Subgroups — C₃₁×#61841 #21518600

Order 7936.

1744 conjugacy classes of subgroups.

The subgroup lattice has 836 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3523	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3388	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3384	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3381	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3376	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3372	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3360	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3238	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3236	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3232	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3229	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#3114	C₂	1	2	yes	maximal
14	C₃₁×#3110	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#3105	C₂	1	1	yes	maximal, characteristic
16	C₃₁×#3101	C₂	1	2	yes	maximal, complemented
17	C₃₁×#2984	C₂	1	1	yes	maximal, characteristic, complemented
18	C₃₁×#557	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#557	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#557	C₂²	1	2	yes	complemented
21	C₃₁×#556	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#556	C₂²	1	2	yes	complemented
23	C₃₁×#554	—	2	1	no	—
24	C₃₁×#554	—	2	1	no	—
25	C₃₁×#554	—	2	2	no	—
26	C₃₁×#554	C₂²	1	2	yes	—
27	C₃₁×#554	C₂²	1	2	yes	—
28	C₃₁×#554	C₂²	1	4	yes	—
29	C₃₁×#552	—	2	1	no	—
30	C₃₁×#552	—	2	2	no	—
31	D₄×Q₈×C₃₁	—	2	1	no	—
32	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#541	—	2	2	no	—
34	C₃₁×#541	—	2	1	no	—
35	C₃₁×#533	C₂²	1	2	yes	—
36	C₃₁×#533	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
38	C₃₁×#529	C₂²	1	1	yes	characteristic
39	C₃₁×#529	C₂²	1	2	yes	—
40	C₂×C₃₁×#125	C₂²	1	2	yes	—
41	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
42	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
43	C₃₁×#519	—	2	1	no	—
44	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
46	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
47	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₃₁×#501	C₂²	1	4	yes	complemented
49	C₃₁×#501	C₂²	1	2	yes	complemented
50	C₃₁×#499	C₂²	1	4	yes	complemented
51	C₃₁×#498	C₂²	1	2	yes	complemented
52	C₃₁×#491	C₂²	1	4	yes	complemented
53	C₃₁×#489	C₂²	1	2	yes	complemented
54	C₃₁×#488	C₂²	1	4	yes	complemented
55	C₃₁×#487	C₂²	1	2	yes	complemented
56	C₃₁×#484	C₂²	1	1	yes	characteristic
57	C₃₁×#484	C₂²	1	2	yes	—
58	C₃₁×#483	C₂²	1	2	yes	—
59	C₃₁×#483	C₂²	1	2	yes	—
60	C₃₁×#483	C₂²	1	2	yes	—
61	C₃₁×#481	C₂²	1	1	yes	characteristic
62	C₃₁×#480	C₂²	1	2	yes	—
63	C₃₁×#479	C₂²	1	4	yes	complemented
64	C₃₁×#479	C₂²	1	4	yes	complemented
65	C₃₁×#479	C₂²	1	4	yes	complemented
66	C₃₁×#478	C₂²	1	2	yes	—
67	C₃₁×#478	C₂²	1	4	yes	—
68	C₃₁×#478	C₂²	1	4	yes	complemented
69	C₃₁×#477	C₂²	1	2	yes	complemented
70	C₃₁×#476	C₂²	1	4	yes	—
71	C₃₁×#476	C₂²	1	2	yes	complemented
72	C₃₁×#475	C₂²	1	4	yes	complemented
73	C₃₁×#474	C₂²	1	2	yes	—
74	C₃₁×#463	C₂²	1	4	yes	—
75	C₃₁×#463	C₂²	1	2	yes	—
76	C₃₁×#461	C₂²	1	4	yes	—
77	C₃₁×#461	C₂²	1	2	yes	—
78	C₃₁×#450	C₂²	1	1	yes	characteristic
79	C₃₁×#450	C₂²	1	2	yes	—
80	C₃₁×#447	C₂²	1	1	yes	characteristic
81	C₃₁×#446	C₂²	1	2	yes	complemented
82	C₃₁×#446	C₂²	1	4	yes	complemented
83	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
84	C₃₁×#442	C₂²	1	4	yes	—
85	C₃₁×#440	C₂²	1	4	yes	—
86	C₃₁×#440	C₂²	1	4	yes	—
87	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
88	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
89	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
90	C₄×D₈×C₃₁	C₂²	1	2	yes	—
91	C₃₁×#432	C₂²	1	1	yes	characteristic
92	C₃₁×#432	C₂²	1	2	yes	—
93	C₃₁×#431	C₂²	1	2	yes	—
94	C₃₁×#431	C₂²	1	1	yes	characteristic
95	C₂²×Q₈×C₃₁	—	2	1	no	—
96	C₂²×Q₈×C₃₁	—	2	1	no	—
97	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
98	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
99	C₂³×C₄×C₃₁	—	2	1	no	—
100	C₂³×C₄×C₃₁	—	2	2	no	—
101	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
102	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
103	C₂×D₈×C₃₁	C₂³	1	2	yes	—
104	C₃₁×C₄⋊Q₈	—	2	1	no	—
105	C₃₁×C₄⋊Q₈	—	2	2	no	—
106	C₃₁×C₄⋊Q₈	—	2	1	no	—
107	C₃₁×C₄⋊Q₈	—	2	1	no	—
108	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
109	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
110	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
111	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
112	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
113	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
114	C₃₁×#126	—	2	2	no	—
115	C₃₁×#126	—	2	2	no	—
116	C₃₁×#126	—	2	2	no	—
117	C₃₁×#126	—	2	2	no	—
118	C₃₁×#126	—	2	1	no	—
119	C₃₁×#125	—	2	2	no	—
120	C₃₁×#125	—	2	2	no	—
121	C₃₁×#125	—	2	2	no	—
122	C₃₁×#125	—	2	1	no	—
123	C₃₁×#125	—	2	2	no	—
124	C₃₁×#125	—	2	2	no	—
125	C₃₁×#125	—	2	2	no	—
126	C₃₁×#125	—	2	2	no	—
127	C₃₁×#125	—	2	1	no	—
128	C₃₁×#125	C₂³	1	2	yes	—
129	C₃₁×#125	C₂³	1	1	yes	characteristic
130	C₃₁×#125	C₂³	1	4	yes	—
131	C₃₁×#125	C₂³	1	4	yes	—
132	C₃₁×#124	C₂³	1	2	yes	—
133	C₃₁×#124	C₂³	1	1	yes	characteristic
134	C₃₁×#123	—	2	1	no	—
135	C₃₁×#123	—	2	2	no	—
136	C₃₁×C₂²⋊Q₈	—	2	2	no	—
137	C₃₁×C₂²⋊Q₈	—	2	2	no	—
138	C₃₁×C₂²⋊Q₈	—	2	4	no	—
139	C₃₁×C₂²⋊Q₈	—	2	1	no	—
140	C₃₁×C₂²⋊Q₈	—	2	2	no	—
141	C₃₁×C₂²⋊Q₈	—	2	1	no	—
142	C₃₁×C₂²⋊Q₈	—	2	2	no	—
143	C₃₁×C₂²⋊Q₈	—	2	4	no	—
144	C₃₁×C₂²⋊Q₈	—	2	4	no	—
145	C₃₁×C₂²⋊Q₈	—	2	4	no	—
146	C₃₁×C₂²⋊Q₈	—	2	4	no	—
147	C₃₁×C₂²⋊Q₈	—	2	1	no	—
148	C₃₁×C₂²⋊Q₈	—	2	2	no	—
149	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
150	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
151	C₃₁×#121	C₂³	1	1	yes	characteristic
152	C₄×Q₈×C₃₁	—	2	1	no	—
153	C₄×Q₈×C₃₁	—	2	1	no	—
154	C₄×Q₈×C₃₁	—	2	2	no	—
155	C₄×Q₈×C₃₁	—	2	2	no	—
156	C₄×Q₈×C₃₁	—	2	1	no	—
157	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
158	C₄×Q₈×C₃₁	D₄	1	2	yes	—
159	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
160	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
161	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
162	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
163	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
164	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
165	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
166	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
167	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
168	C₄×D₄×C₃₁	—	2	1	no	—
169	C₄×D₄×C₃₁	—	2	2	no	—
170	C₄×D₄×C₃₁	—	2	2	no	—
171	C₄×D₄×C₃₁	—	2	1	no	—
172	C₄×D₄×C₃₁	—	2	2	no	—
173	C₄×D₄×C₃₁	—	2	1	no	—
174	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
175	C₄×D₄×C₃₁	C₂³	1	2	yes	—
176	C₄×D₄×C₃₁	C₂³	1	4	yes	—
177	C₄×D₄×C₃₁	C₂³	1	2	yes	—
178	C₄×D₄×C₃₁	C₂³	1	4	yes	—
179	C₄×D₄×C₃₁	C₂³	1	2	yes	—
180	C₃₁×#117	—	2	1	no	—
181	C₃₁×#117	—	2	2	no	—
182	C₃₁×#117	—	2	1	no	—
183	C₃₁×#117	—	2	2	no	—
184	C₃₁×#117	—	2	2	no	—
185	C₃₁×#117	—	2	1	no	—
186	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
193	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
194	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
195	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
196	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
197	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
199	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
200	C₃₁×#107	C₂³	1	2	yes	—
201	C₃₁×#107	C₂³	1	4	yes	—
202	C₃₁×#107	C₂³	1	4	yes	—
203	C₃₁×#106	C₂³	1	4	yes	—
204	C₃₁×#106	C₂³	1	4	yes	—
205	C₃₁×#106	C₂³	1	2	yes	—
206	C₃₁×#106	C₂³	1	4	yes	—
207	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
208	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
209	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
210	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
211	C₃₁×#103	C₂³	1	4	yes	—
212	C₃₁×#103	C₂³	1	4	yes	—
213	C₃₁×#103	C₂³	1	2	yes	—
214	C₃₁×#103	C₂³	1	4	yes	—
215	C₃₁×#103	C₂³	1	4	yes	—
216	C₃₁×#102	C₂³	1	2	yes	—
217	C₃₁×#102	C₂³	1	2	yes	—
218	C₃₁×#102	C₂³	1	2	yes	—
219	C₃₁×#102	C₂³	1	4	yes	—
220	C₃₁×#102	C₂³	1	4	yes	—
221	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
222	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
223	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
224	C₃₁×#97	C₂³	1	2	yes	—
225	C₃₁×#97	C₂³	1	4	yes	—
226	C₄×C₈×C₃₁	C₂³	1	2	yes	—
227	C₄×C₈×C₃₁	C₂³	1	4	yes	—
228	C₂⁴×C₃₁	—	2	1	no	—
229	C₂×Q₈×C₃₁	—	2	2	no	—
230	C₂×Q₈×C₃₁	—	2	1	no	—
231	C₂×Q₈×C₃₁	—	2	2	no	—
232	C₂×Q₈×C₃₁	—	2	2	no	—
233	C₂×Q₈×C₃₁	—	4	2	no	—
234	C₂×Q₈×C₃₁	—	4	4	no	—
235	C₂×Q₈×C₃₁	—	8	2	no	—
236	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
237	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
238	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
239	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
240	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
241	C₂×D₄×C₃₁	—	2	4	no	—
242	C₂×D₄×C₃₁	—	2	1	no	—
243	C₂×D₄×C₃₁	—	2	1	no	—
244	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
245	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
246	C₂²×C₄×C₃₁	—	2	1	no	—
247	C₂²×C₄×C₃₁	—	2	2	no	—
248	C₂²×C₄×C₃₁	—	2	2	no	—
249	C₂²×C₄×C₃₁	—	2	2	no	—
250	C₂²×C₄×C₃₁	—	2	2	no	—
251	C₂²×C₄×C₃₁	—	2	1	no	—
252	C₂²×C₄×C₃₁	—	2	4	no	—
253	C₂²×C₄×C₃₁	—	2	4	no	—
254	C₂²×C₄×C₃₁	—	4	4	no	—
255	C₂²×C₄×C₃₁	—	8	4	no	—
256	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
257	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
258	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
259	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
260	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
261	SD₁₆×C₃₁	—	2	4	no	—
262	SD₁₆×C₃₁	—	2	8	no	—
263	D₈×C₃₁	—	2	4	no	—
264	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
265	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
266	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
267	C₄⋊C₄×C₃₁	—	2	4	no	—
268	C₄⋊C₄×C₃₁	—	2	4	no	—
269	C₄⋊C₄×C₃₁	—	2	2	no	—
270	C₄⋊C₄×C₃₁	—	2	2	no	—
271	C₄⋊C₄×C₃₁	—	2	4	no	—
272	C₄⋊C₄×C₃₁	—	2	4	no	—
273	C₄⋊C₄×C₃₁	—	2	1	no	—
274	C₄⋊C₄×C₃₁	—	2	2	no	—
275	C₄⋊C₄×C₃₁	—	2	2	no	—
276	C₄⋊C₄×C₃₁	—	2	1	no	—
277	C₄⋊C₄×C₃₁	—	2	2	no	—
278	C₄⋊C₄×C₃₁	—	2	1	no	—
279	C₄⋊C₄×C₃₁	—	2	1	no	—
280	C₄⋊C₄×C₃₁	—	2	2	no	—
281	C₄⋊C₄×C₃₁	—	2	1	no	—
282	C₄⋊C₄×C₃₁	—	2	2	no	—
283	C₄⋊C₄×C₃₁	—	2	1	no	—
284	C₄⋊C₄×C₃₁	—	2	2	no	—
285	C₄⋊C₄×C₃₁	—	2	2	no	—
286	C₄⋊C₄×C₃₁	—	2	1	no	—
287	C₄⋊C₄×C₃₁	—	2	2	no	—
288	C₄⋊C₄×C₃₁	—	2	2	no	—
289	C₄⋊C₄×C₃₁	—	2	2	no	—
290	C₄⋊C₄×C₃₁	—	2	1	no	—
291	C₄⋊C₄×C₃₁	—	2	2	no	—
292	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
293	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
294	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
295	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
296	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
297	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
298	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
299	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
300	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
301	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
302	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
303	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
304	C₂²⋊C₄×C₃₁	—	2	2	no	—
305	C₂²⋊C₄×C₃₁	—	2	1	no	—
306	C₂²⋊C₄×C₃₁	—	2	2	no	—
307	C₂²⋊C₄×C₃₁	—	2	2	no	—
308	C₂²⋊C₄×C₃₁	—	2	4	no	—
309	C₂²⋊C₄×C₃₁	—	2	4	no	—
310	C₂²⋊C₄×C₃₁	—	2	1	no	—
311	C₂²⋊C₄×C₃₁	—	2	1	no	—
312	C₂²⋊C₄×C₃₁	—	2	2	no	—
313	C₂²⋊C₄×C₃₁	—	2	2	no	—
314	C₂²⋊C₄×C₃₁	—	2	1	no	—
315	C₂²⋊C₄×C₃₁	—	2	1	no	—
316	C₂²⋊C₄×C₃₁	—	2	2	no	—
317	C₄²×C₃₁	—	2	2	no	—
318	C₄²×C₃₁	—	2	2	no	—
319	C₄²×C₃₁	—	2	2	no	—
320	C₄²×C₃₁	—	2	2	no	—
321	C₄²×C₃₁	—	2	1	no	—
322	C₄²×C₃₁	C₂×D₄	1	2	yes	—
323	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
324	C₄²×C₃₁	C₂×D₄	1	2	yes	—
325	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
326	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
327	C₄²×C₃₁	C₂⁴	1	2	yes	—
328	C₄²×C₃₁	C₂⁴	1	2	yes	—
329	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
330	#61841	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
331	C₂³×C₃₁	—	2	4	no	—
332	C₂³×C₃₁	—	2	2	no	—
333	C₂³×C₃₁	—	2	1	no	—
334	C₂³×C₃₁	—	4	2	no	—
335	C₂³×C₃₁	—	4	2	no	—
336	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
337	Q₈×C₃₁	—	2	2	no	—
338	Q₈×C₃₁	—	2	2	no	—
339	Q₈×C₃₁	—	2	2	no	—
340	Q₈×C₃₁	—	2	4	no	—
341	Q₈×C₃₁	—	4	2	no	—
342	Q₈×C₃₁	—	4	4	no	—
343	Q₈×C₃₁	—	4	2	no	—
344	Q₈×C₃₁	—	8	2	no	—
345	Q₈×C₃₁	—	8	2	no	—
346	D₄×C₃₁	—	2	4	no	—
347	D₄×C₃₁	—	2	2	no	—
348	D₄×C₃₁	—	2	2	no	—
349	D₄×C₃₁	—	4	2	no	—
350	D₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
351	C₂×C₄×C₃₁	—	2	4	no	—
352	C₂×C₄×C₃₁	—	2	2	no	—
353	C₂×C₄×C₃₁	—	2	2	no	—
354	C₂×C₄×C₃₁	—	2	2	no	—
355	C₂×C₄×C₃₁	—	2	2	no	—
356	C₂×C₄×C₃₁	—	2	1	no	—
357	C₂×C₄×C₃₁	—	2	1	no	—
358	C₂×C₄×C₃₁	—	2	2	no	—
359	C₂×C₄×C₃₁	—	2	1	no	—
360	C₂×C₄×C₃₁	—	2	2	no	—
361	C₂×C₄×C₃₁	—	2	4	no	—
362	C₂×C₄×C₃₁	—	4	2	no	—
363	C₂×C₄×C₃₁	—	4	4	no	—
364	C₂×C₄×C₃₁	—	4	4	no	—
365	C₂×C₄×C₃₁	—	4	2	no	—
366	C₂×C₄×C₃₁	—	4	4	no	—
367	C₂×C₄×C₃₁	—	4	4	no	—
368	C₂×C₄×C₃₁	—	4	2	no	—
369	C₂×C₄×C₃₁	—	8	4	no	—
370	C₂×C₄×C₃₁	—	8	4	no	—
371	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
372	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
373	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
374	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
375	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
376	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
377	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
378	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
379	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
380	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
381	C₈×C₃₁	—	2	2	no	—
382	C₈×C₃₁	—	2	2	no	—
383	C₈×C₃₁	—	2	4	no	—
384	#3523	C₂×C₃₁	1	1	yes	characteristic, complemented
385	C₂×#554	C₂×C₃₁	1	1	yes	characteristic, complemented
386	#3388	C₂×C₃₁	1	4	yes	complemented
387	#3384	C₂×C₃₁	1	2	yes	complemented
388	#3381	C₂×C₃₁	1	2	yes	complemented
389	#3376	C₂×C₃₁	1	4	yes	complemented
390	#3372	C₂×C₃₁	1	2	yes	complemented
391	#3360	C₂×C₃₁	1	2	yes	complemented
392	#3238	C₂×C₃₁	1	1	yes	characteristic, complemented
393	#3236	C₂×C₃₁	1	2	yes	complemented
394	#3232	C₂×C₃₁	1	2	yes	complemented
395	#3229	C₂×C₃₁	1	1	yes	characteristic, complemented
396	#3114	C₂×C₃₁	1	2	yes	—
397	#3110	C₂×C₃₁	1	1	yes	characteristic, complemented
398	#3105	C₂×C₃₁	1	1	yes	characteristic
399	#3101	C₂×C₃₁	1	2	yes	complemented
400	#2984	C₂×C₃₁	1	1	yes	characteristic, complemented
401	C₂²×C₃₁	—	2	1	no	—
402	C₂²×C₃₁	—	2	1	no	—
403	C₂²×C₃₁	—	2	4	no	—
404	C₂²×C₃₁	—	4	8	no	—
405	C₂²×C₃₁	—	4	2	no	—
406	C₂²×C₃₁	—	4	2	no	—
407	C₂²×C₃₁	—	4	1	no	—
408	C₂²×C₃₁	—	4	1	no	—
409	C₂²×C₃₁	—	4	1	no	—
410	C₂²×C₃₁	#577	1	2	yes	—
411	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
412	C₄×C₃₁	—	2	1	no	—
413	C₄×C₃₁	—	2	1	no	—
414	C₄×C₃₁	—	2	2	no	—
415	C₄×C₃₁	—	2	2	no	—
416	C₄×C₃₁	—	2	2	no	—
417	C₄×C₃₁	—	2	2	no	—
418	C₄×C₃₁	—	2	1	no	—
419	C₄×C₃₁	—	2	1	no	—
420	C₄×C₃₁	—	4	2	no	—
421	C₄×C₃₁	—	4	1	no	—
422	C₄×C₃₁	—	4	2	no	—
423	C₄×C₃₁	—	4	2	no	—
424	C₄×C₃₁	—	4	1	no	—
425	C₄×C₃₁	—	4	4	no	—
426	C₄×C₃₁	—	4	1	no	—
427	C₄×C₃₁	#575	1	2	yes	—
428	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
429	#557	C₂²×C₃₁	1	1	yes	characteristic, complemented
430	#557	C₂²×C₃₁	1	1	yes	characteristic, complemented
431	#557	C₂²×C₃₁	1	2	yes	complemented
432	#556	C₂²×C₃₁	1	1	yes	characteristic, complemented
433	#556	C₂²×C₃₁	1	2	yes	complemented
434	#554	—	2	1	no	—
435	#554	—	2	1	no	—
436	#554	—	2	2	no	—
437	#554	C₂²×C₃₁	1	2	yes	—
438	#554	C₂²×C₃₁	1	2	yes	—
439	#554	C₂²×C₃₁	1	4	yes	—
440	#552	—	2	1	no	—
441	#552	—	2	2	no	—
442	D₄×Q₈	—	2	1	no	—
443	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
444	#541	—	2	2	no	—
445	#541	—	2	1	no	—
446	#533	C₂²×C₃₁	1	2	yes	—
447	#533	C₂²×C₃₁	1	1	yes	characteristic
448	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
449	#529	C₂²×C₃₁	1	1	yes	characteristic
450	#529	C₂²×C₃₁	1	2	yes	—
451	C₂×#125	C₂²×C₃₁	1	2	yes	—
452	C₂×C₂²⋊Q₈	—	2	1	no	—
453	C₂×C₂²⋊Q₈	—	2	2	no	—
454	#519	—	2	1	no	—
455	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
456	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
457	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
458	C₂²×C₄⋊C₄	—	2	1	no	—
459	#501	C₂²×C₃₁	1	4	yes	complemented
460	#501	C₂²×C₃₁	1	2	yes	complemented
461	#499	C₂²×C₃₁	1	4	yes	complemented
462	#498	C₂²×C₃₁	1	2	yes	complemented
463	#491	C₂²×C₃₁	1	4	yes	complemented
464	#489	C₂²×C₃₁	1	2	yes	complemented
465	#488	C₂²×C₃₁	1	4	yes	complemented
466	#487	C₂²×C₃₁	1	2	yes	complemented
467	#484	C₂²×C₃₁	1	1	yes	characteristic
468	#484	C₂²×C₃₁	1	2	yes	—
469	#483	C₂²×C₃₁	1	2	yes	—
470	#483	C₂²×C₃₁	1	2	yes	—
471	#483	C₂²×C₃₁	1	2	yes	—
472	#481	C₂²×C₃₁	1	1	yes	characteristic
473	#480	C₂²×C₃₁	1	2	yes	—
474	#479	C₂²×C₃₁	1	4	yes	complemented
475	#479	C₂²×C₃₁	1	4	yes	complemented
476	#479	C₂²×C₃₁	1	4	yes	complemented
477	#478	C₂²×C₃₁	1	2	yes	—
478	#478	C₂²×C₃₁	1	4	yes	—
479	#478	C₂²×C₃₁	1	4	yes	complemented
480	#477	C₂²×C₃₁	1	2	yes	complemented
481	#476	C₂²×C₃₁	1	4	yes	—
482	#476	C₂²×C₃₁	1	2	yes	complemented
483	#475	C₂²×C₃₁	1	4	yes	complemented
484	#474	C₂²×C₃₁	1	2	yes	—
485	#463	C₂²×C₃₁	1	4	yes	—
486	#463	C₂²×C₃₁	1	2	yes	—
487	#461	C₂²×C₃₁	1	4	yes	—
488	#461	C₂²×C₃₁	1	2	yes	—
489	#450	C₂²×C₃₁	1	1	yes	characteristic
490	#450	C₂²×C₃₁	1	2	yes	—
491	#447	C₂²×C₃₁	1	1	yes	characteristic
492	#446	C₂²×C₃₁	1	2	yes	complemented
493	#446	C₂²×C₃₁	1	4	yes	complemented
494	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
495	#442	C₂²×C₃₁	1	4	yes	—
496	#440	C₂²×C₃₁	1	4	yes	—
497	#440	C₂²×C₃₁	1	4	yes	—
498	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
499	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
500	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
501	C₄×D₈	C₂²×C₃₁	1	2	yes	—
502	#432	C₂²×C₃₁	1	1	yes	characteristic
503	#432	C₂²×C₃₁	1	2	yes	—
504	#431	C₂²×C₃₁	1	2	yes	—
505	#431	C₂²×C₃₁	1	1	yes	characteristic
506	C₂×C₃₁	—	2	2	no	—
507	C₂×C₃₁	—	4	4	no	—
508	C₂×C₃₁	—	8	1	no	—
509	C₂×C₃₁	#3486	1	1	yes	characteristic
510	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
511	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
512	C₂²×Q₈	—	2	1	no	—
513	C₂²×Q₈	—	2	1	no	—
514	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
515	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
516	C₂³×C₄	—	2	1	no	—
517	C₂³×C₄	—	2	2	no	—
518	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
519	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
520	C₂×D₈	C₂³×C₃₁	1	2	yes	—
521	C₄⋊Q₈	—	2	1	no	—
522	C₄⋊Q₈	—	2	2	no	—
523	C₄⋊Q₈	—	2	1	no	—
524	C₄⋊Q₈	—	2	1	no	—
525	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
526	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
527	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
528	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
529	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
530	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
531	#126	—	2	2	no	—
532	#126	—	2	2	no	—
533	#126	—	2	2	no	—
534	#126	—	2	2	no	—
535	#126	—	2	1	no	—
536	#125	—	2	2	no	—
537	#125	—	2	2	no	—
538	#125	—	2	2	no	—
539	#125	—	2	1	no	—
540	#125	—	2	2	no	—
541	#125	—	2	2	no	—
542	#125	—	2	2	no	—
543	#125	—	2	2	no	—
544	#125	—	2	1	no	—
545	#125	C₂³×C₃₁	1	4	yes	—
546	#125	C₂³×C₃₁	1	4	yes	—
547	#125	C₂³×C₃₁	1	2	yes	—
548	#125	C₂³×C₃₁	1	1	yes	characteristic
549	#124	C₂³×C₃₁	1	2	yes	—
550	#124	C₂³×C₃₁	1	1	yes	characteristic
551	#123	—	2	1	no	—
552	#123	—	2	2	no	—
553	C₂²⋊Q₈	—	2	2	no	—
554	C₂²⋊Q₈	—	2	2	no	—
555	C₂²⋊Q₈	—	2	4	no	—
556	C₂²⋊Q₈	—	2	1	no	—
557	C₂²⋊Q₈	—	2	2	no	—
558	C₂²⋊Q₈	—	2	1	no	—
559	C₂²⋊Q₈	—	2	2	no	—
560	C₂²⋊Q₈	—	2	4	no	—
561	C₂²⋊Q₈	—	2	4	no	—
562	C₂²⋊Q₈	—	2	4	no	—
563	C₂²⋊Q₈	—	2	4	no	—
564	C₂²⋊Q₈	—	2	1	no	—
565	C₂²⋊Q₈	—	2	2	no	—
566	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
567	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
568	#121	C₂³×C₃₁	1	1	yes	characteristic
569	C₄×Q₈	—	2	1	no	—
570	C₄×Q₈	—	2	1	no	—
571	C₄×Q₈	—	2	2	no	—
572	C₄×Q₈	—	2	2	no	—
573	C₄×Q₈	—	2	1	no	—
574	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
575	C₄×Q₈	D₄×C₃₁	1	2	yes	—
576	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
577	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
578	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
579	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
580	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
581	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
582	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
583	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
584	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
585	C₄×D₄	—	2	1	no	—
586	C₄×D₄	—	2	2	no	—
587	C₄×D₄	—	2	2	no	—
588	C₄×D₄	—	2	1	no	—
589	C₄×D₄	—	2	2	no	—
590	C₄×D₄	—	2	1	no	—
591	C₄×D₄	C₂³×C₃₁	1	2	yes	—
592	C₄×D₄	C₂³×C₃₁	1	4	yes	—
593	C₄×D₄	C₂³×C₃₁	1	2	yes	—
594	C₄×D₄	C₂³×C₃₁	1	4	yes	—
595	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
596	C₄×D₄	C₂³×C₃₁	1	2	yes	—
597	#117	—	2	1	no	—
598	#117	—	2	2	no	—
599	#117	—	2	1	no	—
600	#117	—	2	2	no	—
601	#117	—	2	2	no	—
602	#117	—	2	1	no	—
603	C₂×C₄⋊C₄	—	2	2	no	—
604	C₂×C₄⋊C₄	—	2	1	no	—
605	C₂×C₄⋊C₄	—	2	2	no	—
606	C₂×C₄⋊C₄	—	2	2	no	—
607	C₂×C₄⋊C₄	—	2	2	no	—
608	C₂×C₄⋊C₄	—	2	4	no	—
609	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
610	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
611	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
612	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
613	C₂×C₂²⋊C₄	—	2	1	no	—
614	C₂×C₂²⋊C₄	—	2	2	no	—
615	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
616	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
617	#107	C₂³×C₃₁	1	2	yes	—
618	#107	C₂³×C₃₁	1	4	yes	—
619	#107	C₂³×C₃₁	1	4	yes	—
620	#106	C₂³×C₃₁	1	4	yes	—
621	#106	C₂³×C₃₁	1	4	yes	—
622	#106	C₂³×C₃₁	1	2	yes	—
623	#106	C₂³×C₃₁	1	4	yes	—
624	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
625	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
626	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
627	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
628	#103	C₂³×C₃₁	1	4	yes	—
629	#103	C₂³×C₃₁	1	4	yes	—
630	#103	C₂³×C₃₁	1	2	yes	—
631	#103	C₂³×C₃₁	1	4	yes	—
632	#103	C₂³×C₃₁	1	4	yes	—
633	#102	C₂³×C₃₁	1	2	yes	—
634	#102	C₂³×C₃₁	1	2	yes	—
635	#102	C₂³×C₃₁	1	2	yes	—
636	#102	C₂³×C₃₁	1	4	yes	—
637	#102	C₂³×C₃₁	1	4	yes	—
638	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
639	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
640	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
641	#97	C₂³×C₃₁	1	2	yes	—
642	#97	C₂³×C₃₁	1	4	yes	—
643	C₄×C₈	C₂³×C₃₁	1	2	yes	—
644	C₄×C₈	C₂³×C₃₁	1	4	yes	—
645	C₃₁	#61841	1	1	yes	characteristic, direct product factor, Sylow
646	C₂⁴	—	2	1	no	—
647	C₂×Q₈	—	2	2	no	—
648	C₂×Q₈	—	2	1	no	—
649	C₂×Q₈	—	2	2	no	—
650	C₂×Q₈	—	2	2	no	—
651	C₂×Q₈	—	4	2	no	—
652	C₂×Q₈	—	4	4	no	—
653	C₂×Q₈	—	8	2	no	—
654	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
655	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
656	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
657	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
658	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
659	C₂×D₄	—	2	4	no	—
660	C₂×D₄	—	2	1	no	—
661	C₂×D₄	—	2	1	no	—
662	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
663	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
664	C₂²×C₄	—	2	1	no	—
665	C₂²×C₄	—	2	2	no	—
666	C₂²×C₄	—	2	2	no	—
667	C₂²×C₄	—	2	2	no	—
668	C₂²×C₄	—	2	2	no	—
669	C₂²×C₄	—	2	1	no	—
670	C₂²×C₄	—	2	4	no	—
671	C₂²×C₄	—	2	4	no	—
672	C₂²×C₄	—	4	4	no	—
673	C₂²×C₄	—	8	4	no	—
674	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
675	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
676	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
677	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
678	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
679	SD₁₆	—	2	4	no	—
680	SD₁₆	—	2	8	no	—
681	D₈	—	2	4	no	—
682	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
683	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
684	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
685	C₄⋊C₄	—	2	4	no	—
686	C₄⋊C₄	—	2	4	no	—
687	C₄⋊C₄	—	2	2	no	—
688	C₄⋊C₄	—	2	2	no	—
689	C₄⋊C₄	—	2	4	no	—
690	C₄⋊C₄	—	2	4	no	—
691	C₄⋊C₄	—	2	1	no	—
692	C₄⋊C₄	—	2	2	no	—
693	C₄⋊C₄	—	2	2	no	—
694	C₄⋊C₄	—	2	1	no	—
695	C₄⋊C₄	—	2	2	no	—
696	C₄⋊C₄	—	2	1	no	—
697	C₄⋊C₄	—	2	1	no	—
698	C₄⋊C₄	—	2	2	no	—
699	C₄⋊C₄	—	2	1	no	—
700	C₄⋊C₄	—	2	2	no	—
701	C₄⋊C₄	—	2	1	no	—
702	C₄⋊C₄	—	2	2	no	—
703	C₄⋊C₄	—	2	2	no	—
704	C₄⋊C₄	—	2	1	no	—
705	C₄⋊C₄	—	2	2	no	—
706	C₄⋊C₄	—	2	2	no	—
707	C₄⋊C₄	—	2	2	no	—
708	C₄⋊C₄	—	2	1	no	—
709	C₄⋊C₄	—	2	2	no	—
710	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
711	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
712	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
713	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
714	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
715	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
716	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
717	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
718	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
719	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
720	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
721	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
722	C₂²⋊C₄	—	2	2	no	—
723	C₂²⋊C₄	—	2	1	no	—
724	C₂²⋊C₄	—	2	2	no	—
725	C₂²⋊C₄	—	2	2	no	—
726	C₂²⋊C₄	—	2	4	no	—
727	C₂²⋊C₄	—	2	4	no	—
728	C₂²⋊C₄	—	2	1	no	—
729	C₂²⋊C₄	—	2	1	no	—
730	C₂²⋊C₄	—	2	2	no	—
731	C₂²⋊C₄	—	2	2	no	—
732	C₂²⋊C₄	—	2	1	no	—
733	C₂²⋊C₄	—	2	1	no	—
734	C₂²⋊C₄	—	2	2	no	—
735	C₄²	—	2	2	no	—
736	C₄²	—	2	2	no	—
737	C₄²	—	2	2	no	—
738	C₄²	—	2	2	no	—
739	C₄²	—	2	1	no	—
740	C₄²	C₂×D₄×C₃₁	1	2	yes	—
741	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
742	C₄²	C₂×D₄×C₃₁	1	2	yes	—
743	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
744	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
745	C₄²	C₂⁴×C₃₁	1	2	yes	—
746	C₄²	C₂⁴×C₃₁	1	2	yes	—
747	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
748	C₂³	—	2	4	no	—
749	C₂³	—	2	2	no	—
750	C₂³	—	2	1	no	—
751	C₂³	—	4	2	no	—
752	C₂³	—	4	2	no	—
753	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
754	Q₈	—	2	2	no	—
755	Q₈	—	2	2	no	—
756	Q₈	—	2	2	no	—
757	Q₈	—	2	4	no	—
758	Q₈	—	4	2	no	—
759	Q₈	—	4	4	no	—
760	Q₈	—	4	2	no	—
761	Q₈	—	8	2	no	—
762	Q₈	—	8	2	no	—
763	D₄	—	2	4	no	—
764	D₄	—	2	2	no	—
765	D₄	—	2	2	no	—
766	D₄	—	4	2	no	—
767	D₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
768	C₂×C₄	—	2	4	no	—
769	C₂×C₄	—	2	2	no	—
770	C₂×C₄	—	2	2	no	—
771	C₂×C₄	—	2	2	no	—
772	C₂×C₄	—	2	2	no	—
773	C₂×C₄	—	2	1	no	—
774	C₂×C₄	—	2	1	no	—
775	C₂×C₄	—	2	2	no	—
776	C₂×C₄	—	2	1	no	—
777	C₂×C₄	—	2	2	no	—
778	C₂×C₄	—	2	4	no	—
779	C₂×C₄	—	4	2	no	—
780	C₂×C₄	—	4	4	no	—
781	C₂×C₄	—	4	4	no	—
782	C₂×C₄	—	4	2	no	—
783	C₂×C₄	—	4	4	no	—
784	C₂×C₄	—	4	4	no	—
785	C₂×C₄	—	4	2	no	—
786	C₂×C₄	—	8	4	no	—
787	C₂×C₄	—	8	4	no	—
788	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
789	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
790	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
791	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
792	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
793	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
794	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
795	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
796	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
797	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
798	C₈	—	2	2	no	—
799	C₈	—	2	2	no	—
800	C₈	—	2	4	no	—
801	C₂²	—	2	1	no	—
802	C₂²	—	2	1	no	—
803	C₂²	—	2	4	no	—
804	C₂²	—	4	8	no	—
805	C₂²	—	4	2	no	—
806	C₂²	—	4	2	no	—
807	C₂²	—	4	1	no	—
808	C₂²	—	4	1	no	—
809	C₂²	—	4	1	no	—
810	C₂²	C₃₁×#577	1	2	yes	—
811	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
812	C₄	—	2	1	no	—
813	C₄	—	2	1	no	—
814	C₄	—	2	2	no	—
815	C₄	—	2	2	no	—
816	C₄	—	2	2	no	—
817	C₄	—	2	2	no	—
818	C₄	—	2	1	no	—
819	C₄	—	2	1	no	—
820	C₄	—	4	2	no	—
821	C₄	—	4	1	no	—
822	C₄	—	4	2	no	—
823	C₄	—	4	2	no	—
824	C₄	—	4	1	no	—
825	C₄	—	4	4	no	—
826	C₄	—	4	1	no	—
827	C₄	C₃₁×#575	1	2	yes	—
828	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
829	C₂	—	2	2	no	—
830	C₂	—	4	4	no	—
831	C₂	—	8	1	no	—
832	C₂	C₃₁×#3486	1	1	yes	characteristic
833	C₂	C₂×C₃₁×#575	1	1	yes	characteristic
834	C₂	C₂×C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#447	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3523	C₂	semidirect
2. C₂×C₃₁×#554	C₂	semidirect
3. C₃₁×#3388	C₂	semidirect
4. C₃₁×#3384	C₂	semidirect
5. C₃₁×#3381	C₂	semidirect
6. C₃₁×#3376	C₂	semidirect
7. C₃₁×#3372	C₂	semidirect
8. C₃₁×#3360	C₂	semidirect
9. C₃₁×#3238	C₂	semidirect
10. C₃₁×#3236	C₂	semidirect
11. C₃₁×#3232	C₂	semidirect
12. C₃₁×#3229	C₂	semidirect
14. C₃₁×#3110	C₂	semidirect
16. C₃₁×#3101	C₂	semidirect
17. C₃₁×#2984	C₂	semidirect
18. C₃₁×#557	C₂²	semidirect
19. C₃₁×#557	C₂²	semidirect
20. C₃₁×#557	C₂²	semidirect
21. C₃₁×#556	C₂²	semidirect
22. C₃₁×#556	C₂²	semidirect
32. D₄×Q₈×C₃₁	C₂²	semidirect
48. C₃₁×#501	C₂²	semidirect
49. C₃₁×#501	C₂²	semidirect
50. C₃₁×#499	C₂²	semidirect
51. C₃₁×#498	C₂²	semidirect
52. C₃₁×#491	C₂²	semidirect
53. C₃₁×#489	C₂²	semidirect
54. C₃₁×#488	C₂²	semidirect
55. C₃₁×#487	C₂²	semidirect
63. C₃₁×#479	C₂²	semidirect
64. C₃₁×#479	C₂²	semidirect
65. C₃₁×#479	C₂²	semidirect
68. C₃₁×#478	C₂²	semidirect
69. C₃₁×#477	C₂²	semidirect
71. C₃₁×#476	C₂²	semidirect
72. C₃₁×#475	C₂²	semidirect
81. C₃₁×#446	C₂²	semidirect
82. C₃₁×#446	C₂²	semidirect
83. C₈×Q₈×C₃₁	C₂²	semidirect
330. #61841	C₃₁	direct (N × H)
384. #3523	C₂×C₃₁	semidirect
385. C₂×#554	C₂×C₃₁	semidirect
386. #3388	C₂×C₃₁	semidirect
387. #3384	C₂×C₃₁	semidirect
388. #3381	C₂×C₃₁	semidirect
389. #3376	C₂×C₃₁	semidirect
390. #3372	C₂×C₃₁	semidirect
391. #3360	C₂×C₃₁	semidirect
392. #3238	C₂×C₃₁	semidirect
393. #3236	C₂×C₃₁	semidirect
394. #3232	C₂×C₃₁	semidirect
395. #3229	C₂×C₃₁	semidirect
397. #3110	C₂×C₃₁	semidirect
399. #3101	C₂×C₃₁	semidirect
400. #2984	C₂×C₃₁	semidirect
429. #557	C₂²×C₃₁	semidirect
430. #557	C₂²×C₃₁	semidirect
431. #557	C₂²×C₃₁	semidirect
432. #556	C₂²×C₃₁	semidirect
433. #556	C₂²×C₃₁	semidirect
443. D₄×Q₈	C₂²×C₃₁	semidirect
459. #501	C₂²×C₃₁	semidirect
460. #501	C₂²×C₃₁	semidirect
461. #499	C₂²×C₃₁	semidirect
462. #498	C₂²×C₃₁	semidirect
463. #491	C₂²×C₃₁	semidirect
464. #489	C₂²×C₃₁	semidirect
465. #488	C₂²×C₃₁	semidirect
466. #487	C₂²×C₃₁	semidirect
474. #479	C₂²×C₃₁	semidirect
475. #479	C₂²×C₃₁	semidirect
476. #479	C₂²×C₃₁	semidirect
479. #478	C₂²×C₃₁	semidirect
480. #477	C₂²×C₃₁	semidirect
482. #476	C₂²×C₃₁	semidirect
483. #475	C₂²×C₃₁	semidirect
492. #446	C₂²×C₃₁	semidirect
493. #446	C₂²×C₃₁	semidirect
494. C₈×Q₈	C₂²×C₃₁	semidirect
645. C₃₁	#61841	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×#3523	2
C₂×C₃₁×#554	2
C₃₁×#3388	2
C₃₁×#3384	2
C₃₁×#3381	2
C₃₁×#3376	2
C₃₁×#3372	2
C₃₁×#3360	2
C₃₁×#3238	2
C₃₁×#3236	2
C₃₁×#3232	2
C₃₁×#3229	2
C₃₁×#3114	2
C₃₁×#3110	2
C₃₁×#3105	2
C₃₁×#3101	2
C₃₁×#2984	2
#61841	31

Subgroups — C31×#61841 #21518600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61841 #21518600