Subgroups — C₃₁×#61324 #21518083

Order 7936.

1878 conjugacy classes of subgroups.

The subgroup lattice has 880 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3375	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3373	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3371	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3370	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3365	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3361	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3140	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3077	C₂	1	2	yes	maximal, complemented
13	C₃₁×#3040	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic
15	Q₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
18	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
20	C₃₁×#559	—	2	1	no	—
21	C₃₁×#559	—	2	2	no	—
22	C₃₁×#559	—	2	1	no	—
23	C₃₁×#559	C₂²	1	2	yes	—
24	C₃₁×#559	C₂²	1	4	yes	—
25	C₃₁×#559	C₂²	1	2	yes	—
26	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#557	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#557	C₂²	1	2	yes	complemented
29	C₃₁×#554	—	2	1	no	—
30	C₃₁×#554	—	2	2	no	—
31	C₃₁×#554	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#554	C₂²	1	2	yes	complemented
33	D₄×Q₈×C₃₁	—	2	1	no	—
34	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#533	—	2	2	no	—
36	C₃₁×#533	—	2	1	no	—
37	C₃₁×#533	C₂²	1	2	yes	—
38	C₃₁×#533	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#124	—	2	1	no	—
41	C₂×C₃₁×#124	C₂²	1	2	yes	—
42	C₂×C₃₁×#123	—	2	2	no	—
43	C₄×#41×C₃₁	C₂²	1	2	yes	—
44	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
46	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
47	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
48	C₂×C₄×D₄×C₃₁	—	2	1	no	—
49	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
50	C₃₁×#490	C₂²	1	4	yes	—
51	C₃₁×#489	C₂²	1	4	yes	complemented
52	C₃₁×#489	C₂²	1	4	yes	complemented
53	C₃₁×#488	C₂²	1	4	yes	complemented
54	C₃₁×#488	C₂²	1	4	yes	complemented
55	C₃₁×#486	C₂²	1	2	yes	—
56	C₃₁×#486	C₂²	1	2	yes	—
57	C₃₁×#479	C₂²	1	4	yes	complemented
58	C₃₁×#478	C₂²	1	4	yes	complemented
59	C₃₁×#478	C₂²	1	4	yes	complemented
60	C₃₁×#477	C₂²	1	2	yes	complemented
61	C₃₁×#476	C₂²	1	4	yes	complemented
62	C₃₁×#475	C₂²	1	2	yes	complemented
63	C₃₁×#474	C₂²	1	4	yes	complemented
64	C₃₁×#464	C₂²	1	4	yes	complemented
65	C₃₁×#464	C₂²	1	4	yes	complemented
66	C₃₁×#463	C₂²	1	4	yes	complemented
67	C₃₁×#463	C₂²	1	4	yes	complemented
68	C₃₁×#460	C₂²	1	2	yes	—
69	C₃₁×#460	C₂²	1	2	yes	—
70	C₃₁×#459	C₂²	1	2	yes	—
71	C₃₁×#459	C₂²	1	2	yes	—
72	C₃₁×#446	C₂²	1	4	yes	complemented
73	C₃₁×#446	C₂²	1	2	yes	complemented
74	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
75	C₃₁×#442	C₂²	1	4	yes	complemented
76	C₃₁×#441	C₂²	1	2	yes	complemented
77	C₃₁×#440	C₂²	1	2	yes	complemented
78	C₃₁×#440	C₂²	1	4	yes	complemented
79	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
80	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
81	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
82	C₄×D₈×C₃₁	C₂²	1	4	yes	complemented
83	C₃₁×#428	C₂²	1	1	yes	characteristic
84	C₃₁×#428	C₂²	1	2	yes	—
85	C₃₁×#423	C₂²	1	2	yes	—
86	C₃₁×#423	C₂²	1	1	yes	characteristic
87	C₃₁×#419	C₂²	1	2	yes	—
88	C₃₁×#419	C₂²	1	1	yes	characteristic
89	C₃₁×#418	C₂²	1	2	yes	—
90	C₃₁×#418	C₂²	1	2	yes	—
91	C₃₁×#418	C₂²	1	2	yes	—
92	C₃₁×#417	C₂²	1	1	yes	characteristic
93	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
94	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
95	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
96	C₂²×Q₈×C₃₁	—	2	1	no	—
97	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
98	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₂²×D₄×C₃₁	—	2	1	no	—
100	C₂²×D₄×C₃₁	—	2	1	no	—
101	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
102	C₂³×C₄×C₃₁	—	2	2	no	—
103	C₃₁×#137	—	2	4	no	—
104	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
105	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
106	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
107	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
108	C₂×D₈×C₃₁	C₂³	1	4	yes	—
109	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
112	C₃₁×C₄⋊Q₈	—	2	1	no	—
113	C₃₁×C₄⋊Q₈	—	2	2	no	—
114	C₃₁×C₄⋊Q₈	—	2	1	no	—
115	C₃₁×C₄⋊Q₈	—	2	2	no	—
116	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
117	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
118	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
119	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
120	C₃₁×#127	—	2	1	no	—
121	C₃₁×#127	—	2	1	no	—
122	C₃₁×#127	C₂³	1	2	yes	—
123	C₃₁×#127	C₂³	1	2	yes	—
124	C₃₁×#125	—	2	2	no	—
125	C₃₁×#125	—	2	2	no	—
126	C₃₁×#125	—	2	2	no	—
127	C₃₁×#125	C₂³	1	2	yes	—
128	C₃₁×#125	C₂³	1	2	yes	—
129	C₃₁×#125	C₂³	1	2	yes	—
130	C₃₁×#124	—	2	2	no	—
131	C₃₁×#124	—	2	2	no	—
132	C₃₁×#124	—	2	4	no	—
133	C₃₁×#124	—	2	2	no	—
134	C₃₁×#124	—	2	2	no	—
135	C₃₁×#124	C₂³	1	4	yes	—
136	C₃₁×#124	C₂³	1	4	yes	—
137	C₃₁×#123	—	2	4	no	—
138	C₃₁×#123	—	2	4	no	—
139	C₃₁×#123	—	2	4	no	—
140	C₃₁×#123	—	2	4	no	—
141	C₃₁×C₂²⋊Q₈	—	2	1	no	—
142	C₃₁×C₂²⋊Q₈	—	2	2	no	—
143	C₃₁×C₂²⋊Q₈	—	2	1	no	—
144	C₃₁×C₂²⋊Q₈	—	2	2	no	—
145	C₃₁×C₂²⋊Q₈	—	2	2	no	—
146	C₃₁×C₂²⋊Q₈	—	2	1	no	—
147	C₃₁×C₂²⋊Q₈	—	2	1	no	—
148	C₃₁×C₂²⋊Q₈	—	2	2	no	—
149	C₃₁×C₂²⋊Q₈	—	2	2	no	—
150	C₃₁×C₂²⋊Q₈	—	2	2	no	—
151	C₃₁×C₂²⋊Q₈	—	2	2	no	—
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	—	2	1	no	—
158	C₄×Q₈×C₃₁	—	2	1	no	—
159	C₄×Q₈×C₃₁	—	2	2	no	—
160	C₄×Q₈×C₃₁	—	2	1	no	—
161	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
162	C₄×Q₈×C₃₁	D₄	1	2	yes	—
163	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
164	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
165	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
166	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
167	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
168	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
169	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
170	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
171	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
172	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
173	C₄×D₄×C₃₁	—	2	1	no	—
174	C₄×D₄×C₃₁	—	2	2	no	—
175	C₄×D₄×C₃₁	—	2	2	no	—
176	C₄×D₄×C₃₁	—	2	1	no	—
177	C₄×D₄×C₃₁	—	2	2	no	—
178	C₄×D₄×C₃₁	—	2	4	no	—
179	C₄×D₄×C₃₁	—	2	2	no	—
180	C₄×D₄×C₃₁	—	2	2	no	—
181	C₄×D₄×C₃₁	—	2	2	no	—
182	C₄×D₄×C₃₁	C₂³	1	2	yes	—
183	C₄×D₄×C₃₁	C₂³	1	4	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	4	yes	—
187	C₄×D₄×C₃₁	C₂³	1	2	yes	—
188	C₃₁×#117	—	2	2	no	—
189	C₃₁×#117	—	2	1	no	—
190	C₃₁×#117	C₂³	1	2	yes	—
191	C₃₁×#117	C₂³	1	1	yes	characteristic
192	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
195	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
196	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
197	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
198	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
199	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
200	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
201	C₂×C₄²×C₃₁	—	2	1	no	—
202	C₂×C₄²×C₃₁	—	2	2	no	—
203	C₂×C₄²×C₃₁	—	2	1	no	—
204	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
205	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
206	C₃₁×#107	C₂³	1	4	yes	—
207	C₃₁×#107	C₂³	1	2	yes	—
208	C₃₁×#106	C₂³	1	2	yes	—
209	C₃₁×#106	C₂³	1	4	yes	—
210	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
211	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
212	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
213	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
214	C₃₁×#103	C₂³	1	2	yes	—
215	C₃₁×#103	C₂³	1	2	yes	—
216	C₃₁×#103	C₂³	1	2	yes	—
217	C₃₁×#103	C₂³	1	4	yes	—
218	C₃₁×#103	C₂³	1	4	yes	—
219	C₃₁×#102	C₂³	1	2	yes	—
220	C₃₁×#102	C₂³	1	4	yes	—
221	C₃₁×#102	C₂³	1	4	yes	—
222	C₃₁×#102	C₂³	1	4	yes	—
223	C₃₁×#102	C₂³	1	4	yes	—
224	C₃₁×#102	C₂³	1	4	yes	—
225	C₃₁×#102	C₂³	1	4	yes	—
226	C₃₁×#97	C₂³	1	2	yes	—
227	C₃₁×#97	C₂³	1	4	yes	—
228	C₄×C₈×C₃₁	C₂³	1	4	yes	—
229	C₄×C₈×C₃₁	C₂³	1	2	yes	—
230	C₂⁴×C₃₁	—	2	2	no	—
231	#41×C₃₁	—	2	2	no	—
232	#41×C₃₁	#41	1	4	yes	—
233	C₂×Q₈×C₃₁	—	2	2	no	—
234	C₂×Q₈×C₃₁	—	2	1	no	—
235	C₂×Q₈×C₃₁	—	2	1	no	—
236	C₂×Q₈×C₃₁	—	2	2	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	—	4	2	no	—
241	C₂×Q₈×C₃₁	—	4	4	no	—
242	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
243	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
244	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
245	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
246	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
247	C₂×D₄×C₃₁	—	2	1	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	2	no	—
253	C₂×D₄×C₃₁	—	2	4	no	—
254	C₂×D₄×C₃₁	—	2	4	no	—
255	C₂×D₄×C₃₁	—	4	4	no	—
256	C₂×D₄×C₃₁	—	8	2	no	—
257	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
258	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
259	C₂²×C₄×C₃₁	—	2	1	no	—
260	C₂²×C₄×C₃₁	—	2	1	no	—
261	C₂²×C₄×C₃₁	—	2	1	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	1	no	—
266	C₂²×C₄×C₃₁	—	2	4	no	—
267	C₂²×C₄×C₃₁	—	2	4	no	—
268	C₂²×C₄×C₃₁	—	8	4	no	—
269	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
270	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
271	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
272	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
273	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
274	Q₁₆×C₃₁	—	2	4	no	—
275	SD₁₆×C₃₁	—	2	8	no	—
276	SD₁₆×C₃₁	—	2	4	no	—
277	D₈×C₃₁	—	2	8	no	—
278	M₁₆×C₃₁	—	2	4	no	—
279	M₁₆×C₃₁	—	2	2	no	—
280	M₁₆×C₃₁	—	2	2	no	—
281	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
282	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
283	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
284	C₄⋊C₄×C₃₁	—	2	2	no	—
285	C₄⋊C₄×C₃₁	—	2	2	no	—
286	C₄⋊C₄×C₃₁	—	2	2	no	—
287	C₄⋊C₄×C₃₁	—	2	2	no	—
288	C₄⋊C₄×C₃₁	—	2	1	no	—
289	C₄⋊C₄×C₃₁	—	2	2	no	—
290	C₄⋊C₄×C₃₁	—	2	2	no	—
291	C₄⋊C₄×C₃₁	—	2	1	no	—
292	C₄⋊C₄×C₃₁	—	2	1	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	2	no	—
299	C₄⋊C₄×C₃₁	—	2	2	no	—
300	C₄⋊C₄×C₃₁	—	2	2	no	—
301	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
302	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
303	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
304	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
305	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
306	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
307	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
308	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
309	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
310	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
311	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
312	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
313	C₂²⋊C₄×C₃₁	—	2	2	no	—
314	C₂²⋊C₄×C₃₁	—	2	1	no	—
315	C₂²⋊C₄×C₃₁	—	2	4	no	—
316	C₂²⋊C₄×C₃₁	—	2	4	no	—
317	C₂²⋊C₄×C₃₁	—	2	4	no	—
318	C₂²⋊C₄×C₃₁	—	2	4	no	—
319	C₂²⋊C₄×C₃₁	—	2	4	no	—
320	C₂²⋊C₄×C₃₁	—	2	4	no	—
321	C₂²⋊C₄×C₃₁	—	2	1	no	—
322	C₂²⋊C₄×C₃₁	—	2	2	no	—
323	C₂²⋊C₄×C₃₁	—	2	2	no	—
324	C₂²⋊C₄×C₃₁	—	2	1	no	—
325	C₄²×C₃₁	—	2	2	no	—
326	C₄²×C₃₁	—	2	1	no	—
327	C₄²×C₃₁	—	2	2	no	—
328	C₄²×C₃₁	—	2	2	no	—
329	C₄²×C₃₁	—	2	2	no	—
330	C₄²×C₃₁	—	2	1	no	—
331	C₄²×C₃₁	—	2	2	no	—
332	C₄²×C₃₁	—	2	1	no	—
333	C₄²×C₃₁	C₂×D₄	1	2	yes	—
334	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
335	C₄²×C₃₁	C₂×D₄	1	2	yes	—
336	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
337	C₄²×C₃₁	C₂⁴	1	2	yes	—
338	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
339	C₄²×C₃₁	C₂⁴	1	2	yes	—
340	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
341	#61324	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
342	C₂³×C₃₁	—	2	1	no	—
343	C₂³×C₃₁	—	2	4	no	—
344	C₂³×C₃₁	—	4	4	no	—
345	C₂³×C₃₁	—	4	4	no	—
346	C₂³×C₃₁	—	4	4	no	—
347	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
348	Q₈×C₃₁	—	2	2	no	—
349	Q₈×C₃₁	—	2	2	no	—
350	Q₈×C₃₁	—	2	1	no	—
351	Q₈×C₃₁	—	2	2	no	—
352	Q₈×C₃₁	—	2	2	no	—
353	Q₈×C₃₁	—	4	2	no	—
354	Q₈×C₃₁	—	4	4	no	—
355	Q₈×C₃₁	—	4	2	no	—
356	Q₈×C₃₁	—	4	2	no	—
357	Q₈×C₃₁	—	8	2	no	—
358	Q₈×C₃₁	—	8	1	no	—
359	Q₈×C₃₁	C₂×#41	1	2	yes	—
360	D₄×C₃₁	—	2	4	no	—
361	D₄×C₃₁	—	2	4	no	—
362	D₄×C₃₁	—	2	1	no	—
363	D₄×C₃₁	—	2	4	no	—
364	D₄×C₃₁	—	4	4	no	—
365	D₄×C₃₁	—	4	4	no	—
366	D₄×C₃₁	—	4	4	no	—
367	D₄×C₃₁	—	8	2	no	—
368	D₄×C₃₁	—	8	2	no	—
369	D₄×C₃₁	C₂×#41	1	2	yes	—
370	C₂×C₄×C₃₁	—	2	1	no	—
371	C₂×C₄×C₃₁	—	2	4	no	—
372	C₂×C₄×C₃₁	—	2	2	no	—
373	C₂×C₄×C₃₁	—	2	2	no	—
374	C₂×C₄×C₃₁	—	2	2	no	—
375	C₂×C₄×C₃₁	—	2	2	no	—
376	C₂×C₄×C₃₁	—	2	1	no	—
377	C₂×C₄×C₃₁	—	2	2	no	—
378	C₂×C₄×C₃₁	—	2	1	no	—
379	C₂×C₄×C₃₁	—	4	2	no	—
380	C₂×C₄×C₃₁	—	4	1	no	—
381	C₂×C₄×C₃₁	—	4	2	no	—
382	C₂×C₄×C₃₁	—	4	4	no	—
383	C₂×C₄×C₃₁	—	4	2	no	—
384	C₂×C₄×C₃₁	—	4	2	no	—
385	C₂×C₄×C₃₁	—	4	2	no	—
386	C₂×C₄×C₃₁	—	8	2	no	—
387	C₂×C₄×C₃₁	—	8	1	no	—
388	C₂×C₄×C₃₁	—	8	4	no	—
389	C₂×C₄×C₃₁	—	8	4	no	—
390	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
391	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
392	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
393	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
394	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
395	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
396	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
397	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
398	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
399	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
400	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
401	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
402	C₈×C₃₁	—	2	4	no	—
403	C₈×C₃₁	—	2	2	no	—
404	C₈×C₃₁	—	2	2	no	—
405	C₂×#559	C₂×C₃₁	1	1	yes	characteristic, complemented
406	Q₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
407	#3375	C₂×C₃₁	1	2	yes	complemented
408	#3373	C₂×C₃₁	1	2	yes	complemented
409	#3371	C₂×C₃₁	1	4	yes	complemented
410	#3370	C₂×C₃₁	1	4	yes	complemented
411	#3365	C₂×C₃₁	1	2	yes	complemented
412	#3361	C₂×C₃₁	1	2	yes	complemented
413	#3140	C₂×C₃₁	1	2	yes	complemented
414	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
415	#3078	C₂×C₃₁	1	1	yes	characteristic, complemented
416	#3077	C₂×C₃₁	1	2	yes	complemented
417	#3040	C₂×C₃₁	1	2	yes	complemented
418	#3038	C₂×C₃₁	1	1	yes	characteristic
419	Q₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
420	C₄×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
421	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	2	yes	complemented
422	C₂²×C₃₁	—	2	1	no	—
423	C₂²×C₃₁	—	2	1	no	—
424	C₂²×C₃₁	—	2	2	no	—
425	C₂²×C₃₁	—	4	1	no	—
426	C₂²×C₃₁	—	4	1	no	—
427	C₂²×C₃₁	—	4	4	no	—
428	C₂²×C₃₁	—	4	4	no	—
429	C₂²×C₃₁	—	4	4	no	—
430	C₂²×C₃₁	—	8	8	no	—
431	C₂²×C₃₁	#548	1	2	yes	—
432	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
433	C₄×C₃₁	—	2	2	no	—
434	C₄×C₃₁	—	2	2	no	—
435	C₄×C₃₁	—	2	1	no	—
436	C₄×C₃₁	—	2	1	no	—
437	C₄×C₃₁	—	2	2	no	—
438	C₄×C₃₁	—	2	1	no	—
439	C₄×C₃₁	—	2	2	no	—
440	C₄×C₃₁	—	2	2	no	—
441	C₄×C₃₁	—	4	2	no	—
442	C₄×C₃₁	—	4	2	no	—
443	C₄×C₃₁	—	4	1	no	—
444	C₄×C₃₁	—	4	2	no	—
445	C₄×C₃₁	—	4	2	no	—
446	C₄×C₃₁	—	4	1	no	—
447	C₄×C₃₁	#548	1	2	yes	—
448	C₄×C₃₁	#575	1	2	yes	—
449	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
450	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
451	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
452	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
453	#559	—	2	1	no	—
454	#559	—	2	2	no	—
455	#559	—	2	1	no	—
456	#559	C₂²×C₃₁	1	2	yes	—
457	#559	C₂²×C₃₁	1	4	yes	—
458	#559	C₂²×C₃₁	1	2	yes	—
459	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
460	#557	C₂²×C₃₁	1	1	yes	characteristic, complemented
461	#557	C₂²×C₃₁	1	2	yes	complemented
462	#554	—	2	1	no	—
463	#554	—	2	2	no	—
464	#554	C₂²×C₃₁	1	1	yes	characteristic, complemented
465	#554	C₂²×C₃₁	1	2	yes	complemented
466	D₄×Q₈	—	2	1	no	—
467	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
468	#533	—	2	2	no	—
469	#533	—	2	1	no	—
470	#533	C₂²×C₃₁	1	2	yes	—
471	#533	C₂²×C₃₁	1	1	yes	characteristic
472	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
473	C₂×#124	—	2	1	no	—
474	C₂×#124	C₂²×C₃₁	1	2	yes	—
475	C₂×#123	—	2	2	no	—
476	C₄×#41	C₂²×C₃₁	1	2	yes	—
477	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
478	C₂×C₄×Q₈	—	2	1	no	—
479	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
480	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
481	C₂×C₄×D₄	—	2	1	no	—
482	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
483	#490	C₂²×C₃₁	1	4	yes	—
484	#489	C₂²×C₃₁	1	4	yes	complemented
485	#489	C₂²×C₃₁	1	4	yes	complemented
486	#488	C₂²×C₃₁	1	4	yes	complemented
487	#488	C₂²×C₃₁	1	4	yes	complemented
488	#486	C₂²×C₃₁	1	2	yes	—
489	#486	C₂²×C₃₁	1	2	yes	—
490	#479	C₂²×C₃₁	1	4	yes	complemented
491	#478	C₂²×C₃₁	1	4	yes	complemented
492	#478	C₂²×C₃₁	1	4	yes	complemented
493	#477	C₂²×C₃₁	1	2	yes	complemented
494	#476	C₂²×C₃₁	1	4	yes	complemented
495	#475	C₂²×C₃₁	1	2	yes	complemented
496	#474	C₂²×C₃₁	1	4	yes	complemented
497	#464	C₂²×C₃₁	1	4	yes	complemented
498	#464	C₂²×C₃₁	1	4	yes	complemented
499	#463	C₂²×C₃₁	1	4	yes	complemented
500	#463	C₂²×C₃₁	1	4	yes	complemented
501	#460	C₂²×C₃₁	1	2	yes	—
502	#460	C₂²×C₃₁	1	2	yes	—
503	#459	C₂²×C₃₁	1	2	yes	—
504	#459	C₂²×C₃₁	1	2	yes	—
505	#446	C₂²×C₃₁	1	4	yes	complemented
506	#446	C₂²×C₃₁	1	2	yes	complemented
507	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
508	#442	C₂²×C₃₁	1	4	yes	complemented
509	#441	C₂²×C₃₁	1	2	yes	complemented
510	#440	C₂²×C₃₁	1	2	yes	complemented
511	#440	C₂²×C₃₁	1	4	yes	complemented
512	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
513	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
514	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
515	C₄×D₈	C₂²×C₃₁	1	4	yes	complemented
516	#428	C₂²×C₃₁	1	1	yes	characteristic
517	#428	C₂²×C₃₁	1	2	yes	—
518	#423	C₂²×C₃₁	1	2	yes	—
519	#423	C₂²×C₃₁	1	1	yes	characteristic
520	#419	C₂²×C₃₁	1	2	yes	—
521	#419	C₂²×C₃₁	1	1	yes	characteristic
522	#418	C₂²×C₃₁	1	2	yes	—
523	#418	C₂²×C₃₁	1	2	yes	—
524	#418	C₂²×C₃₁	1	2	yes	—
525	#417	C₂²×C₃₁	1	1	yes	characteristic
526	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
527	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
528	C₂×C₃₁	—	2	2	no	—
529	C₂×C₃₁	—	4	2	no	—
530	C₂×C₃₁	—	8	4	no	—
531	C₂×C₃₁	C₂×#548	1	1	yes	characteristic
532	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
533	C₂×C₃₁	#3585	1	1	yes	characteristic
534	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
535	C₂²×Q₈	—	2	1	no	—
536	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
537	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
538	C₂²×D₄	—	2	1	no	—
539	C₂²×D₄	—	2	1	no	—
540	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
541	C₂³×C₄	—	2	2	no	—
542	#137	—	2	4	no	—
543	AGL(1,ℤ/8)	—	2	8	no	—
544	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
545	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
546	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
547	C₂×D₈	C₂³×C₃₁	1	4	yes	—
548	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
549	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
550	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
551	C₄⋊Q₈	—	2	1	no	—
552	C₄⋊Q₈	—	2	2	no	—
553	C₄⋊Q₈	—	2	1	no	—
554	C₄⋊Q₈	—	2	2	no	—
555	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
556	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
557	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
558	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
559	#127	—	2	1	no	—
560	#127	—	2	1	no	—
561	#127	C₂³×C₃₁	1	2	yes	—
562	#127	C₂³×C₃₁	1	2	yes	—
563	#125	—	2	2	no	—
564	#125	—	2	2	no	—
565	#125	—	2	2	no	—
566	#125	C₂³×C₃₁	1	2	yes	—
567	#125	C₂³×C₃₁	1	2	yes	—
568	#125	C₂³×C₃₁	1	2	yes	—
569	#124	—	2	2	no	—
570	#124	—	2	2	no	—
571	#124	—	2	4	no	—
572	#124	—	2	2	no	—
573	#124	—	2	2	no	—
574	#124	C₂³×C₃₁	1	4	yes	—
575	#124	C₂³×C₃₁	1	4	yes	—
576	#123	—	2	4	no	—
577	#123	—	2	4	no	—
578	#123	—	2	4	no	—
579	#123	—	2	4	no	—
580	C₂²⋊Q₈	—	2	1	no	—
581	C₂²⋊Q₈	—	2	2	no	—
582	C₂²⋊Q₈	—	2	1	no	—
583	C₂²⋊Q₈	—	2	2	no	—
584	C₂²⋊Q₈	—	2	2	no	—
585	C₂²⋊Q₈	—	2	1	no	—
586	C₂²⋊Q₈	—	2	1	no	—
587	C₂²⋊Q₈	—	2	2	no	—
588	C₂²⋊Q₈	—	2	2	no	—
589	C₂²⋊Q₈	—	2	2	no	—
590	C₂²⋊Q₈	—	2	2	no	—
591	C₄×Q₈	—	2	1	no	—
592	C₄×Q₈	—	2	1	no	—
593	C₄×Q₈	—	2	2	no	—
594	C₄×Q₈	—	2	2	no	—
595	C₄×Q₈	—	2	1	no	—
596	C₄×Q₈	—	2	1	no	—
597	C₄×Q₈	—	2	1	no	—
598	C₄×Q₈	—	2	2	no	—
599	C₄×Q₈	—	2	1	no	—
600	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
601	C₄×Q₈	D₄×C₃₁	1	2	yes	—
602	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
603	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
604	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
605	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
606	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
607	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
608	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
609	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
610	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
611	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
612	C₄×D₄	—	2	1	no	—
613	C₄×D₄	—	2	2	no	—
614	C₄×D₄	—	2	2	no	—
615	C₄×D₄	—	2	1	no	—
616	C₄×D₄	—	2	2	no	—
617	C₄×D₄	—	2	4	no	—
618	C₄×D₄	—	2	2	no	—
619	C₄×D₄	—	2	2	no	—
620	C₄×D₄	—	2	2	no	—
621	C₄×D₄	C₂³×C₃₁	1	4	yes	—
622	C₄×D₄	C₂³×C₃₁	1	4	yes	—
623	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
624	C₄×D₄	C₂³×C₃₁	1	2	yes	—
625	C₄×D₄	C₂³×C₃₁	1	2	yes	—
626	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
627	#117	—	2	2	no	—
628	#117	—	2	1	no	—
629	#117	C₂³×C₃₁	1	2	yes	—
630	#117	C₂³×C₃₁	1	1	yes	characteristic
631	C₂×C₄⋊C₄	—	2	1	no	—
632	C₂×C₄⋊C₄	—	2	1	no	—
633	C₂×C₄⋊C₄	—	2	2	no	—
634	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
635	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
636	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
637	C₂×C₂²⋊C₄	—	2	2	no	—
638	C₂×C₂²⋊C₄	—	2	2	no	—
639	C₂×C₂²⋊C₄	—	2	2	no	—
640	C₂×C₄²	—	2	1	no	—
641	C₂×C₄²	—	2	2	no	—
642	C₂×C₄²	—	2	1	no	—
643	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
644	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
645	#107	C₂³×C₃₁	1	4	yes	—
646	#107	C₂³×C₃₁	1	2	yes	—
647	#106	C₂³×C₃₁	1	2	yes	—
648	#106	C₂³×C₃₁	1	4	yes	—
649	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
650	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
651	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
652	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
653	#103	C₂³×C₃₁	1	2	yes	—
654	#103	C₂³×C₃₁	1	2	yes	—
655	#103	C₂³×C₃₁	1	2	yes	—
656	#103	C₂³×C₃₁	1	4	yes	—
657	#103	C₂³×C₃₁	1	4	yes	—
658	#102	C₂³×C₃₁	1	2	yes	—
659	#102	C₂³×C₃₁	1	4	yes	—
660	#102	C₂³×C₃₁	1	4	yes	—
661	#102	C₂³×C₃₁	1	4	yes	—
662	#102	C₂³×C₃₁	1	4	yes	—
663	#102	C₂³×C₃₁	1	4	yes	—
664	#102	C₂³×C₃₁	1	4	yes	—
665	#97	C₂³×C₃₁	1	2	yes	—
666	#97	C₂³×C₃₁	1	4	yes	—
667	C₄×C₈	C₂³×C₃₁	1	4	yes	—
668	C₄×C₈	C₂³×C₃₁	1	2	yes	—
669	C₃₁	#61324	1	1	yes	characteristic, direct product factor, Sylow
670	C₂⁴	—	2	2	no	—
671	#41	—	2	2	no	—
672	#41	#41×C₃₁	1	4	yes	—
673	C₂×Q₈	—	2	2	no	—
674	C₂×Q₈	—	2	1	no	—
675	C₂×Q₈	—	2	1	no	—
676	C₂×Q₈	—	2	2	no	—
677	C₂×Q₈	—	2	1	no	—
678	C₂×Q₈	—	2	2	no	—
679	C₂×Q₈	—	2	2	no	—
680	C₂×Q₈	—	4	2	no	—
681	C₂×Q₈	—	4	4	no	—
682	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
683	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
684	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
685	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
686	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
687	C₂×D₄	—	2	1	no	—
688	C₂×D₄	—	2	2	no	—
689	C₂×D₄	—	2	2	no	—
690	C₂×D₄	—	2	2	no	—
691	C₂×D₄	—	2	2	no	—
692	C₂×D₄	—	2	2	no	—
693	C₂×D₄	—	2	4	no	—
694	C₂×D₄	—	2	4	no	—
695	C₂×D₄	—	4	4	no	—
696	C₂×D₄	—	8	2	no	—
697	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
698	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
699	C₂²×C₄	—	2	1	no	—
700	C₂²×C₄	—	2	1	no	—
701	C₂²×C₄	—	2	1	no	—
702	C₂²×C₄	—	2	2	no	—
703	C₂²×C₄	—	2	1	no	—
704	C₂²×C₄	—	2	2	no	—
705	C₂²×C₄	—	2	1	no	—
706	C₂²×C₄	—	2	4	no	—
707	C₂²×C₄	—	2	4	no	—
708	C₂²×C₄	—	8	4	no	—
709	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
710	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
711	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
712	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
713	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
714	Q₁₆	—	2	4	no	—
715	SD₁₆	—	2	8	no	—
716	SD₁₆	—	2	4	no	—
717	D₈	—	2	8	no	—
718	M₁₆	—	2	4	no	—
719	M₁₆	—	2	2	no	—
720	M₁₆	—	2	2	no	—
721	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
722	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
723	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
724	C₄⋊C₄	—	2	2	no	—
725	C₄⋊C₄	—	2	2	no	—
726	C₄⋊C₄	—	2	2	no	—
727	C₄⋊C₄	—	2	2	no	—
728	C₄⋊C₄	—	2	1	no	—
729	C₄⋊C₄	—	2	2	no	—
730	C₄⋊C₄	—	2	2	no	—
731	C₄⋊C₄	—	2	1	no	—
732	C₄⋊C₄	—	2	1	no	—
733	C₄⋊C₄	—	2	2	no	—
734	C₄⋊C₄	—	2	2	no	—
735	C₄⋊C₄	—	2	1	no	—
736	C₄⋊C₄	—	2	2	no	—
737	C₄⋊C₄	—	2	1	no	—
738	C₄⋊C₄	—	2	2	no	—
739	C₄⋊C₄	—	2	2	no	—
740	C₄⋊C₄	—	2	2	no	—
741	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
742	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
743	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
744	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
745	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
746	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
747	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
748	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
749	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
750	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
751	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
752	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
753	C₂²⋊C₄	—	2	2	no	—
754	C₂²⋊C₄	—	2	1	no	—
755	C₂²⋊C₄	—	2	4	no	—
756	C₂²⋊C₄	—	2	4	no	—
757	C₂²⋊C₄	—	2	4	no	—
758	C₂²⋊C₄	—	2	4	no	—
759	C₂²⋊C₄	—	2	4	no	—
760	C₂²⋊C₄	—	2	4	no	—
761	C₂²⋊C₄	—	2	1	no	—
762	C₂²⋊C₄	—	2	2	no	—
763	C₂²⋊C₄	—	2	2	no	—
764	C₂²⋊C₄	—	2	1	no	—
765	C₄²	—	2	2	no	—
766	C₄²	—	2	1	no	—
767	C₄²	—	2	2	no	—
768	C₄²	—	2	2	no	—
769	C₄²	—	2	2	no	—
770	C₄²	—	2	1	no	—
771	C₄²	—	2	2	no	—
772	C₄²	—	2	1	no	—
773	C₄²	C₂×D₄×C₃₁	1	2	yes	—
774	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
775	C₄²	C₂×D₄×C₃₁	1	2	yes	—
776	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
777	C₄²	C₂⁴×C₃₁	1	2	yes	—
778	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
779	C₄²	C₂⁴×C₃₁	1	2	yes	—
780	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
781	C₂³	—	2	1	no	—
782	C₂³	—	2	4	no	—
783	C₂³	—	4	4	no	—
784	C₂³	—	4	4	no	—
785	C₂³	—	4	4	no	—
786	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
787	Q₈	—	2	2	no	—
788	Q₈	—	2	2	no	—
789	Q₈	—	2	1	no	—
790	Q₈	—	2	2	no	—
791	Q₈	—	2	2	no	—
792	Q₈	—	4	2	no	—
793	Q₈	—	4	4	no	—
794	Q₈	—	4	2	no	—
795	Q₈	—	4	2	no	—
796	Q₈	—	8	2	no	—
797	Q₈	—	8	1	no	—
798	Q₈	C₂×#41×C₃₁	1	2	yes	—
799	D₄	—	2	4	no	—
800	D₄	—	2	4	no	—
801	D₄	—	2	1	no	—
802	D₄	—	2	4	no	—
803	D₄	—	4	4	no	—
804	D₄	—	4	4	no	—
805	D₄	—	4	4	no	—
806	D₄	—	8	2	no	—
807	D₄	—	8	2	no	—
808	D₄	C₂×#41×C₃₁	1	2	yes	—
809	C₂×C₄	—	2	1	no	—
810	C₂×C₄	—	2	4	no	—
811	C₂×C₄	—	2	2	no	—
812	C₂×C₄	—	2	2	no	—
813	C₂×C₄	—	2	2	no	—
814	C₂×C₄	—	2	2	no	—
815	C₂×C₄	—	2	1	no	—
816	C₂×C₄	—	2	2	no	—
817	C₂×C₄	—	2	1	no	—
818	C₂×C₄	—	4	2	no	—
819	C₂×C₄	—	4	1	no	—
820	C₂×C₄	—	4	2	no	—
821	C₂×C₄	—	4	4	no	—
822	C₂×C₄	—	4	2	no	—
823	C₂×C₄	—	4	2	no	—
824	C₂×C₄	—	4	2	no	—
825	C₂×C₄	—	8	2	no	—
826	C₂×C₄	—	8	1	no	—
827	C₂×C₄	—	8	4	no	—
828	C₂×C₄	—	8	4	no	—
829	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
830	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
831	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
832	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
833	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
834	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
835	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
836	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
837	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
838	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
839	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
840	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
841	C₈	—	2	4	no	—
842	C₈	—	2	2	no	—
843	C₈	—	2	2	no	—
844	C₂²	—	2	1	no	—
845	C₂²	—	2	1	no	—
846	C₂²	—	2	2	no	—
847	C₂²	—	4	1	no	—
848	C₂²	—	4	1	no	—
849	C₂²	—	4	4	no	—
850	C₂²	—	4	4	no	—
851	C₂²	—	4	4	no	—
852	C₂²	—	8	8	no	—
853	C₂²	C₃₁×#548	1	2	yes	—
854	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
855	C₄	—	2	2	no	—
856	C₄	—	2	1	no	—
857	C₄	—	2	2	no	—
858	C₄	—	2	2	no	—
859	C₄	—	2	2	no	—
860	C₄	—	2	2	no	—
861	C₄	—	2	1	no	—
862	C₄	—	2	1	no	—
863	C₄	—	4	2	no	—
864	C₄	—	4	2	no	—
865	C₄	—	4	1	no	—
866	C₄	—	4	2	no	—
867	C₄	—	4	2	no	—
868	C₄	—	4	1	no	—
869	C₄	C₃₁×#548	1	2	yes	—
870	C₄	C₃₁×#575	1	2	yes	—
871	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
872	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
873	C₂	—	2	2	no	—
874	C₂	—	4	2	no	—
875	C₂	—	8	4	no	—
876	C₂	C₂×C₃₁×#548	1	1	yes	characteristic
877	C₂	C₂×C₃₁×#575	1	1	yes	characteristic
878	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3038	C₂
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. C₂×C₃₁×#559	C₂	semidirect
2. Q₈×#41×C₃₁	C₂	semidirect
3. C₃₁×#3375	C₂	semidirect
4. C₃₁×#3373	C₂	semidirect
5. C₃₁×#3371	C₂	semidirect
6. C₃₁×#3370	C₂	semidirect
7. C₃₁×#3365	C₂	semidirect
8. C₃₁×#3361	C₂	semidirect
9. C₃₁×#3140	C₂	semidirect
10. C₃₁×#3138	C₂	semidirect
11. C₃₁×#3078	C₂	semidirect
12. C₃₁×#3077	C₂	semidirect
13. C₃₁×#3040	C₂	semidirect
15. Q₈×M₁₆×C₃₁	C₂	semidirect
16. C₄×C₃₁×#137	C₂	semidirect
17. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
26. Q₈²×C₃₁	C₂²	semidirect
27. C₃₁×#557	C₂²	semidirect
28. C₃₁×#557	C₂²	semidirect
31. C₃₁×#554	C₂²	semidirect
32. C₃₁×#554	C₂²	semidirect
34. D₄×Q₈×C₃₁	C₂²	semidirect
51. C₃₁×#489	C₂²	semidirect
52. C₃₁×#489	C₂²	semidirect
53. C₃₁×#488	C₂²	semidirect
54. C₃₁×#488	C₂²	semidirect
57. C₃₁×#479	C₂²	semidirect
58. C₃₁×#478	C₂²	semidirect
59. C₃₁×#478	C₂²	semidirect
60. C₃₁×#477	C₂²	semidirect
61. C₃₁×#476	C₂²	semidirect
62. C₃₁×#475	C₂²	semidirect
63. C₃₁×#474	C₂²	semidirect
64. C₃₁×#464	C₂²	semidirect
65. C₃₁×#464	C₂²	semidirect
66. C₃₁×#463	C₂²	semidirect
67. C₃₁×#463	C₂²	semidirect
72. C₃₁×#446	C₂²	semidirect
73. C₃₁×#446	C₂²	semidirect
74. C₈×Q₈×C₃₁	C₂²	semidirect
75. C₃₁×#442	C₂²	semidirect
76. C₃₁×#441	C₂²	semidirect
77. C₃₁×#440	C₂²	semidirect
78. C₃₁×#440	C₂²	semidirect
79. C₄×Q₁₆×C₃₁	C₂²	semidirect
80. C₄×SD₁₆×C₃₁	C₂²	semidirect
81. C₄×SD₁₆×C₃₁	C₂²	semidirect
82. C₄×D₈×C₃₁	C₂²	semidirect
341. #61324	C₃₁	direct (N × H)
405. C₂×#559	C₂×C₃₁	semidirect
406. Q₈×#41	C₂×C₃₁	semidirect
407. #3375	C₂×C₃₁	semidirect
408. #3373	C₂×C₃₁	semidirect
409. #3371	C₂×C₃₁	semidirect
410. #3370	C₂×C₃₁	semidirect
411. #3365	C₂×C₃₁	semidirect
412. #3361	C₂×C₃₁	semidirect
413. #3140	C₂×C₃₁	semidirect
414. #3138	C₂×C₃₁	semidirect
415. #3078	C₂×C₃₁	semidirect
416. #3077	C₂×C₃₁	semidirect
417. #3040	C₂×C₃₁	semidirect
419. Q₈×M₁₆	C₂×C₃₁	semidirect
420. C₄×#137	C₂×C₃₁	semidirect
421. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
459. Q₈²	C₂²×C₃₁	semidirect
460. #557	C₂²×C₃₁	semidirect
461. #557	C₂²×C₃₁	semidirect
464. #554	C₂²×C₃₁	semidirect
465. #554	C₂²×C₃₁	semidirect
467. D₄×Q₈	C₂²×C₃₁	semidirect
484. #489	C₂²×C₃₁	semidirect
485. #489	C₂²×C₃₁	semidirect
486. #488	C₂²×C₃₁	semidirect
487. #488	C₂²×C₃₁	semidirect
490. #479	C₂²×C₃₁	semidirect
491. #478	C₂²×C₃₁	semidirect
492. #478	C₂²×C₃₁	semidirect
493. #477	C₂²×C₃₁	semidirect
494. #476	C₂²×C₃₁	semidirect
495. #475	C₂²×C₃₁	semidirect
496. #474	C₂²×C₃₁	semidirect
497. #464	C₂²×C₃₁	semidirect
498. #464	C₂²×C₃₁	semidirect
499. #463	C₂²×C₃₁	semidirect
500. #463	C₂²×C₃₁	semidirect
505. #446	C₂²×C₃₁	semidirect
506. #446	C₂²×C₃₁	semidirect
507. C₈×Q₈	C₂²×C₃₁	semidirect
508. #442	C₂²×C₃₁	semidirect
509. #441	C₂²×C₃₁	semidirect
510. #440	C₂²×C₃₁	semidirect
511. #440	C₂²×C₃₁	semidirect
512. C₄×Q₁₆	C₂²×C₃₁	semidirect
513. C₄×SD₁₆	C₂²×C₃₁	semidirect
514. C₄×SD₁₆	C₂²×C₃₁	semidirect
515. C₄×D₈	C₂²×C₃₁	semidirect
669. C₃₁	#61324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#559	2
Q₈×#41×C₃₁	2
C₃₁×#3375	2
C₃₁×#3373	2
C₃₁×#3371	2
C₃₁×#3370	2
C₃₁×#3365	2
C₃₁×#3361	2
C₃₁×#3140	2
C₃₁×#3138	2
C₃₁×#3078	2
C₃₁×#3077	2
C₃₁×#3040	2
C₃₁×#3038	2
Q₈×M₁₆×C₃₁	2
C₄×C₃₁×#137	2
C₄×C₃₁×AGL(1,ℤ/8)	2
#61324	31

Subgroups — C31×#61324 #21518083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61324 #21518083