Subgroups — C₃₁×#62109 #21518868

Order 7936.

1974 conjugacy classes of subgroups.

The subgroup lattice has 916 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3386	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3368	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3353	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3343	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3324	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3310	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3254	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3246	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3235	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3226	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#3110	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#3077	C₂	1	2	yes	maximal, complemented
16	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#2989	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₃₁	—	2	1	no	—
19	C₃₁×#557	—	2	1	no	—
20	C₃₁×#557	C₂²	1	2	yes	complemented
21	C₃₁×#556	—	2	1	no	—
22	C₃₁×#556	C₂²	1	2	yes	complemented
23	C₃₁×#552	—	2	2	no	—
24	C₃₁×#552	C₂²	1	4	yes	complemented
25	C₃₁×#550	C₂²	1	1	yes	characteristic
26	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#548	—	2	1	no	—
28	C₃₁×#547	C₂²	1	2	yes	—
29	C₃₁×#547	C₂²	1	1	yes	characteristic
30	C₃₁×#546	—	2	2	no	—
31	C₃₁×#546	—	2	1	no	—
32	D₄²×C₃₁	C₂²	1	2	yes	—
33	D₄²×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#541	—	2	2	no	—
35	C₃₁×#536	—	2	1	no	—
36	C₃₁×#533	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#533	C₂²	1	2	yes	complemented
38	C₃₁×#533	C₂²	1	2	yes	complemented
39	C₃₁×#532	C₂²	1	1	yes	characteristic
40	C₃₁×#532	C₂²	1	2	yes	—
41	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×#525	—	2	1	no	—
43	C₃₁×#525	—	2	2	no	—
44	C₃₁×#519	—	2	1	no	—
45	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
46	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
47	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
48	C₃₁×#501	C₂²	1	4	yes	complemented
49	C₃₁×#500	C₂²	1	2	yes	complemented
50	C₃₁×#490	C₂²	1	4	yes	complemented
51	C₃₁×#486	C₂²	1	2	yes	complemented
52	C₃₁×#484	C₂²	1	4	yes	complemented
53	C₃₁×#483	C₂²	1	2	yes	complemented
54	C₃₁×#482	C₂²	1	4	yes	complemented
55	C₃₁×#482	C₂²	1	4	yes	complemented
56	C₃₁×#480	C₂²	1	4	yes	complemented
57	C₃₁×#478	C₂²	1	4	yes	—
58	C₃₁×#478	C₂²	1	4	yes	complemented
59	C₃₁×#477	C₂²	1	4	yes	complemented
60	C₃₁×#476	C₂²	1	1	yes	characteristic
61	C₃₁×#476	C₂²	1	2	yes	complemented
62	C₃₁×#476	C₂²	1	1	yes	characteristic, complemented
63	C₃₁×#475	C₂²	1	2	yes	complemented
64	C₃₁×#474	C₂²	1	2	yes	—
65	C₃₁×#474	C₂²	1	2	yes	—
66	C₃₁×#474	C₂²	1	2	yes	complemented
67	C₃₁×#474	C₂²	1	2	yes	complemented
68	C₃₁×#474	C₂²	1	2	yes	complemented
69	C₃₁×#474	C₂²	1	2	yes	complemented
70	C₃₁×#468	C₂²	1	4	yes	complemented
71	C₃₁×#468	C₂²	1	4	yes	complemented
72	C₃₁×#465	C₂²	1	4	yes	complemented
73	C₃₁×#461	C₂²	1	1	yes	characteristic
74	C₃₁×#461	C₂²	1	1	yes	characteristic, complemented
75	C₃₁×#461	C₂²	1	2	yes	complemented
76	C₃₁×#460	C₂²	1	2	yes	complemented
77	C₃₁×#460	C₂²	1	4	yes	complemented
78	C₃₁×#450	C₂²	1	2	yes	—
79	C₃₁×#450	C₂²	1	4	yes	—
80	C₃₁×#446	C₂²	1	2	yes	complemented
81	C₃₁×#440	C₂²	1	4	yes	—
82	C₃₁×#440	C₂²	1	4	yes	complemented
83	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
84	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
85	C₃₁×#436	C₂²	1	2	yes	complemented
86	C₃₁×#435	C₂²	1	4	yes	complemented
87	C₃₁×#432	C₂²	1	2	yes	complemented
88	C₃₁×#431	C₂²	1	1	yes	characteristic, complemented
89	C₃₁×#428	C₂²	1	4	yes	complemented
90	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
91	C₃₁×#423	C₂²	1	2	yes	complemented
92	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
93	C₃₁×#418	C₂²	1	4	yes	—
94	C₂×C₃₁×#102	C₂²	1	2	yes	—
95	C₂×#41×C₃₁	—	2	1	no	—
96	C₂×#41×C₃₁	—	2	2	no	—
97	C₂×#41×C₃₁	—	2	1	no	—
98	C₂×#41×C₃₁	—	2	2	no	—
99	C₂×#41×C₃₁	—	2	1	no	—
100	C₂×#41×C₃₁	—	4	8	no	—
101	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
102	C₂²×Q₈×C₃₁	—	2	1	no	—
103	C₂²×Q₈×C₃₁	—	2	1	no	—
104	C₂²×D₄×C₃₁	—	2	2	no	—
105	C₂²×D₄×C₃₁	—	2	1	no	—
106	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
107	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
108	C₂³×C₄×C₃₁	—	2	2	no	—
109	C₂³×C₄×C₃₁	—	2	1	no	—
110	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
111	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
112	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
113	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
114	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
115	C₃₁×C₄⋊Q₈	—	2	2	no	—
116	C₃₁×C₄⋊Q₈	—	2	1	no	—
117	C₃₁×C₄⋊Q₈	—	2	1	no	—
118	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
119	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
120	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
121	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
122	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
123	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
124	C₃₁×#127	C₂³	1	1	yes	characteristic
125	C₃₁×#127	C₂³	1	2	yes	—
126	C₃₁×#126	—	2	4	no	—
127	C₃₁×#126	—	2	4	no	—
128	C₃₁×#125	—	2	4	no	—
129	C₃₁×#125	—	2	4	no	—
130	C₃₁×#125	—	2	1	no	—
131	C₃₁×#125	—	2	2	no	—
132	C₃₁×#125	C₂³	1	4	yes	—
133	C₃₁×#124	—	2	2	no	—
134	C₃₁×#124	—	2	2	no	—
135	C₃₁×#124	—	2	1	no	—
136	C₃₁×#123	—	2	4	no	—
137	C₃₁×#123	—	2	2	no	—
138	C₃₁×#123	—	2	2	no	—
139	C₃₁×#123	—	2	2	no	—
140	C₃₁×C₂²⋊Q₈	—	2	2	no	—
141	C₃₁×C₂²⋊Q₈	—	2	2	no	—
142	C₃₁×C₂²⋊Q₈	—	2	1	no	—
143	C₃₁×C₂²⋊Q₈	—	2	2	no	—
144	C₃₁×C₂²⋊Q₈	—	2	1	no	—
145	C₃₁×C₂²⋊Q₈	—	2	2	no	—
146	C₃₁×C₂²⋊Q₈	—	2	1	no	—
147	C₃₁×C₂²⋊Q₈	—	2	2	no	—
148	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
149	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
150	C₃₁×#121	—	2	1	no	—
151	C₃₁×#121	—	2	2	no	—
152	C₃₁×#121	—	2	1	no	—
153	C₃₁×#121	—	2	2	no	—
154	C₃₁×#121	—	2	1	no	—
155	C₃₁×#121	—	2	2	no	—
156	C₃₁×#121	C₂³	1	2	yes	—
157	C₃₁×#121	C₂³	1	2	yes	—
158	C₃₁×#121	C₂³	1	2	yes	—
159	C₃₁×#120	—	2	2	no	—
160	C₃₁×#120	—	2	2	no	—
161	C₃₁×#120	—	2	2	no	—
162	C₄×Q₈×C₃₁	—	2	2	no	—
163	C₄×Q₈×C₃₁	—	2	2	no	—
164	C₄×Q₈×C₃₁	—	2	2	no	—
165	C₄×Q₈×C₃₁	—	2	2	no	—
166	C₄×Q₈×C₃₁	—	2	2	no	—
167	C₄×Q₈×C₃₁	—	2	1	no	—
168	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
169	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
170	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
171	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
172	C₄×D₄×C₃₁	—	2	2	no	—
173	C₄×D₄×C₃₁	—	2	1	no	—
174	C₄×D₄×C₃₁	—	2	2	no	—
175	C₄×D₄×C₃₁	—	2	1	no	—
176	C₄×D₄×C₃₁	—	2	2	no	—
177	C₄×D₄×C₃₁	—	2	1	no	—
178	C₄×D₄×C₃₁	D₄	1	2	yes	—
179	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
180	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
181	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
182	C₄×D₄×C₃₁	C₂³	1	2	yes	—
183	C₄×D₄×C₃₁	C₂³	1	2	yes	—
184	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
185	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
186	C₄×D₄×C₃₁	C₂³	1	2	yes	—
187	C₄×D₄×C₃₁	C₂³	1	2	yes	—
188	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
189	C₃₁×#117	—	2	1	no	—
190	C₃₁×#117	—	2	2	no	—
191	C₃₁×#117	—	2	4	no	—
192	C₃₁×#117	D₄	1	1	yes	characteristic
193	C₃₁×#117	D₄	1	2	yes	—
194	C₃₁×#117	C₂³	1	4	yes	—
195	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
196	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
197	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
198	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
199	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
200	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
201	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
202	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
203	C₃₁×#107	C₂³	1	4	yes	—
204	C₃₁×#107	C₂³	1	4	yes	—
205	C₃₁×#107	C₂³	1	4	yes	—
206	C₃₁×#107	C₂³	1	4	yes	—
207	C₃₁×#106	C₂³	1	2	yes	—
208	C₃₁×#106	C₂³	1	4	yes	—
209	C₃₁×#106	C₂³	1	2	yes	—
210	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
211	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
212	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
213	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
214	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
215	C₃₁×#103	C₂³	1	2	yes	—
216	C₃₁×#103	C₂³	1	2	yes	—
217	C₃₁×#103	C₂³	1	4	yes	—
218	C₃₁×#102	C₂³	1	4	yes	—
219	C₃₁×#102	C₂³	1	4	yes	—
220	C₃₁×#102	C₂³	1	2	yes	—
221	C₃₁×#102	C₂³	1	4	yes	—
222	C₃₁×#102	C₂³	1	4	yes	—
223	C₃₁×#102	C₂³	1	2	yes	—
224	C₃₁×#102	C₂³	1	4	yes	—
225	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
226	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
227	C₃₁×#97	C₂³	1	4	yes	—
228	C₄×C₈×C₃₁	C₂³	1	2	yes	—
229	C₂⁴×C₃₁	—	2	2	no	—
230	C₂⁴×C₃₁	—	2	1	no	—
231	#41×C₃₁	—	2	4	no	—
232	#41×C₃₁	—	4	8	no	—
233	#41×C₃₁	—	4	8	no	—
234	#41×C₃₁	—	4	4	no	—
235	#41×C₃₁	—	4	4	no	—
236	#41×C₃₁	—	4	4	no	—
237	C₂×Q₈×C₃₁	—	2	1	no	—
238	C₂×Q₈×C₃₁	—	2	2	no	—
239	C₂×Q₈×C₃₁	—	2	2	no	—
240	C₂×Q₈×C₃₁	—	2	2	no	—
241	C₂×Q₈×C₃₁	—	2	1	no	—
242	C₂×Q₈×C₃₁	—	4	4	no	—
243	C₂×Q₈×C₃₁	—	4	4	no	—
244	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
245	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
246	C₂×D₄×C₃₁	—	2	2	no	—
247	C₂×D₄×C₃₁	—	2	4	no	—
248	C₂×D₄×C₃₁	—	2	2	no	—
249	C₂×D₄×C₃₁	—	2	2	no	—
250	C₂×D₄×C₃₁	—	2	2	no	—
251	C₂×D₄×C₃₁	—	2	2	no	—
252	C₂×D₄×C₃₁	—	2	1	no	—
253	C₂×D₄×C₃₁	—	2	2	no	—
254	C₂×D₄×C₃₁	—	2	1	no	—
255	C₂×D₄×C₃₁	—	2	2	no	—
256	C₂×D₄×C₃₁	—	2	1	no	—
257	C₂×D₄×C₃₁	—	4	1	no	—
258	C₂×D₄×C₃₁	—	4	2	no	—
259	C₂×D₄×C₃₁	—	4	8	no	—
260	C₂×D₄×C₃₁	—	4	4	no	—
261	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
262	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
263	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
264	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
265	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
266	C₂²×C₄×C₃₁	—	2	2	no	—
267	C₂²×C₄×C₃₁	—	2	1	no	—
268	C₂²×C₄×C₃₁	—	2	2	no	—
269	C₂²×C₄×C₃₁	—	2	1	no	—
270	C₂²×C₄×C₃₁	—	2	2	no	—
271	C₂²×C₄×C₃₁	—	2	1	no	—
272	C₂²×C₄×C₃₁	—	2	2	no	—
273	C₂²×C₄×C₃₁	—	2	1	no	—
274	C₂²×C₄×C₃₁	—	2	2	no	—
275	C₂²×C₄×C₃₁	—	2	1	no	—
276	C₂²×C₄×C₃₁	—	4	8	no	—
277	C₂²×C₄×C₃₁	—	4	4	no	—
278	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
279	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
280	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
281	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
282	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
283	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
284	SD₁₆×C₃₁	—	2	4	no	—
285	SD₁₆×C₃₁	—	4	4	no	—
286	SD₁₆×C₃₁	—	4	2	no	—
287	M₁₆×C₃₁	—	4	4	no	—
288	C₂×C₈×C₃₁	—	4	2	no	—
289	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
290	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
291	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
292	C₄⋊C₄×C₃₁	—	2	2	no	—
293	C₄⋊C₄×C₃₁	—	2	2	no	—
294	C₄⋊C₄×C₃₁	—	2	2	no	—
295	C₄⋊C₄×C₃₁	—	2	1	no	—
296	C₄⋊C₄×C₃₁	—	2	2	no	—
297	C₄⋊C₄×C₃₁	—	2	1	no	—
298	C₄⋊C₄×C₃₁	—	2	1	no	—
299	C₄⋊C₄×C₃₁	—	2	1	no	—
300	C₄⋊C₄×C₃₁	—	2	4	no	—
301	C₄⋊C₄×C₃₁	—	2	4	no	—
302	C₄⋊C₄×C₃₁	—	2	2	no	—
303	C₄⋊C₄×C₃₁	—	2	1	no	—
304	C₄⋊C₄×C₃₁	—	2	4	no	—
305	C₄⋊C₄×C₃₁	—	2	2	no	—
306	C₄⋊C₄×C₃₁	—	2	4	no	—
307	C₄⋊C₄×C₃₁	—	2	2	no	—
308	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
309	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
310	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
311	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
312	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
313	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
314	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
315	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
316	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
317	C₂²⋊C₄×C₃₁	—	2	2	no	—
318	C₂²⋊C₄×C₃₁	—	2	4	no	—
319	C₂²⋊C₄×C₃₁	—	2	4	no	—
320	C₂²⋊C₄×C₃₁	—	2	1	no	—
321	C₂²⋊C₄×C₃₁	—	2	2	no	—
322	C₂²⋊C₄×C₃₁	—	2	1	no	—
323	C₂²⋊C₄×C₃₁	—	2	2	no	—
324	C₂²⋊C₄×C₃₁	—	2	1	no	—
325	C₂²⋊C₄×C₃₁	—	2	1	no	—
326	C₂²⋊C₄×C₃₁	—	2	2	no	—
327	C₂²⋊C₄×C₃₁	—	2	2	no	—
328	C₂²⋊C₄×C₃₁	—	2	2	no	—
329	C₂²⋊C₄×C₃₁	—	2	2	no	—
330	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
331	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
332	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
333	C₄²×C₃₁	—	2	4	no	—
334	C₄²×C₃₁	—	2	4	no	—
335	C₄²×C₃₁	—	2	2	no	—
336	C₄²×C₃₁	C₂×D₄	1	2	yes	—
337	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
338	C₄²×C₃₁	C₂×D₄	1	2	yes	—
339	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
340	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
341	C₄²×C₃₁	C₂⁴	1	2	yes	—
342	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
343	#62109	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
344	C₂³×C₃₁	—	2	1	no	—
345	C₂³×C₃₁	—	2	2	no	—
346	C₂³×C₃₁	—	2	1	no	—
347	C₂³×C₃₁	—	4	2	no	—
348	C₂³×C₃₁	—	4	4	no	—
349	C₂³×C₃₁	—	4	4	no	—
350	C₂³×C₃₁	—	4	2	no	—
351	C₂³×C₃₁	—	8	1	no	—
352	C₂³×C₃₁	—	8	2	no	—
353	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
354	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
355	Q₈×C₃₁	—	2	2	no	—
356	Q₈×C₃₁	—	2	2	no	—
357	Q₈×C₃₁	—	2	2	no	—
358	Q₈×C₃₁	—	4	2	no	—
359	Q₈×C₃₁	—	4	4	no	—
360	Q₈×C₃₁	—	4	4	no	—
361	Q₈×C₃₁	—	4	4	no	—
362	Q₈×C₃₁	—	4	2	no	—
363	D₄×C₃₁	—	2	4	no	—
364	D₄×C₃₁	—	2	2	no	—
365	D₄×C₃₁	—	2	1	no	—
366	D₄×C₃₁	—	2	2	no	—
367	D₄×C₃₁	—	2	1	no	—
368	D₄×C₃₁	—	2	2	no	—
369	D₄×C₃₁	—	2	1	no	—
370	D₄×C₃₁	—	4	2	no	—
371	D₄×C₃₁	—	4	4	no	—
372	D₄×C₃₁	—	4	4	no	—
373	D₄×C₃₁	—	4	4	no	—
374	D₄×C₃₁	—	4	4	no	—
375	D₄×C₃₁	—	4	4	no	—
376	D₄×C₃₁	—	4	2	no	—
377	D₄×C₃₁	—	8	2	no	—
378	D₄×C₃₁	—	8	1	no	—
379	D₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
380	C₂×C₄×C₃₁	—	2	4	no	—
381	C₂×C₄×C₃₁	—	2	1	no	—
382	C₂×C₄×C₃₁	—	2	2	no	—
383	C₂×C₄×C₃₁	—	2	1	no	—
384	C₂×C₄×C₃₁	—	2	2	no	—
385	C₂×C₄×C₃₁	—	2	2	no	—
386	C₂×C₄×C₃₁	—	2	2	no	—
387	C₂×C₄×C₃₁	—	2	4	no	—
388	C₂×C₄×C₃₁	—	2	1	no	—
389	C₂×C₄×C₃₁	—	2	2	no	—
390	C₂×C₄×C₃₁	—	2	1	no	—
391	C₂×C₄×C₃₁	—	4	1	no	—
392	C₂×C₄×C₃₁	—	4	2	no	—
393	C₂×C₄×C₃₁	—	4	4	no	—
394	C₂×C₄×C₃₁	—	4	4	no	—
395	C₂×C₄×C₃₁	—	4	4	no	—
396	C₂×C₄×C₃₁	—	4	2	no	—
397	C₂×C₄×C₃₁	—	4	4	no	—
398	C₂×C₄×C₃₁	—	4	2	no	—
399	C₂×C₄×C₃₁	—	4	2	no	—
400	C₂×C₄×C₃₁	—	4	2	no	—
401	C₂×C₄×C₃₁	—	4	4	no	—
402	C₂×C₄×C₃₁	—	4	2	no	—
403	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
404	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
405	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
406	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
407	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
408	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
409	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
410	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
411	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
412	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
413	C₈×C₃₁	—	2	2	no	—
414	C₈×C₃₁	—	2	4	no	—
415	C₈×C₃₁	—	2	2	no	—
416	#3509	C₂×C₃₁	1	1	yes	characteristic, complemented
417	D₄×#41	C₂×C₃₁	1	1	yes	characteristic
418	#3386	C₂×C₃₁	1	2	yes	complemented
419	#3368	C₂×C₃₁	1	2	yes	complemented
420	#3353	C₂×C₃₁	1	2	yes	complemented
421	#3343	C₂×C₃₁	1	4	yes	complemented
422	#3324	C₂×C₃₁	1	2	yes	complemented
423	#3310	C₂×C₃₁	1	4	yes	complemented
424	#3254	C₂×C₃₁	1	2	yes	complemented
425	#3246	C₂×C₃₁	1	1	yes	characteristic, complemented
426	#3235	C₂×C₃₁	1	2	yes	complemented
427	#3226	C₂×C₃₁	1	1	yes	characteristic, complemented
428	#3110	C₂×C₃₁	1	2	yes	complemented
429	#3101	C₂×C₃₁	1	1	yes	characteristic, complemented
430	#3077	C₂×C₃₁	1	2	yes	complemented
431	C₂×#474	C₂×C₃₁	1	1	yes	characteristic, complemented
432	#2989	C₂×C₃₁	1	1	yes	characteristic, complemented
433	C₂²×C₃₁	—	2	1	no	—
434	C₂²×C₃₁	—	2	1	no	—
435	C₂²×C₃₁	—	2	2	no	—
436	C₂²×C₃₁	—	2	2	no	—
437	C₂²×C₃₁	—	2	2	no	—
438	C₂²×C₃₁	—	2	2	no	—
439	C₂²×C₃₁	—	4	1	no	—
440	C₂²×C₃₁	—	4	1	no	—
441	C₂²×C₃₁	—	4	1	no	—
442	C₂²×C₃₁	—	4	2	no	—
443	C₂²×C₃₁	—	4	1	no	—
444	C₂²×C₃₁	—	4	2	no	—
445	C₂²×C₃₁	—	4	2	no	—
446	C₂²×C₃₁	—	4	1	no	—
447	C₂²×C₃₁	—	8	4	no	—
448	C₂²×C₃₁	—	8	8	no	—
449	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
450	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
451	C₄×C₃₁	—	2	1	no	—
452	C₄×C₃₁	—	2	2	no	—
453	C₄×C₃₁	—	2	1	no	—
454	C₄×C₃₁	—	2	1	no	—
455	C₄×C₃₁	—	2	2	no	—
456	C₄×C₃₁	—	2	2	no	—
457	C₄×C₃₁	—	4	2	no	—
458	C₄×C₃₁	—	4	2	no	—
459	C₄×C₃₁	—	4	1	no	—
460	C₄×C₃₁	—	4	2	no	—
461	C₄×C₃₁	—	4	1	no	—
462	C₄×C₃₁	—	4	4	no	—
463	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
464	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
465	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
466	C₄×C₃₁	#585	1	1	yes	characteristic
467	C₂²×#41	—	2	1	no	—
468	#557	—	2	1	no	—
469	#557	C₂²×C₃₁	1	2	yes	complemented
470	#556	—	2	1	no	—
471	#556	C₂²×C₃₁	1	2	yes	complemented
472	#552	—	2	2	no	—
473	#552	C₂²×C₃₁	1	4	yes	complemented
474	#550	C₂²×C₃₁	1	1	yes	characteristic
475	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
476	#548	—	2	1	no	—
477	#547	C₂²×C₃₁	1	2	yes	—
478	#547	C₂²×C₃₁	1	1	yes	characteristic
479	#546	—	2	2	no	—
480	#546	—	2	1	no	—
481	D₄²	C₂²×C₃₁	1	2	yes	—
482	D₄²	C₂²×C₃₁	1	1	yes	characteristic
483	#541	—	2	2	no	—
484	#536	—	2	1	no	—
485	#533	C₂²×C₃₁	1	1	yes	characteristic, complemented
486	#533	C₂²×C₃₁	1	2	yes	complemented
487	#533	C₂²×C₃₁	1	2	yes	complemented
488	#532	C₂²×C₃₁	1	1	yes	characteristic
489	#532	C₂²×C₃₁	1	2	yes	—
490	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
491	#525	—	2	1	no	—
492	#525	—	2	2	no	—
493	#519	—	2	1	no	—
494	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
495	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
496	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
497	#501	C₂²×C₃₁	1	4	yes	complemented
498	#500	C₂²×C₃₁	1	2	yes	complemented
499	#490	C₂²×C₃₁	1	4	yes	complemented
500	#486	C₂²×C₃₁	1	2	yes	complemented
501	#484	C₂²×C₃₁	1	4	yes	complemented
502	#483	C₂²×C₃₁	1	2	yes	complemented
503	#482	C₂²×C₃₁	1	4	yes	complemented
504	#482	C₂²×C₃₁	1	4	yes	complemented
505	#480	C₂²×C₃₁	1	4	yes	complemented
506	#478	C₂²×C₃₁	1	4	yes	—
507	#478	C₂²×C₃₁	1	4	yes	complemented
508	#477	C₂²×C₃₁	1	4	yes	complemented
509	#476	C₂²×C₃₁	1	1	yes	characteristic
510	#476	C₂²×C₃₁	1	2	yes	complemented
511	#476	C₂²×C₃₁	1	1	yes	characteristic, complemented
512	#475	C₂²×C₃₁	1	2	yes	complemented
513	#474	C₂²×C₃₁	1	2	yes	—
514	#474	C₂²×C₃₁	1	2	yes	—
515	#474	C₂²×C₃₁	1	2	yes	complemented
516	#474	C₂²×C₃₁	1	2	yes	complemented
517	#474	C₂²×C₃₁	1	2	yes	complemented
518	#474	C₂²×C₃₁	1	2	yes	complemented
519	#468	C₂²×C₃₁	1	4	yes	complemented
520	#468	C₂²×C₃₁	1	4	yes	complemented
521	#465	C₂²×C₃₁	1	4	yes	complemented
522	#461	C₂²×C₃₁	1	1	yes	characteristic
523	#461	C₂²×C₃₁	1	1	yes	characteristic, complemented
524	#461	C₂²×C₃₁	1	2	yes	complemented
525	#460	C₂²×C₃₁	1	2	yes	complemented
526	#460	C₂²×C₃₁	1	4	yes	complemented
527	#450	C₂²×C₃₁	1	2	yes	—
528	#450	C₂²×C₃₁	1	4	yes	—
529	#446	C₂²×C₃₁	1	2	yes	complemented
530	#440	C₂²×C₃₁	1	4	yes	—
531	#440	C₂²×C₃₁	1	4	yes	complemented
532	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
533	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
534	#436	C₂²×C₃₁	1	2	yes	complemented
535	#435	C₂²×C₃₁	1	4	yes	complemented
536	#432	C₂²×C₃₁	1	2	yes	complemented
537	#431	C₂²×C₃₁	1	1	yes	characteristic, complemented
538	#428	C₂²×C₃₁	1	4	yes	complemented
539	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
540	#423	C₂²×C₃₁	1	2	yes	complemented
541	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
542	#418	C₂²×C₃₁	1	4	yes	—
543	C₂×#102	C₂²×C₃₁	1	2	yes	—
544	C₂×C₃₁	—	2	2	no	—
545	C₂×C₃₁	—	4	2	no	—
546	C₂×C₃₁	—	4	2	no	—
547	C₂×C₃₁	—	8	1	no	—
548	C₂×C₃₁	—	8	2	no	—
549	C₂×C₃₁	#3489	1	1	yes	characteristic
550	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
551	C₂×C₃₁	#3586	1	1	yes	characteristic
552	C₂×#41	—	2	1	no	—
553	C₂×#41	—	2	2	no	—
554	C₂×#41	—	2	1	no	—
555	C₂×#41	—	2	2	no	—
556	C₂×#41	—	2	1	no	—
557	C₂×#41	—	4	8	no	—
558	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
559	C₂²×Q₈	—	2	1	no	—
560	C₂²×Q₈	—	2	1	no	—
561	C₂²×D₄	—	2	2	no	—
562	C₂²×D₄	—	2	1	no	—
563	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
564	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
565	C₂³×C₄	—	2	2	no	—
566	C₂³×C₄	—	2	1	no	—
567	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
568	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
569	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
570	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
571	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
572	C₄⋊Q₈	—	2	2	no	—
573	C₄⋊Q₈	—	2	1	no	—
574	C₄⋊Q₈	—	2	1	no	—
575	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
576	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
577	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
578	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
579	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
580	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
581	#127	C₂³×C₃₁	1	1	yes	characteristic
582	#127	C₂³×C₃₁	1	2	yes	—
583	#126	—	2	4	no	—
584	#126	—	2	4	no	—
585	#125	—	2	4	no	—
586	#125	—	2	4	no	—
587	#125	—	2	1	no	—
588	#125	—	2	2	no	—
589	#125	C₂³×C₃₁	1	4	yes	—
590	#124	—	2	2	no	—
591	#124	—	2	2	no	—
592	#124	—	2	1	no	—
593	#123	—	2	4	no	—
594	#123	—	2	2	no	—
595	#123	—	2	2	no	—
596	#123	—	2	2	no	—
597	C₂²⋊Q₈	—	2	2	no	—
598	C₂²⋊Q₈	—	2	2	no	—
599	C₂²⋊Q₈	—	2	1	no	—
600	C₂²⋊Q₈	—	2	2	no	—
601	C₂²⋊Q₈	—	2	1	no	—
602	C₂²⋊Q₈	—	2	2	no	—
603	C₂²⋊Q₈	—	2	1	no	—
604	C₂²⋊Q₈	—	2	2	no	—
605	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
606	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
607	#121	—	2	1	no	—
608	#121	—	2	2	no	—
609	#121	—	2	1	no	—
610	#121	—	2	2	no	—
611	#121	—	2	1	no	—
612	#121	—	2	2	no	—
613	#121	C₂³×C₃₁	1	2	yes	—
614	#121	C₂³×C₃₁	1	2	yes	—
615	#121	C₂³×C₃₁	1	2	yes	—
616	#120	—	2	2	no	—
617	#120	—	2	2	no	—
618	#120	—	2	2	no	—
619	C₄×Q₈	—	2	2	no	—
620	C₄×Q₈	—	2	2	no	—
621	C₄×Q₈	—	2	2	no	—
622	C₄×Q₈	—	2	2	no	—
623	C₄×Q₈	—	2	2	no	—
624	C₄×Q₈	—	2	1	no	—
625	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
626	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
627	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
628	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
629	C₄×D₄	—	2	2	no	—
630	C₄×D₄	—	2	1	no	—
631	C₄×D₄	—	2	2	no	—
632	C₄×D₄	—	2	1	no	—
633	C₄×D₄	—	2	2	no	—
634	C₄×D₄	—	2	1	no	—
635	C₄×D₄	D₄×C₃₁	1	2	yes	—
636	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
637	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
638	C₄×D₄	C₂³×C₃₁	1	2	yes	—
639	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
640	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
641	C₄×D₄	C₂³×C₃₁	1	2	yes	—
642	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
643	C₄×D₄	C₂³×C₃₁	1	2	yes	—
644	C₄×D₄	C₂³×C₃₁	1	2	yes	—
645	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
646	#117	—	2	1	no	—
647	#117	—	2	2	no	—
648	#117	—	2	4	no	—
649	#117	D₄×C₃₁	1	1	yes	characteristic
650	#117	D₄×C₃₁	1	2	yes	—
651	#117	C₂³×C₃₁	1	4	yes	—
652	C₂×C₄⋊C₄	—	2	1	no	—
653	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
654	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
655	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
656	C₂×C₂²⋊C₄	—	2	1	no	—
657	C₂×C₂²⋊C₄	—	2	2	no	—
658	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
659	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
660	#107	C₂³×C₃₁	1	4	yes	—
661	#107	C₂³×C₃₁	1	4	yes	—
662	#107	C₂³×C₃₁	1	4	yes	—
663	#107	C₂³×C₃₁	1	4	yes	—
664	#106	C₂³×C₃₁	1	2	yes	—
665	#106	C₂³×C₃₁	1	4	yes	—
666	#106	C₂³×C₃₁	1	2	yes	—
667	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
668	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
669	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
670	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
671	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
672	#103	C₂³×C₃₁	1	2	yes	—
673	#103	C₂³×C₃₁	1	2	yes	—
674	#103	C₂³×C₃₁	1	4	yes	—
675	#102	C₂³×C₃₁	1	4	yes	—
676	#102	C₂³×C₃₁	1	4	yes	—
677	#102	C₂³×C₃₁	1	2	yes	—
678	#102	C₂³×C₃₁	1	4	yes	—
679	#102	C₂³×C₃₁	1	4	yes	—
680	#102	C₂³×C₃₁	1	2	yes	—
681	#102	C₂³×C₃₁	1	4	yes	—
682	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
683	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
684	#97	C₂³×C₃₁	1	4	yes	—
685	C₄×C₈	C₂³×C₃₁	1	2	yes	—
686	C₃₁	#62109	1	1	yes	characteristic, direct product factor, Sylow
687	C₂⁴	—	2	2	no	—
688	C₂⁴	—	2	1	no	—
689	#41	—	2	4	no	—
690	#41	—	4	8	no	—
691	#41	—	4	8	no	—
692	#41	—	4	4	no	—
693	#41	—	4	4	no	—
694	#41	—	4	4	no	—
695	C₂×Q₈	—	2	1	no	—
696	C₂×Q₈	—	2	2	no	—
697	C₂×Q₈	—	2	2	no	—
698	C₂×Q₈	—	2	2	no	—
699	C₂×Q₈	—	2	1	no	—
700	C₂×Q₈	—	4	4	no	—
701	C₂×Q₈	—	4	4	no	—
702	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
703	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
704	C₂×D₄	—	2	2	no	—
705	C₂×D₄	—	2	4	no	—
706	C₂×D₄	—	2	2	no	—
707	C₂×D₄	—	2	2	no	—
708	C₂×D₄	—	2	2	no	—
709	C₂×D₄	—	2	2	no	—
710	C₂×D₄	—	2	1	no	—
711	C₂×D₄	—	2	2	no	—
712	C₂×D₄	—	2	1	no	—
713	C₂×D₄	—	2	2	no	—
714	C₂×D₄	—	2	1	no	—
715	C₂×D₄	—	4	1	no	—
716	C₂×D₄	—	4	2	no	—
717	C₂×D₄	—	4	8	no	—
718	C₂×D₄	—	4	4	no	—
719	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
720	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
721	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
722	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
723	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
724	C₂²×C₄	—	2	2	no	—
725	C₂²×C₄	—	2	1	no	—
726	C₂²×C₄	—	2	2	no	—
727	C₂²×C₄	—	2	1	no	—
728	C₂²×C₄	—	2	2	no	—
729	C₂²×C₄	—	2	1	no	—
730	C₂²×C₄	—	2	2	no	—
731	C₂²×C₄	—	2	1	no	—
732	C₂²×C₄	—	2	2	no	—
733	C₂²×C₄	—	2	1	no	—
734	C₂²×C₄	—	4	8	no	—
735	C₂²×C₄	—	4	4	no	—
736	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
737	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
738	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
739	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
740	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
741	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
742	SD₁₆	—	2	4	no	—
743	SD₁₆	—	4	4	no	—
744	SD₁₆	—	4	2	no	—
745	M₁₆	—	4	4	no	—
746	C₂×C₈	—	4	2	no	—
747	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
748	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
749	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
750	C₄⋊C₄	—	2	2	no	—
751	C₄⋊C₄	—	2	2	no	—
752	C₄⋊C₄	—	2	2	no	—
753	C₄⋊C₄	—	2	1	no	—
754	C₄⋊C₄	—	2	2	no	—
755	C₄⋊C₄	—	2	1	no	—
756	C₄⋊C₄	—	2	1	no	—
757	C₄⋊C₄	—	2	1	no	—
758	C₄⋊C₄	—	2	4	no	—
759	C₄⋊C₄	—	2	4	no	—
760	C₄⋊C₄	—	2	2	no	—
761	C₄⋊C₄	—	2	1	no	—
762	C₄⋊C₄	—	2	4	no	—
763	C₄⋊C₄	—	2	2	no	—
764	C₄⋊C₄	—	2	4	no	—
765	C₄⋊C₄	—	2	2	no	—
766	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
767	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
768	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
769	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
770	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
771	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
772	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
773	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
774	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
775	C₂²⋊C₄	—	2	2	no	—
776	C₂²⋊C₄	—	2	4	no	—
777	C₂²⋊C₄	—	2	4	no	—
778	C₂²⋊C₄	—	2	1	no	—
779	C₂²⋊C₄	—	2	2	no	—
780	C₂²⋊C₄	—	2	1	no	—
781	C₂²⋊C₄	—	2	2	no	—
782	C₂²⋊C₄	—	2	1	no	—
783	C₂²⋊C₄	—	2	1	no	—
784	C₂²⋊C₄	—	2	2	no	—
785	C₂²⋊C₄	—	2	2	no	—
786	C₂²⋊C₄	—	2	2	no	—
787	C₂²⋊C₄	—	2	2	no	—
788	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
789	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
790	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
791	C₄²	—	2	4	no	—
792	C₄²	—	2	4	no	—
793	C₄²	—	2	2	no	—
794	C₄²	C₂×D₄×C₃₁	1	2	yes	—
795	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
796	C₄²	C₂×D₄×C₃₁	1	2	yes	—
797	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
798	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
799	C₄²	C₂⁴×C₃₁	1	2	yes	—
800	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
801	C₂³	—	2	1	no	—
802	C₂³	—	2	2	no	—
803	C₂³	—	2	1	no	—
804	C₂³	—	4	2	no	—
805	C₂³	—	4	4	no	—
806	C₂³	—	4	4	no	—
807	C₂³	—	4	2	no	—
808	C₂³	—	8	1	no	—
809	C₂³	—	8	2	no	—
810	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
811	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
812	Q₈	—	2	2	no	—
813	Q₈	—	2	2	no	—
814	Q₈	—	2	2	no	—
815	Q₈	—	4	2	no	—
816	Q₈	—	4	4	no	—
817	Q₈	—	4	4	no	—
818	Q₈	—	4	4	no	—
819	Q₈	—	4	2	no	—
820	D₄	—	2	4	no	—
821	D₄	—	2	2	no	—
822	D₄	—	2	1	no	—
823	D₄	—	2	2	no	—
824	D₄	—	2	1	no	—
825	D₄	—	2	2	no	—
826	D₄	—	2	1	no	—
827	D₄	—	4	2	no	—
828	D₄	—	4	4	no	—
829	D₄	—	4	4	no	—
830	D₄	—	4	4	no	—
831	D₄	—	4	4	no	—
832	D₄	—	4	4	no	—
833	D₄	—	4	2	no	—
834	D₄	—	8	2	no	—
835	D₄	—	8	1	no	—
836	D₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
837	C₂×C₄	—	2	4	no	—
838	C₂×C₄	—	2	1	no	—
839	C₂×C₄	—	2	2	no	—
840	C₂×C₄	—	2	1	no	—
841	C₂×C₄	—	2	2	no	—
842	C₂×C₄	—	2	2	no	—
843	C₂×C₄	—	2	2	no	—
844	C₂×C₄	—	2	4	no	—
845	C₂×C₄	—	2	1	no	—
846	C₂×C₄	—	2	2	no	—
847	C₂×C₄	—	2	1	no	—
848	C₂×C₄	—	4	1	no	—
849	C₂×C₄	—	4	2	no	—
850	C₂×C₄	—	4	4	no	—
851	C₂×C₄	—	4	4	no	—
852	C₂×C₄	—	4	4	no	—
853	C₂×C₄	—	4	2	no	—
854	C₂×C₄	—	4	4	no	—
855	C₂×C₄	—	4	2	no	—
856	C₂×C₄	—	4	2	no	—
857	C₂×C₄	—	4	2	no	—
858	C₂×C₄	—	4	4	no	—
859	C₂×C₄	—	4	2	no	—
860	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
861	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
862	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
863	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
864	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
865	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
866	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
867	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
868	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
869	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
870	C₈	—	2	2	no	—
871	C₈	—	2	4	no	—
872	C₈	—	2	2	no	—
873	C₂²	—	2	1	no	—
874	C₂²	—	2	1	no	—
875	C₂²	—	2	2	no	—
876	C₂²	—	2	2	no	—
877	C₂²	—	2	2	no	—
878	C₂²	—	2	2	no	—
879	C₂²	—	4	1	no	—
880	C₂²	—	4	1	no	—
881	C₂²	—	4	1	no	—
882	C₂²	—	4	2	no	—
883	C₂²	—	4	2	no	—
884	C₂²	—	4	1	no	—
885	C₂²	—	4	1	no	—
886	C₂²	—	4	2	no	—
887	C₂²	—	8	4	no	—
888	C₂²	—	8	8	no	—
889	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
890	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
891	C₄	—	2	1	no	—
892	C₄	—	2	2	no	—
893	C₄	—	2	2	no	—
894	C₄	—	2	1	no	—
895	C₄	—	2	2	no	—
896	C₄	—	2	1	no	—
897	C₄	—	4	2	no	—
898	C₄	—	4	1	no	—
899	C₄	—	4	2	no	—
900	C₄	—	4	1	no	—
901	C₄	—	4	4	no	—
902	C₄	—	4	2	no	—
903	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
904	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
905	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
906	C₄	C₃₁×#585	1	1	yes	characteristic
907	C₂	—	2	2	no	—
908	C₂	—	4	2	no	—
909	C₂	—	4	2	no	—
910	C₂	—	8	1	no	—
911	C₂	—	8	2	no	—
912	C₂	C₃₁×#3489	1	1	yes	characteristic
913	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
914	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	D₄×#41×C₃₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3509	C₂	semidirect
3. C₃₁×#3386	C₂	semidirect
4. C₃₁×#3368	C₂	semidirect
5. C₃₁×#3353	C₂	semidirect
6. C₃₁×#3343	C₂	semidirect
7. C₃₁×#3324	C₂	semidirect
8. C₃₁×#3310	C₂	semidirect
9. C₃₁×#3254	C₂	semidirect
10. C₃₁×#3246	C₂	semidirect
11. C₃₁×#3235	C₂	semidirect
12. C₃₁×#3226	C₂	semidirect
13. C₃₁×#3110	C₂	semidirect
14. C₃₁×#3101	C₂	semidirect
15. C₃₁×#3077	C₂	semidirect
16. C₂×C₃₁×#474	C₂	semidirect
17. C₃₁×#2989	C₂	semidirect
20. C₃₁×#557	C₂²	semidirect
22. C₃₁×#556	C₂²	semidirect
24. C₃₁×#552	C₂²	semidirect
36. C₃₁×#533	C₂²	semidirect
37. C₃₁×#533	C₂²	semidirect
38. C₃₁×#533	C₂²	semidirect
41. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
48. C₃₁×#501	C₂²	semidirect
49. C₃₁×#500	C₂²	semidirect
50. C₃₁×#490	C₂²	semidirect
51. C₃₁×#486	C₂²	semidirect
52. C₃₁×#484	C₂²	semidirect
53. C₃₁×#483	C₂²	semidirect
54. C₃₁×#482	C₂²	semidirect
55. C₃₁×#482	C₂²	semidirect
56. C₃₁×#480	C₂²	semidirect
58. C₃₁×#478	C₂²	semidirect
59. C₃₁×#477	C₂²	semidirect
61. C₃₁×#476	C₂²	semidirect
62. C₃₁×#476	C₂²	semidirect
63. C₃₁×#475	C₂²	semidirect
66. C₃₁×#474	C₂²	semidirect
67. C₃₁×#474	C₂²	semidirect
68. C₃₁×#474	C₂²	semidirect
69. C₃₁×#474	C₂²	semidirect
70. C₃₁×#468	C₂²	semidirect
71. C₃₁×#468	C₂²	semidirect
72. C₃₁×#465	C₂²	semidirect
74. C₃₁×#461	C₂²	semidirect
75. C₃₁×#461	C₂²	semidirect
76. C₃₁×#460	C₂²	semidirect
77. C₃₁×#460	C₂²	semidirect
80. C₃₁×#446	C₂²	semidirect
82. C₃₁×#440	C₂²	semidirect
84. C₄×SD₁₆×C₃₁	C₂²	semidirect
85. C₃₁×#436	C₂²	semidirect
86. C₃₁×#435	C₂²	semidirect
87. C₃₁×#432	C₂²	semidirect
88. C₃₁×#431	C₂²	semidirect
89. C₃₁×#428	C₂²	semidirect
90. C₂×C₃₁×#107	C₂²	semidirect
91. C₃₁×#423	C₂²	semidirect
92. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
343. #62109	C₃₁	direct (N × H)
416. #3509	C₂×C₃₁	semidirect
418. #3386	C₂×C₃₁	semidirect
419. #3368	C₂×C₃₁	semidirect
420. #3353	C₂×C₃₁	semidirect
421. #3343	C₂×C₃₁	semidirect
422. #3324	C₂×C₃₁	semidirect
423. #3310	C₂×C₃₁	semidirect
424. #3254	C₂×C₃₁	semidirect
425. #3246	C₂×C₃₁	semidirect
426. #3235	C₂×C₃₁	semidirect
427. #3226	C₂×C₃₁	semidirect
428. #3110	C₂×C₃₁	semidirect
429. #3101	C₂×C₃₁	semidirect
430. #3077	C₂×C₃₁	semidirect
431. C₂×#474	C₂×C₃₁	semidirect
432. #2989	C₂×C₃₁	semidirect
469. #557	C₂²×C₃₁	semidirect
471. #556	C₂²×C₃₁	semidirect
473. #552	C₂²×C₃₁	semidirect
485. #533	C₂²×C₃₁	semidirect
486. #533	C₂²×C₃₁	semidirect
487. #533	C₂²×C₃₁	semidirect
490. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
497. #501	C₂²×C₃₁	semidirect
498. #500	C₂²×C₃₁	semidirect
499. #490	C₂²×C₃₁	semidirect
500. #486	C₂²×C₃₁	semidirect
501. #484	C₂²×C₃₁	semidirect
502. #483	C₂²×C₃₁	semidirect
503. #482	C₂²×C₃₁	semidirect
504. #482	C₂²×C₃₁	semidirect
505. #480	C₂²×C₃₁	semidirect
507. #478	C₂²×C₃₁	semidirect
508. #477	C₂²×C₃₁	semidirect
510. #476	C₂²×C₃₁	semidirect
511. #476	C₂²×C₃₁	semidirect
512. #475	C₂²×C₃₁	semidirect
515. #474	C₂²×C₃₁	semidirect
516. #474	C₂²×C₃₁	semidirect
517. #474	C₂²×C₃₁	semidirect
518. #474	C₂²×C₃₁	semidirect
519. #468	C₂²×C₃₁	semidirect
520. #468	C₂²×C₃₁	semidirect
521. #465	C₂²×C₃₁	semidirect
523. #461	C₂²×C₃₁	semidirect
524. #461	C₂²×C₃₁	semidirect
525. #460	C₂²×C₃₁	semidirect
526. #460	C₂²×C₃₁	semidirect
529. #446	C₂²×C₃₁	semidirect
531. #440	C₂²×C₃₁	semidirect
533. C₄×SD₁₆	C₂²×C₃₁	semidirect
534. #436	C₂²×C₃₁	semidirect
535. #435	C₂²×C₃₁	semidirect
536. #432	C₂²×C₃₁	semidirect
537. #431	C₂²×C₃₁	semidirect
538. #428	C₂²×C₃₁	semidirect
539. C₂×#107	C₂²×C₃₁	semidirect
540. #423	C₂²×C₃₁	semidirect
541. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
686. C₃₁	#62109	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×#3509	2
D₄×#41×C₃₁	2
C₃₁×#3386	2
C₃₁×#3368	2
C₃₁×#3353	2
C₃₁×#3343	2
C₃₁×#3324	2
C₃₁×#3310	2
C₃₁×#3254	2
C₃₁×#3246	2
C₃₁×#3235	2
C₃₁×#3226	2
C₃₁×#3110	2
C₃₁×#3101	2
C₃₁×#3077	2
C₂×C₃₁×#474	2
C₃₁×#2989	2
#62109	31

Subgroups — C31×#62109 #21518868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62109 #21518868