₈	—	2	1	no	—
982	Q₈	—	2	1	no	—
983	Q₈	—	2	2	no	—
984	Q₈	—	2	1	no	—
985	Q₈	—	2	1	no	—
986	Q₈	—	2	2	no	—
987	Q₈	—	4	4	no	—
988	Q₈	—	4	2	no	—
989	Q₈	—	8	1	no	—
990	Q₈	—	8	1	no	—
991	Q₈	—	8	1	no	—
992	Q₈	—	8	2	no	—
993	Q₈	—	8	2	no	—
994	Q₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
995	D₄	—	2	2	no	—
996	D₄	—	2	1	no	—
997	D₄	—	2	1	no	—
998	D₄	—	4	2	no	—
999	D₄	—	4	2	no	—
1000	D₄	—	8	2	no	—
1001	C₂×C₄	—	2	1	no	—
1002	C₂×C₄	—	2	2	no	—
1003	C₂×C₄	—	2	2	no	—
1004	C₂×C₄	—	2	1	no	—
1005	C₂×C₄	—	2	1	no	—
1006	C₂×C₄	—	2	2	no	—
1007	C₂×C₄	—	2	1	no	—
1008	C₂×C₄	—	2	1	no	—
1009	C₂×C₄	—	2	1	no	—
1010	C₂×C₄	—	2	2	no	—
1011	C₂×C₄	—	2	2	no	—
1012	C₂×C₄	—	2	1	no	—
1013	C₂×C₄	—	2	2	no	—
1014	C₂×C₄	—	2	1	no	—
1015	C₂×C₄	—	2	2	no	—
1016	C₂×C₄	—	2	1	no	—
1017	C₂×C₄	—	4	1	no	—
1018	C₂×C₄	—	4	2	no	—
1019	C₂×C₄	—	4	4	no	—
1020	C₂×C₄	—	4	2	no	—
1021	C₂×C₄	—	8	1	no	—
1022	C₂×C₄	—	8	2	no	—
1023	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1024	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1025	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1026	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1027	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1028	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1029	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1030	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1031	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
1032	C₈	—	2	1	no	—
1033	C₈	—	2	1	no	—
1034	C₈	—	2	1	no	—
1035	C₈	—	2	2	no	—
1036	C₈	—	2	2	no	—
1037	C₈	—	2	1	no	—
1038	C₂²	—	2	2	no	—
1039	C₂²	—	2	1	no	—
1040	C₂²	—	2	1	no	—
1041	C₂²	—	2	2	no	—
1042	C₂²	—	2	1	no	—
1043	C₂²	—	2	2	no	—
1044	C₂²	—	4	1	no	—
1045	C₂²	—	4	1	no	—
1046	C₂²	—	4	1	no	—
1047	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
1048	C₄	—	2	1	no	—
1049	C₄	—	2	2	no	—
1050	C₄	—	2	2	no	—
1051	C₄	—	2	1	no	—
1052	C₄	—	2	2	no	—
1053	C₄	—	2	1	no	—
1054	C₄	—	2	1	no	—
1055	C₄	—	2	2	no	—
1056	C₄	—	4	2	no	—
1057	C₄	—	4	1	no	—
1058	C₄	—	4	1	no	—
1059	C₄	—	4	1	no	—
1060	C₄	—	4	2	no	—
1061	C₄	—	4	1	no	—
1062	C₄	—	4	1	no	—
1063	C₄	—	4	2	no	—
1064	C₄	—	4	2	no	—
1065	C₄	—	4	1	no	—
1066	C₄	C₃₁×#578	1	2	yes	—
1067	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
1068	C₄	C₃₁×#585	1	1	yes	characteristic
1069	C₂	—	4	2	no	—
1070	C₂	—	4	1	no	—
1071	C₂	—	8	1	no	—
1072	C₂	C₃₁×#3489	1	1	yes	characteristic
1073	C₂	C₂×C₃₁×#578	1	1	yes	characteristic
1074	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×#3518	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×#3369	C₂	semidirect
6. C₃₁×#3364	C₂	semidirect
7. C₃₁×#3357	C₂	semidirect
8. C₃₁×#3349	C₂	semidirect
9. C₃₁×#3342	C₂	semidirect
10. C₃₁×#3340	C₂	semidirect
11. C₃₁×#3328	C₂	semidirect
12. C₃₁×#3313	C₂	semidirect
13. C₃₁×#3300	C₂	semidirect
14. C₃₁×#3299	C₂	semidirect
15. C₃₁×#3258	C₂	semidirect
16. C₃₁×#3249	C₂	semidirect
17. C₃₁×#3243	C₂	semidirect
18. C₃₁×#3231	C₂	semidirect
19. C₃₁×#3127	C₂	semidirect
20. C₃₁×#3107	C₂	semidirect
21. C₃₁×#3082	C₂	semidirect
22. C₃₁×#3076	C₂	semidirect
23. C₃₁×#2994	C₂	semidirect
438. #62220	C₃₁	direct (N × H)
501. #3555	C₂×C₃₁	semidirect
503. #3387	C₂×C₃₁	semidirect
504. Q₈×Q₁₆	C₂×C₃₁	semidirect
505. #3369	C₂×C₃₁	semidirect
506. #3364	C₂×C₃₁	semidirect
507. #3357	C₂×C₃₁	semidirect
508. #3349	C₂×C₃₁	semidirect
509. #3342	C₂×C₃₁	semidirect
510. #3340	C₂×C₃₁	semidirect
511. #3328	C₂×C₃₁	semidirect
512. #3313	C₂×C₃₁	semidirect
513. #3300	C₂×C₃₁	semidirect
514. #3299	C₂×C₃₁	semidirect
515. #3258	C₂×C₃₁	semidirect
516. #3249	C₂×C₃₁	semidirect
517. #3243	C₂×C₃₁	semidirect
518. #3231	C₂×C₃₁	semidirect
519. #3127	C₂×C₃₁	semidirect
520. #3107	C₂×C₃₁	semidirect
521. #3082	C₂×C₃₁	semidirect
522. #3076	C₂×C₃₁	semidirect
523. #2994	C₂×C₃₁	semidirect
849. C₃₁	#62220	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×#3518	2
C₃₁×#3387	2
Q₈×Q₁₆×C₃₁	2
C₃₁×#3369	2
C₃₁×#3364	2
C₃₁×#3357	2
C₃₁×#3349	2
C₃₁×#3342	2
C₃₁×#3340	2
C₃₁×#3328	2
C₃₁×#3313	2
C₃₁×#3300	2
C₃₁×#3299	2
C₃₁×#3258	2
C₃₁×#3249	2
C₃₁×#3243	2
C₃₁×#3231	2
C₃₁×#3127	2
C₃₁×#3107	2
C₃₁×#3082	2
C₃₁×#3076	2
C₃₁×#2994	2
#62220	31

Subgroups — C31×#62220 #21518979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62220 #21518979