Subgroups — C₃₁×#61252 #21518011

Order 7936.

2088 conjugacy classes of subgroups.

The subgroup lattice has 918 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3475	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3354	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3353	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3345	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3343	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3335	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3334	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3094	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3092	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#3077	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3055	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
16	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×C₃₁×#428	C₂	1	2	yes	maximal, complemented
18	C₂²×#41×C₃₁	—	2	1	no	—
19	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#556	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#555	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#554	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#553	C₂²	1	2	yes	complemented
25	C₃₁×#552	—	2	1	no	—
26	C₃₁×#552	C₂²	1	2	yes	complemented
27	C₃₁×#551	—	2	2	no	—
28	C₃₁×#547	—	2	1	no	—
29	C₃₁×#547	—	2	2	no	—
30	C₃₁×#547	—	2	1	no	—
31	C₃₁×#547	C₂²	1	2	yes	—
32	C₃₁×#547	C₂²	1	4	yes	—
33	C₃₁×#547	C₂²	1	2	yes	—
34	C₃₁×#547	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#546	—	2	1	no	—
36	C₃₁×#533	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
38	C₃₁×#529	—	2	2	no	—
39	C₃₁×#529	—	2	1	no	—
40	C₃₁×#525	C₂²	1	2	yes	—
41	C₂×C₃₁×#123	—	2	2	no	—
42	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
43	C₂×C₃₁×#121	C₂²	1	2	yes	—
44	C₄×#41×C₃₁	—	2	1	no	—
45	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
46	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
47	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#117	C₂²	1	2	yes	—
49	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
50	C₃₁×#501	C₂²	1	4	yes	complemented
51	C₃₁×#500	C₂²	1	2	yes	complemented
52	C₃₁×#498	C₂²	1	2	yes	complemented
53	C₃₁×#484	C₂²	1	4	yes	complemented
54	C₃₁×#483	C₂²	1	4	yes	complemented
55	C₃₁×#482	C₂²	1	4	yes	complemented
56	C₃₁×#482	C₂²	1	4	yes	complemented
57	C₃₁×#481	C₂²	1	4	yes	complemented
58	C₃₁×#481	C₂²	1	4	yes	complemented
59	C₃₁×#480	C₂²	1	4	yes	complemented
60	C₃₁×#480	C₂²	1	4	yes	complemented
61	C₃₁×#478	C₂²	1	4	yes	complemented
62	C₃₁×#477	C₂²	1	2	yes	complemented
63	C₃₁×#476	C₂²	1	2	yes	complemented
64	C₃₁×#476	C₂²	1	2	yes	complemented
65	C₃₁×#475	C₂²	1	2	yes	complemented
66	C₃₁×#474	C₂²	1	2	yes	complemented
67	C₃₁×#474	C₂²	1	2	yes	complemented
68	C₃₁×#469	C₂²	1	4	yes	complemented
69	C₃₁×#468	C₂²	1	4	yes	complemented
70	C₃₁×#466	C₂²	1	4	yes	complemented
71	C₃₁×#465	C₂²	1	4	yes	complemented
72	C₃₁×#442	C₂²	1	2	yes	complemented
73	C₃₁×#440	C₂²	1	2	yes	complemented
74	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
75	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
76	C₃₁×#436	C₂²	1	2	yes	complemented
77	C₃₁×#435	C₂²	1	4	yes	complemented
78	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
79	C₃₁×#428	C₂²	1	2	yes	—
80	C₃₁×#428	C₂²	1	2	yes	—
81	C₃₁×#428	C₂²	1	2	yes	—
82	C₃₁×#428	C₂²	1	2	yes	—
83	C₃₁×#428	C₂²	1	1	yes	characteristic
84	C₃₁×#428	C₂²	1	4	yes	complemented
85	C₃₁×#428	C₂²	1	4	yes	complemented
86	C₂×C₃₁×#107	C₂²	1	4	yes	complemented
87	C₂×C₃₁×#106	C₂²	1	4	yes	complemented
88	C₃₁×#423	C₂²	1	1	yes	characteristic
89	C₃₁×#418	C₂²	1	2	yes	—
90	C₃₁×#418	C₂²	1	1	yes	characteristic
91	C₃₁×#417	C₂²	1	2	yes	—
92	C₃₁×#417	C₂²	1	1	yes	characteristic
93	C₃₁×#407	C₂²	1	2	yes	—
94	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
95	C₂×#41×C₃₁	—	2	1	no	—
96	C₂×#41×C₃₁	—	2	4	no	—
97	C₂×#41×C₃₁	—	2	2	no	—
98	C₂×#41×C₃₁	—	2	2	no	—
99	C₂×#41×C₃₁	—	4	8	no	—
100	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
101	C₂²×Q₈×C₃₁	—	2	1	no	—
102	C₂²×D₄×C₃₁	—	2	2	no	—
103	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
104	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂³×C₄×C₃₁	—	2	2	no	—
106	C₂³×C₄×C₃₁	—	2	1	no	—
107	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
108	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
109	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
110	C₂×D₈×C₃₁	C₂³	1	2	yes	—
111	C₂×M₁₆×C₃₁	—	2	8	no	—
112	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
114	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
115	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
116	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
117	C₃₁×C₄⋊Q₈	—	2	1	no	—
118	C₃₁×C₄⋊Q₈	—	2	1	no	—
119	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
120	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
121	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
122	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
123	C₃₁×#126	—	2	2	no	—
124	C₃₁×#126	—	2	2	no	—
125	C₃₁×#126	—	2	2	no	—
126	C₃₁×#125	—	2	2	no	—
127	C₃₁×#125	—	2	1	no	—
128	C₃₁×#125	C₂³	1	2	yes	—
129	C₃₁×#124	—	2	1	no	—
130	C₃₁×#124	—	2	2	no	—
131	C₃₁×#123	—	2	2	no	—
132	C₃₁×#123	—	2	2	no	—
133	C₃₁×#123	—	2	4	no	—
134	C₃₁×#123	—	2	4	no	—
135	C₃₁×#123	—	2	4	no	—
136	C₃₁×#123	—	2	4	no	—
137	C₃₁×#123	—	2	2	no	—
138	C₃₁×#123	—	2	2	no	—
139	C₃₁×C₂²⋊Q₈	—	2	1	no	—
140	C₃₁×C₂²⋊Q₈	—	2	1	no	—
141	C₃₁×C₂²⋊Q₈	—	2	2	no	—
142	C₃₁×C₂²⋊Q₈	—	2	2	no	—
143	C₃₁×C₂²⋊Q₈	—	2	4	no	—
144	C₃₁×C₂²⋊Q₈	—	2	1	no	—
145	C₃₁×C₂²⋊Q₈	—	2	2	no	—
146	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
147	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
148	C₃₁×#121	—	2	2	no	—
149	C₃₁×#121	—	2	1	no	—
150	C₃₁×#121	—	2	2	no	—
151	C₃₁×#121	—	2	2	no	—
152	C₃₁×#121	C₂³	1	4	yes	—
153	C₃₁×#121	C₂³	1	2	yes	—
154	C₃₁×#121	C₂³	1	4	yes	—
155	C₃₁×#121	C₂³	1	2	yes	—
156	C₃₁×#120	—	2	2	no	—
157	C₄×Q₈×C₃₁	—	2	1	no	—
158	C₄×Q₈×C₃₁	—	2	1	no	—
159	C₄×Q₈×C₃₁	—	2	1	no	—
160	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
161	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
162	C₄×D₄×C₃₁	—	2	1	no	—
163	C₄×D₄×C₃₁	—	2	2	no	—
164	C₄×D₄×C₃₁	—	2	1	no	—
165	C₄×D₄×C₃₁	—	2	1	no	—
166	C₄×D₄×C₃₁	—	2	2	no	—
167	C₄×D₄×C₃₁	—	2	2	no	—
168	C₄×D₄×C₃₁	—	2	1	no	—
169	C₄×D₄×C₃₁	—	2	1	no	—
170	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
171	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
172	C₄×D₄×C₃₁	D₄	1	2	yes	—
173	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
174	C₄×D₄×C₃₁	C₂³	1	2	yes	—
175	C₄×D₄×C₃₁	C₂³	1	2	yes	—
176	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
177	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
178	C₄×D₄×C₃₁	C₂³	1	2	yes	—
179	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
180	C₃₁×#117	—	2	2	no	—
181	C₃₁×#117	—	2	2	no	—
182	C₃₁×#117	—	2	1	no	—
183	C₃₁×#117	—	2	2	no	—
184	C₃₁×#117	—	2	1	no	—
185	C₃₁×#117	C₂³	1	1	yes	characteristic
186	C₃₁×#117	C₂³	1	2	yes	—
187	C₃₁×#117	C₂³	1	2	yes	—
188	C₃₁×#117	C₂³	1	2	yes	—
189	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
194	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
195	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
196	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
197	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
198	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
199	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
200	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
201	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
202	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
204	C₂×C₄²×C₃₁	—	2	2	no	—
205	C₂×C₄²×C₃₁	—	2	1	no	—
206	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
207	C₃₁×#107	C₂³	1	2	yes	—
208	C₃₁×#107	C₂³	1	4	yes	—
209	C₃₁×#107	C₂³	1	4	yes	—
210	C₃₁×#107	C₂³	1	4	yes	—
211	C₃₁×#107	C₂³	1	4	yes	—
212	C₃₁×#106	C₂³	1	2	yes	—
213	C₃₁×#106	C₂³	1	4	yes	—
214	C₃₁×#106	C₂³	1	4	yes	—
215	C₃₁×#106	C₂³	1	4	yes	—
216	C₃₁×#106	C₂³	1	4	yes	—
217	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
218	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
219	C₃₁×#103	C₂³	1	2	yes	—
220	C₃₁×#103	C₂³	1	4	yes	—
221	C₃₁×#102	C₂³	1	2	yes	—
222	C₃₁×#102	C₂³	1	4	yes	—
223	C₃₁×#102	C₂³	1	2	yes	—
224	C₃₁×#102	C₂³	1	2	yes	—
225	C₃₁×#102	C₂³	1	4	yes	—
226	C₃₁×#102	C₂³	1	4	yes	—
227	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
228	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
229	C₃₁×#97	C₂³	1	2	yes	—
230	C₄×C₈×C₃₁	C₂³	1	2	yes	—
231	C₂⁴×C₃₁	—	2	1	no	—
232	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
233	#41×C₃₁	—	2	2	no	—
234	#41×C₃₁	—	2	2	no	—
235	#41×C₃₁	—	4	16	no	—
236	#41×C₃₁	—	4	8	no	—
237	#41×C₃₁	—	4	4	no	—
238	#41×C₃₁	—	4	8	no	—
239	C₂×Q₈×C₃₁	—	2	1	no	—
240	C₂×Q₈×C₃₁	—	2	2	no	—
241	C₂×Q₈×C₃₁	—	2	2	no	—
242	C₂×Q₈×C₃₁	—	2	1	no	—
243	C₂×Q₈×C₃₁	—	4	4	no	—
244	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
245	C₂×D₄×C₃₁	—	2	4	no	—
246	C₂×D₄×C₃₁	—	2	1	no	—
247	C₂×D₄×C₃₁	—	2	2	no	—
248	C₂×D₄×C₃₁	—	2	4	no	—
249	C₂×D₄×C₃₁	—	2	4	no	—
250	C₂×D₄×C₃₁	—	2	2	no	—
251	C₂×D₄×C₃₁	—	2	1	no	—
252	C₂×D₄×C₃₁	—	2	2	no	—
253	C₂×D₄×C₃₁	—	2	2	no	—
254	C₂×D₄×C₃₁	—	4	8	no	—
255	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
256	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
257	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
258	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
259	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
260	C₂²×C₄×C₃₁	—	2	4	no	—
261	C₂²×C₄×C₃₁	—	2	4	no	—
262	C₂²×C₄×C₃₁	—	2	2	no	—
263	C₂²×C₄×C₃₁	—	2	1	no	—
264	C₂²×C₄×C₃₁	—	2	2	no	—
265	C₂²×C₄×C₃₁	—	2	4	no	—
266	C₂²×C₄×C₃₁	—	2	4	no	—
267	C₂²×C₄×C₃₁	—	2	2	no	—
268	C₂²×C₄×C₃₁	—	2	1	no	—
269	C₂²×C₄×C₃₁	—	2	1	no	—
270	C₂²×C₄×C₃₁	—	2	2	no	—
271	C₂²×C₄×C₃₁	—	2	2	no	—
272	C₂²×C₄×C₃₁	—	2	1	no	—
273	C₂²×C₄×C₃₁	—	4	4	no	—
274	C₂²×C₄×C₃₁	—	4	8	no	—
275	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
276	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
277	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
278	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
279	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
280	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
281	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
282	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
283	SD₁₆×C₃₁	—	2	4	no	—
284	D₈×C₃₁	—	2	4	no	—
285	M₁₆×C₃₁	—	2	4	no	—
286	M₁₆×C₃₁	—	2	2	no	—
287	M₁₆×C₃₁	—	2	8	no	—
288	M₁₆×C₃₁	#41	1	4	yes	complemented
289	C₂×C₈×C₃₁	—	2	8	no	—
290	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
291	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
292	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
293	C₄⋊C₄×C₃₁	—	2	1	no	—
294	C₄⋊C₄×C₃₁	—	2	2	no	—
295	C₄⋊C₄×C₃₁	—	2	4	no	—
296	C₄⋊C₄×C₃₁	—	2	4	no	—
297	C₄⋊C₄×C₃₁	—	2	2	no	—
298	C₄⋊C₄×C₃₁	—	2	2	no	—
299	C₄⋊C₄×C₃₁	—	2	2	no	—
300	C₄⋊C₄×C₃₁	—	2	2	no	—
301	C₄⋊C₄×C₃₁	—	2	1	no	—
302	C₄⋊C₄×C₃₁	—	2	2	no	—
303	C₄⋊C₄×C₃₁	—	2	2	no	—
304	C₄⋊C₄×C₃₁	—	2	1	no	—
305	C₄⋊C₄×C₃₁	—	2	1	no	—
306	C₄⋊C₄×C₃₁	—	2	2	no	—
307	C₄⋊C₄×C₃₁	—	2	2	no	—
308	C₄⋊C₄×C₃₁	—	2	1	no	—
309	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
313	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
314	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
315	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
316	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
317	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
318	C₂²⋊C₄×C₃₁	—	2	1	no	—
319	C₂²⋊C₄×C₃₁	—	2	2	no	—
320	C₂²⋊C₄×C₃₁	—	2	2	no	—
321	C₂²⋊C₄×C₃₁	—	2	2	no	—
322	C₂²⋊C₄×C₃₁	—	2	2	no	—
323	C₂²⋊C₄×C₃₁	—	2	2	no	—
324	C₂²⋊C₄×C₃₁	—	2	2	no	—
325	C₂²⋊C₄×C₃₁	—	2	2	no	—
326	C₂²⋊C₄×C₃₁	—	2	4	no	—
327	C₂²⋊C₄×C₃₁	—	2	4	no	—
328	C₂²⋊C₄×C₃₁	—	2	2	no	—
329	C₂²⋊C₄×C₃₁	—	2	1	no	—
330	C₂²⋊C₄×C₃₁	—	2	1	no	—
331	C₂²⋊C₄×C₃₁	—	2	2	no	—
332	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
333	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
334	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
335	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
336	C₄²×C₃₁	—	2	1	no	—
337	C₄²×C₃₁	—	2	2	no	—
338	C₄²×C₃₁	—	2	1	no	—
339	C₄²×C₃₁	—	2	2	no	—
340	C₄²×C₃₁	—	2	2	no	—
341	C₄²×C₃₁	—	2	1	no	—
342	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
343	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
344	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
345	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
346	#61252	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
347	C₂³×C₃₁	—	2	4	no	—
348	C₂³×C₃₁	—	2	4	no	—
349	C₂³×C₃₁	—	2	1	no	—
350	C₂³×C₃₁	—	2	2	no	—
351	C₂³×C₃₁	—	4	2	no	—
352	C₂³×C₃₁	—	4	2	no	—
353	C₂³×C₃₁	—	4	2	no	—
354	C₂³×C₃₁	—	4	2	no	—
355	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
356	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
357	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
358	Q₈×C₃₁	—	2	1	no	—
359	Q₈×C₃₁	—	2	1	no	—
360	Q₈×C₃₁	—	4	4	no	—
361	Q₈×C₃₁	—	4	4	no	—
362	Q₈×C₃₁	—	4	2	no	—
363	Q₈×C₃₁	—	8	1	no	—
364	D₄×C₃₁	—	2	4	no	—
365	D₄×C₃₁	—	2	2	no	—
366	D₄×C₃₁	—	2	1	no	—
367	D₄×C₃₁	—	2	2	no	—
368	D₄×C₃₁	—	2	2	no	—
369	D₄×C₃₁	—	2	1	no	—
370	D₄×C₃₁	—	4	8	no	—
371	D₄×C₃₁	—	4	2	no	—
372	D₄×C₃₁	—	4	8	no	—
373	D₄×C₃₁	—	4	4	no	—
374	D₄×C₃₁	—	4	2	no	—
375	D₄×C₃₁	—	8	2	no	—
376	D₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	complemented
377	C₂×C₄×C₃₁	—	2	4	no	—
378	C₂×C₄×C₃₁	—	2	4	no	—
379	C₂×C₄×C₃₁	—	2	4	no	—
380	C₂×C₄×C₃₁	—	2	2	no	—
381	C₂×C₄×C₃₁	—	2	2	no	—
382	C₂×C₄×C₃₁	—	2	2	no	—
383	C₂×C₄×C₃₁	—	2	2	no	—
384	C₂×C₄×C₃₁	—	2	2	no	—
385	C₂×C₄×C₃₁	—	2	1	no	—
386	C₂×C₄×C₃₁	—	2	1	no	—
387	C₂×C₄×C₃₁	—	2	1	no	—
388	C₂×C₄×C₃₁	—	2	2	no	—
389	C₂×C₄×C₃₁	—	2	2	no	—
390	C₂×C₄×C₃₁	—	2	2	no	—
391	C₂×C₄×C₃₁	—	2	4	no	—
392	C₂×C₄×C₃₁	—	4	1	no	—
393	C₂×C₄×C₃₁	—	4	4	no	—
394	C₂×C₄×C₃₁	—	4	2	no	—
395	C₂×C₄×C₃₁	—	4	4	no	—
396	C₂×C₄×C₃₁	—	4	8	no	—
397	C₂×C₄×C₃₁	—	4	8	no	—
398	C₂×C₄×C₃₁	—	4	4	no	—
399	C₂×C₄×C₃₁	—	4	2	no	—
400	C₂×C₄×C₃₁	—	4	4	no	—
401	C₂×C₄×C₃₁	—	4	2	no	—
402	C₂×C₄×C₃₁	—	8	1	no	—
403	C₂×C₄×C₃₁	—	8	2	no	—
404	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
405	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
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	2	yes	—
409	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
410	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
411	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
412	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
413	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
414	C₈×C₃₁	—	2	4	no	—
415	C₈×C₃₁	—	2	2	no	—
416	C₈×C₃₁	C₂×#41	1	4	yes	complemented
417	#3475	C₂×C₃₁	1	1	yes	characteristic, complemented
418	C₂×#547	C₂×C₃₁	1	1	yes	characteristic, complemented
419	#3354	C₂×C₃₁	1	2	yes	complemented
420	#3353	C₂×C₃₁	1	2	yes	complemented
421	#3345	C₂×C₃₁	1	4	yes	complemented
422	#3343	C₂×C₃₁	1	4	yes	complemented
423	#3335	C₂×C₃₁	1	2	yes	complemented
424	#3334	C₂×C₃₁	1	2	yes	complemented
425	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
426	#3094	C₂×C₃₁	1	2	yes	complemented
427	#3092	C₂×C₃₁	1	2	yes	complemented
428	#3079	C₂×C₃₁	1	1	yes	characteristic, complemented
429	#3077	C₂×C₃₁	1	1	yes	characteristic, complemented
430	#3055	C₂×C₃₁	1	2	yes	complemented
431	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic, complemented
432	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
433	C₂×#428	C₂×C₃₁	1	2	yes	complemented
434	C₂²×C₃₁	—	2	1	no	—
435	C₂²×C₃₁	—	2	2	no	—
436	C₂²×C₃₁	—	2	2	no	—
437	C₂²×C₃₁	—	2	1	no	—
438	C₂²×C₃₁	—	2	2	no	—
439	C₂²×C₃₁	—	2	2	no	—
440	C₂²×C₃₁	—	2	2	no	—
441	C₂²×C₃₁	—	4	8	no	—
442	C₂²×C₃₁	—	4	1	no	—
443	C₂²×C₃₁	—	4	2	no	—
444	C₂²×C₃₁	—	4	1	no	—
445	C₂²×C₃₁	—	4	2	no	—
446	C₂²×C₃₁	—	4	2	no	—
447	C₂²×C₃₁	—	4	1	no	—
448	C₂²×C₃₁	—	8	4	no	—
449	C₂²×C₃₁	#547	1	2	yes	—
450	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
451	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
452	C₄×C₃₁	—	2	1	no	—
453	C₄×C₃₁	—	2	1	no	—
454	C₄×C₃₁	—	2	2	no	—
455	C₄×C₃₁	—	2	2	no	—
456	C₄×C₃₁	—	2	1	no	—
457	C₄×C₃₁	—	4	1	no	—
458	C₄×C₃₁	—	4	2	no	—
459	C₄×C₃₁	—	4	1	no	—
460	C₄×C₃₁	—	4	1	no	—
461	C₄×C₃₁	—	4	2	no	—
462	C₄×C₃₁	—	4	4	no	—
463	C₄×C₃₁	—	4	2	no	—
464	C₄×C₃₁	#547	1	2	yes	—
465	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
466	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
467	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
468	C₂²×#41	—	2	1	no	—
469	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
470	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
471	#556	C₂²×C₃₁	1	1	yes	characteristic, complemented
472	#555	C₂²×C₃₁	1	1	yes	characteristic, complemented
473	#554	C₂²×C₃₁	1	1	yes	characteristic, complemented
474	#553	C₂²×C₃₁	1	2	yes	complemented
475	#552	—	2	1	no	—
476	#552	C₂²×C₃₁	1	2	yes	complemented
477	#551	—	2	2	no	—
478	#547	—	2	1	no	—
479	#547	—	2	2	no	—
480	#547	—	2	1	no	—
481	#547	C₂²×C₃₁	1	2	yes	—
482	#547	C₂²×C₃₁	1	4	yes	—
483	#547	C₂²×C₃₁	1	2	yes	—
484	#547	C₂²×C₃₁	1	1	yes	characteristic, complemented
485	#546	—	2	1	no	—
486	#533	C₂²×C₃₁	1	1	yes	characteristic
487	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
488	#529	—	2	2	no	—
489	#529	—	2	1	no	—
490	#525	C₂²×C₃₁	1	2	yes	—
491	C₂×#123	—	2	2	no	—
492	C₂×C₂²⋊Q₈	—	2	1	no	—
493	C₂×#121	C₂²×C₃₁	1	2	yes	—
494	C₄×#41	—	2	1	no	—
495	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
496	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
497	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
498	C₂×#117	C₂²×C₃₁	1	2	yes	—
499	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
500	#501	C₂²×C₃₁	1	4	yes	complemented
501	#500	C₂²×C₃₁	1	2	yes	complemented
502	#498	C₂²×C₃₁	1	2	yes	complemented
503	#484	C₂²×C₃₁	1	4	yes	complemented
504	#483	C₂²×C₃₁	1	4	yes	complemented
505	#482	C₂²×C₃₁	1	4	yes	complemented
506	#482	C₂²×C₃₁	1	4	yes	complemented
507	#481	C₂²×C₃₁	1	4	yes	complemented
508	#481	C₂²×C₃₁	1	4	yes	complemented
509	#480	C₂²×C₃₁	1	4	yes	complemented
510	#480	C₂²×C₃₁	1	4	yes	complemented
511	#478	C₂²×C₃₁	1	4	yes	complemented
512	#477	C₂²×C₃₁	1	2	yes	complemented
513	#476	C₂²×C₃₁	1	2	yes	complemented
514	#476	C₂²×C₃₁	1	2	yes	complemented
515	#475	C₂²×C₃₁	1	2	yes	complemented
516	#474	C₂²×C₃₁	1	2	yes	complemented
517	#474	C₂²×C₃₁	1	2	yes	complemented
518	#469	C₂²×C₃₁	1	4	yes	complemented
519	#468	C₂²×C₃₁	1	4	yes	complemented
520	#466	C₂²×C₃₁	1	4	yes	complemented
521	#465	C₂²×C₃₁	1	4	yes	complemented
522	#442	C₂²×C₃₁	1	2	yes	complemented
523	#440	C₂²×C₃₁	1	2	yes	complemented
524	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
525	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
526	#436	C₂²×C₃₁	1	2	yes	complemented
527	#435	C₂²×C₃₁	1	4	yes	complemented
528	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
529	#428	C₂²×C₃₁	1	2	yes	—
530	#428	C₂²×C₃₁	1	2	yes	—
531	#428	C₂²×C₃₁	1	2	yes	—
532	#428	C₂²×C₃₁	1	2	yes	—
533	#428	C₂²×C₃₁	1	1	yes	characteristic
534	#428	C₂²×C₃₁	1	4	yes	complemented
535	#428	C₂²×C₃₁	1	4	yes	complemented
536	C₂×#107	C₂²×C₃₁	1	4	yes	complemented
537	C₂×#106	C₂²×C₃₁	1	4	yes	complemented
538	#423	C₂²×C₃₁	1	1	yes	characteristic
539	#418	C₂²×C₃₁	1	2	yes	—
540	#418	C₂²×C₃₁	1	1	yes	characteristic
541	#417	C₂²×C₃₁	1	2	yes	—
542	#417	C₂²×C₃₁	1	1	yes	characteristic
543	#407	C₂²×C₃₁	1	2	yes	—
544	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
545	C₂×C₃₁	—	2	2	no	—
546	C₂×C₃₁	—	2	4	no	—
547	C₂×C₃₁	—	4	2	no	—
548	C₂×C₃₁	—	8	1	no	—
549	C₂×C₃₁	—	8	2	no	—
550	C₂×C₃₁	C₂×#547	1	1	yes	characteristic
551	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
552	C₂×C₃₁	#3586	1	1	yes	characteristic
553	C₂×#41	—	2	1	no	—
554	C₂×#41	—	2	4	no	—
555	C₂×#41	—	2	2	no	—
556	C₂×#41	—	2	2	no	—
557	C₂×#41	—	4	8	no	—
558	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
559	C₂²×Q₈	—	2	1	no	—
560	C₂²×D₄	—	2	2	no	—
561	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
562	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
563	C₂³×C₄	—	2	2	no	—
564	C₂³×C₄	—	2	1	no	—
565	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
566	AGL(1,ℤ/8)	—	2	4	no	—
567	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
568	C₂×D₈	C₂³×C₃₁	1	2	yes	—
569	C₂×M₁₆	—	2	8	no	—
570	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
571	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
572	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
573	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
574	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
575	C₄⋊Q₈	—	2	1	no	—
576	C₄⋊Q₈	—	2	1	no	—
577	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
578	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
579	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
580	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
581	#126	—	2	2	no	—
582	#126	—	2	2	no	—
583	#126	—	2	2	no	—
584	#125	—	2	2	no	—
585	#125	—	2	1	no	—
586	#125	C₂³×C₃₁	1	2	yes	—
587	#124	—	2	1	no	—
588	#124	—	2	2	no	—
589	#123	—	2	2	no	—
590	#123	—	2	2	no	—
591	#123	—	2	4	no	—
592	#123	—	2	4	no	—
593	#123	—	2	4	no	—
594	#123	—	2	4	no	—
595	#123	—	2	2	no	—
596	#123	—	2	2	no	—
597	C₂²⋊Q₈	—	2	1	no	—
598	C₂²⋊Q₈	—	2	1	no	—
599	C₂²⋊Q₈	—	2	2	no	—
600	C₂²⋊Q₈	—	2	2	no	—
601	C₂²⋊Q₈	—	2	4	no	—
602	C₂²⋊Q₈	—	2	1	no	—
603	C₂²⋊Q₈	—	2	2	no	—
604	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
605	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
606	#121	—	2	2	no	—
607	#121	—	2	1	no	—
608	#121	—	2	2	no	—
609	#121	—	2	2	no	—
610	#121	C₂³×C₃₁	1	4	yes	—
611	#121	C₂³×C₃₁	1	4	yes	—
612	#121	C₂³×C₃₁	1	2	yes	—
613	#121	C₂³×C₃₁	1	2	yes	—
614	#120	—	2	2	no	—
615	C₄×Q₈	—	2	1	no	—
616	C₄×Q₈	—	2	1	no	—
617	C₄×Q₈	—	2	1	no	—
618	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
619	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
620	C₄×D₄	—	2	1	no	—
621	C₄×D₄	—	2	2	no	—
622	C₄×D₄	—	2	1	no	—
623	C₄×D₄	—	2	1	no	—
624	C₄×D₄	—	2	2	no	—
625	C₄×D₄	—	2	2	no	—
626	C₄×D₄	—	2	1	no	—
627	C₄×D₄	—	2	1	no	—
628	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
629	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
630	C₄×D₄	D₄×C₃₁	1	2	yes	—
631	C₄×D₄	C₂³×C₃₁	1	2	yes	—
632	C₄×D₄	C₂³×C₃₁	1	2	yes	—
633	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
634	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
635	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
636	C₄×D₄	C₂³×C₃₁	1	2	yes	—
637	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
638	#117	—	2	2	no	—
639	#117	—	2	2	no	—
640	#117	—	2	1	no	—
641	#117	—	2	2	no	—
642	#117	—	2	1	no	—
643	#117	C₂³×C₃₁	1	1	yes	characteristic
644	#117	C₂³×C₃₁	1	2	yes	—
645	#117	C₂³×C₃₁	1	2	yes	—
646	#117	C₂³×C₃₁	1	2	yes	—
647	C₂×C₄⋊C₄	—	2	2	no	—
648	C₂×C₄⋊C₄	—	2	2	no	—
649	C₂×C₄⋊C₄	—	2	2	no	—
650	C₂×C₄⋊C₄	—	2	1	no	—
651	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
652	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
653	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
654	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
655	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
656	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
657	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
658	C₂×C₂²⋊C₄	—	2	2	no	—
659	C₂×C₂²⋊C₄	—	2	1	no	—
660	C₂×C₂²⋊C₄	—	2	2	no	—
661	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
662	C₂×C₄²	—	2	2	no	—
663	C₂×C₄²	—	2	1	no	—
664	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
665	#107	C₂³×C₃₁	1	2	yes	—
666	#107	C₂³×C₃₁	1	4	yes	—
667	#107	C₂³×C₃₁	1	4	yes	—
668	#107	C₂³×C₃₁	1	4	yes	—
669	#107	C₂³×C₃₁	1	4	yes	—
670	#106	C₂³×C₃₁	1	2	yes	—
671	#106	C₂³×C₃₁	1	4	yes	—
672	#106	C₂³×C₃₁	1	4	yes	—
673	#106	C₂³×C₃₁	1	4	yes	—
674	#106	C₂³×C₃₁	1	4	yes	—
675	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
676	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
677	#103	C₂³×C₃₁	1	2	yes	—
678	#103	C₂³×C₃₁	1	4	yes	—
679	#102	C₂³×C₃₁	1	2	yes	—
680	#102	C₂³×C₃₁	1	4	yes	—
681	#102	C₂³×C₃₁	1	2	yes	—
682	#102	C₂³×C₃₁	1	2	yes	—
683	#102	C₂³×C₃₁	1	4	yes	—
684	#102	C₂³×C₃₁	1	4	yes	—
685	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
686	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
687	#97	C₂³×C₃₁	1	2	yes	—
688	C₄×C₈	C₂³×C₃₁	1	2	yes	—
689	C₃₁	#61252	1	1	yes	characteristic, direct product factor, Sylow
690	C₂⁴	—	2	1	no	—
691	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
692	#41	—	2	2	no	—
693	#41	—	2	2	no	—
694	#41	—	4	16	no	—
695	#41	—	4	8	no	—
696	#41	—	4	4	no	—
697	#41	—	4	8	no	—
698	C₂×Q₈	—	2	1	no	—
699	C₂×Q₈	—	2	2	no	—
700	C₂×Q₈	—	2	2	no	—
701	C₂×Q₈	—	2	1	no	—
702	C₂×Q₈	—	4	4	no	—
703	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
704	C₂×D₄	—	2	4	no	—
705	C₂×D₄	—	2	1	no	—
706	C₂×D₄	—	2	2	no	—
707	C₂×D₄	—	2	4	no	—
708	C₂×D₄	—	2	4	no	—
709	C₂×D₄	—	2	2	no	—
710	C₂×D₄	—	2	1	no	—
711	C₂×D₄	—	2	2	no	—
712	C₂×D₄	—	2	2	no	—
713	C₂×D₄	—	4	8	no	—
714	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
715	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
716	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
717	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
718	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
719	C₂²×C₄	—	2	4	no	—
720	C₂²×C₄	—	2	4	no	—
721	C₂²×C₄	—	2	2	no	—
722	C₂²×C₄	—	2	1	no	—
723	C₂²×C₄	—	2	2	no	—
724	C₂²×C₄	—	2	4	no	—
725	C₂²×C₄	—	2	4	no	—
726	C₂²×C₄	—	2	2	no	—
727	C₂²×C₄	—	2	1	no	—
728	C₂²×C₄	—	2	1	no	—
729	C₂²×C₄	—	2	2	no	—
730	C₂²×C₄	—	2	2	no	—
731	C₂²×C₄	—	2	1	no	—
732	C₂²×C₄	—	4	4	no	—
733	C₂²×C₄	—	4	8	no	—
734	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
735	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
736	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
737	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
738	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
739	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
740	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
741	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
742	SD₁₆	—	2	4	no	—
743	D₈	—	2	4	no	—
744	M₁₆	—	2	4	no	—
745	M₁₆	—	2	2	no	—
746	M₁₆	—	2	8	no	—
747	M₁₆	#41×C₃₁	1	4	yes	complemented
748	C₂×C₈	—	2	8	no	—
749	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
750	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
751	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
752	C₄⋊C₄	—	2	1	no	—
753	C₄⋊C₄	—	2	2	no	—
754	C₄⋊C₄	—	2	4	no	—
755	C₄⋊C₄	—	2	4	no	—
756	C₄⋊C₄	—	2	2	no	—
757	C₄⋊C₄	—	2	2	no	—
758	C₄⋊C₄	—	2	2	no	—
759	C₄⋊C₄	—	2	2	no	—
760	C₄⋊C₄	—	2	1	no	—
761	C₄⋊C₄	—	2	2	no	—
762	C₄⋊C₄	—	2	2	no	—
763	C₄⋊C₄	—	2	1	no	—
764	C₄⋊C₄	—	2	1	no	—
765	C₄⋊C₄	—	2	2	no	—
766	C₄⋊C₄	—	2	2	no	—
767	C₄⋊C₄	—	2	1	no	—
768	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
769	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
770	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
771	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
772	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
773	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
774	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
775	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
776	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
777	C₂²⋊C₄	—	2	1	no	—
778	C₂²⋊C₄	—	2	2	no	—
779	C₂²⋊C₄	—	2	2	no	—
780	C₂²⋊C₄	—	2	2	no	—
781	C₂²⋊C₄	—	2	2	no	—
782	C₂²⋊C₄	—	2	2	no	—
783	C₂²⋊C₄	—	2	2	no	—
784	C₂²⋊C₄	—	2	2	no	—
785	C₂²⋊C₄	—	2	4	no	—
786	C₂²⋊C₄	—	2	4	no	—
787	C₂²⋊C₄	—	2	2	no	—
788	C₂²⋊C₄	—	2	1	no	—
789	C₂²⋊C₄	—	2	1	no	—
790	C₂²⋊C₄	—	2	2	no	—
791	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
792	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
793	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
794	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
795	C₄²	—	2	1	no	—
796	C₄²	—	2	2	no	—
797	C₄²	—	2	1	no	—
798	C₄²	—	2	2	no	—
799	C₄²	—	2	2	no	—
800	C₄²	—	2	1	no	—
801	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
802	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
803	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
804	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
805	C₂³	—	2	4	no	—
806	C₂³	—	2	4	no	—
807	C₂³	—	2	1	no	—
808	C₂³	—	2	2	no	—
809	C₂³	—	4	2	no	—
810	C₂³	—	4	2	no	—
811	C₂³	—	4	2	no	—
812	C₂³	—	4	2	no	—
813	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
814	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
815	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
816	Q₈	—	2	1	no	—
817	Q₈	—	2	1	no	—
818	Q₈	—	4	4	no	—
819	Q₈	—	4	4	no	—
820	Q₈	—	4	2	no	—
821	Q₈	—	8	1	no	—
822	D₄	—	2	4	no	—
823	D₄	—	2	2	no	—
824	D₄	—	2	1	no	—
825	D₄	—	2	2	no	—
826	D₄	—	2	2	no	—
827	D₄	—	2	1	no	—
828	D₄	—	4	8	no	—
829	D₄	—	4	2	no	—
830	D₄	—	4	8	no	—
831	D₄	—	4	4	no	—
832	D₄	—	4	2	no	—
833	D₄	—	8	2	no	—
834	D₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	complemented
835	C₂×C₄	—	2	4	no	—
836	C₂×C₄	—	2	4	no	—
837	C₂×C₄	—	2	4	no	—
838	C₂×C₄	—	2	2	no	—
839	C₂×C₄	—	2	2	no	—
840	C₂×C₄	—	2	2	no	—
841	C₂×C₄	—	2	2	no	—
842	C₂×C₄	—	2	2	no	—
843	C₂×C₄	—	2	1	no	—
844	C₂×C₄	—	2	1	no	—
845	C₂×C₄	—	2	1	no	—
846	C₂×C₄	—	2	2	no	—
847	C₂×C₄	—	2	2	no	—
848	C₂×C₄	—	2	2	no	—
849	C₂×C₄	—	2	4	no	—
850	C₂×C₄	—	4	1	no	—
851	C₂×C₄	—	4	4	no	—
852	C₂×C₄	—	4	2	no	—
853	C₂×C₄	—	4	4	no	—
854	C₂×C₄	—	4	8	no	—
855	C₂×C₄	—	4	8	no	—
856	C₂×C₄	—	4	4	no	—
857	C₂×C₄	—	4	2	no	—
858	C₂×C₄	—	4	4	no	—
859	C₂×C₄	—	4	2	no	—
860	C₂×C₄	—	8	1	no	—
861	C₂×C₄	—	8	2	no	—
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	C₂×#41×C₃₁	1	2	yes	—
870	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
871	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
872	C₈	—	2	4	no	—
873	C₈	—	2	2	no	—
874	C₈	C₂×#41×C₃₁	1	4	yes	complemented
875	C₂²	—	2	2	no	—
876	C₂²	—	2	1	no	—
877	C₂²	—	2	1	no	—
878	C₂²	—	2	2	no	—
879	C₂²	—	2	2	no	—
880	C₂²	—	2	2	no	—
881	C₂²	—	2	2	no	—
882	C₂²	—	4	8	no	—
883	C₂²	—	4	1	no	—
884	C₂²	—	4	2	no	—
885	C₂²	—	4	2	no	—
886	C₂²	—	4	1	no	—
887	C₂²	—	4	1	no	—
888	C₂²	—	4	2	no	—
889	C₂²	—	8	4	no	—
890	C₂²	C₃₁×#547	1	2	yes	—
891	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
892	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
893	C₄	—	2	2	no	—
894	C₄	—	2	2	no	—
895	C₄	—	2	1	no	—
896	C₄	—	2	1	no	—
897	C₄	—	2	1	no	—
898	C₄	—	4	1	no	—
899	C₄	—	4	2	no	—
900	C₄	—	4	4	no	—
901	C₄	—	4	1	no	—
902	C₄	—	4	2	no	—
903	C₄	—	4	1	no	—
904	C₄	—	4	2	no	—
905	C₄	C₃₁×#547	1	2	yes	—
906	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
907	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
908	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
909	C₂	—	2	2	no	—
910	C₂	—	2	4	no	—
911	C₂	—	4	2	no	—
912	C₂	—	8	2	no	—
913	C₂	—	8	1	no	—
914	C₂	C₂×C₃₁×#547	1	1	yes	characteristic
915	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
916	C₂	C₃₁×#3586	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3475	C₂	semidirect
2. C₂×C₃₁×#547	C₂	semidirect
3. C₃₁×#3354	C₂	semidirect
4. C₃₁×#3353	C₂	semidirect
5. C₃₁×#3345	C₂	semidirect
6. C₃₁×#3343	C₂	semidirect
7. C₃₁×#3335	C₂	semidirect
8. C₃₁×#3334	C₂	semidirect
9. C₃₁×#3165	C₂	semidirect
10. C₃₁×#3094	C₂	semidirect
11. C₃₁×#3092	C₂	semidirect
12. C₃₁×#3079	C₂	semidirect
13. C₃₁×#3077	C₂	semidirect
14. C₃₁×#3055	C₂	semidirect
15. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
16. D₄×M₁₆×C₃₁	C₂	semidirect
17. C₂×C₃₁×#428	C₂	semidirect
21. C₃₁×#556	C₂²	semidirect
22. C₃₁×#555	C₂²	semidirect
23. C₃₁×#554	C₂²	semidirect
24. C₃₁×#553	C₂²	semidirect
26. C₃₁×#552	C₂²	semidirect
34. C₃₁×#547	C₂²	semidirect
50. C₃₁×#501	C₂²	semidirect
51. C₃₁×#500	C₂²	semidirect
52. C₃₁×#498	C₂²	semidirect
53. C₃₁×#484	C₂²	semidirect
54. C₃₁×#483	C₂²	semidirect
55. C₃₁×#482	C₂²	semidirect
56. C₃₁×#482	C₂²	semidirect
57. C₃₁×#481	C₂²	semidirect
58. C₃₁×#481	C₂²	semidirect
59. C₃₁×#480	C₂²	semidirect
60. C₃₁×#480	C₂²	semidirect
61. C₃₁×#478	C₂²	semidirect
62. C₃₁×#477	C₂²	semidirect
63. C₃₁×#476	C₂²	semidirect
64. C₃₁×#476	C₂²	semidirect
65. C₃₁×#475	C₂²	semidirect
66. C₃₁×#474	C₂²	semidirect
67. C₃₁×#474	C₂²	semidirect
68. C₃₁×#469	C₂²	semidirect
69. C₃₁×#468	C₂²	semidirect
70. C₃₁×#466	C₂²	semidirect
71. C₃₁×#465	C₂²	semidirect
72. C₃₁×#442	C₂²	semidirect
73. C₃₁×#440	C₂²	semidirect
74. C₄×SD₁₆×C₃₁	C₂²	semidirect
75. C₄×D₈×C₃₁	C₂²	semidirect
76. C₃₁×#436	C₂²	semidirect
77. C₃₁×#435	C₂²	semidirect
78. C₈×D₄×C₃₁	C₂²	semidirect
84. C₃₁×#428	C₂²	semidirect
85. C₃₁×#428	C₂²	semidirect
86. C₂×C₃₁×#107	C₂²	semidirect
87. C₂×C₃₁×#106	C₂²	semidirect
288. M₁₆×C₃₁	#41	semidirect
346. #61252	C₃₁	direct (N × H)
376. D₄×C₃₁	AGL(1,ℤ/8)	semidirect
416. C₈×C₃₁	C₂×#41	semidirect
417. #3475	C₂×C₃₁	semidirect
418. C₂×#547	C₂×C₃₁	semidirect
419. #3354	C₂×C₃₁	semidirect
420. #3353	C₂×C₃₁	semidirect
421. #3345	C₂×C₃₁	semidirect
422. #3343	C₂×C₃₁	semidirect
423. #3335	C₂×C₃₁	semidirect
424. #3334	C₂×C₃₁	semidirect
425. #3165	C₂×C₃₁	semidirect
426. #3094	C₂×C₃₁	semidirect
427. #3092	C₂×C₃₁	semidirect
428. #3079	C₂×C₃₁	semidirect
429. #3077	C₂×C₃₁	semidirect
430. #3055	C₂×C₃₁	semidirect
431. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
432. D₄×M₁₆	C₂×C₃₁	semidirect
433. C₂×#428	C₂×C₃₁	semidirect
471. #556	C₂²×C₃₁	semidirect
472. #555	C₂²×C₃₁	semidirect
473. #554	C₂²×C₃₁	semidirect
474. #553	C₂²×C₃₁	semidirect
476. #552	C₂²×C₃₁	semidirect
484. #547	C₂²×C₃₁	semidirect
500. #501	C₂²×C₃₁	semidirect
501. #500	C₂²×C₃₁	semidirect
502. #498	C₂²×C₃₁	semidirect
503. #484	C₂²×C₃₁	semidirect
504. #483	C₂²×C₃₁	semidirect
505. #482	C₂²×C₃₁	semidirect
506. #482	C₂²×C₃₁	semidirect
507. #481	C₂²×C₃₁	semidirect
508. #481	C₂²×C₃₁	semidirect
509. #480	C₂²×C₃₁	semidirect
510. #480	C₂²×C₃₁	semidirect
511. #478	C₂²×C₃₁	semidirect
512. #477	C₂²×C₃₁	semidirect
513. #476	C₂²×C₃₁	semidirect
514. #476	C₂²×C₃₁	semidirect
515. #475	C₂²×C₃₁	semidirect
516. #474	C₂²×C₃₁	semidirect
517. #474	C₂²×C₃₁	semidirect
518. #469	C₂²×C₃₁	semidirect
519. #468	C₂²×C₃₁	semidirect
520. #466	C₂²×C₃₁	semidirect
521. #465	C₂²×C₃₁	semidirect
522. #442	C₂²×C₃₁	semidirect
523. #440	C₂²×C₃₁	semidirect
524. C₄×SD₁₆	C₂²×C₃₁	semidirect
525. C₄×D₈	C₂²×C₃₁	semidirect
526. #436	C₂²×C₃₁	semidirect
527. #435	C₂²×C₃₁	semidirect
528. C₈×D₄	C₂²×C₃₁	semidirect
534. #428	C₂²×C₃₁	semidirect
535. #428	C₂²×C₃₁	semidirect
536. C₂×#107	C₂²×C₃₁	semidirect
537. C₂×#106	C₂²×C₃₁	semidirect
689. C₃₁	#61252	direct (N × H)
747. M₁₆	#41×C₃₁	semidirect
834. D₄	C₃₁×AGL(1,ℤ/8)	semidirect
874. C₈	C₂×#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3475	2
C₂×C₃₁×#547	2
C₃₁×#3354	2
C₃₁×#3353	2
C₃₁×#3345	2
C₃₁×#3343	2
C₃₁×#3335	2
C₃₁×#3334	2
C₃₁×#3165	2
C₃₁×#3094	2
C₃₁×#3092	2
C₃₁×#3079	2
C₃₁×#3077	2
C₃₁×#3055	2
C₄×C₃₁×AGL(1,ℤ/8)	2
D₄×M₁₆×C₃₁	2
C₂×C₃₁×#428	2
#61252	31

Subgroups — C31×#61252 #21518011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61252 #21518011