Subgroups — C₃₁×#61371 #21518130

Order 7936.

1750 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3501	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3388	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3386	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3385	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3381	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3381	C₂	1	4	yes	maximal, complemented
8	Q₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3163	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3082	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3081	C₂	1	2	yes	maximal
13	C₃₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3077	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#2949	C₂	1	2	yes	maximal
17	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
18	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	Q₈²×C₃₁	—	2	1	no	—
20	Q₈²×C₃₁	C₂²	1	2	yes	complemented
21	C₃₁×#557	—	2	1	no	—
22	C₃₁×#557	—	2	2	no	—
23	C₃₁×#557	C₂²	1	2	yes	complemented
24	C₃₁×#557	C₂²	1	4	yes	complemented
25	C₃₁×#554	—	2	1	no	—
26	C₃₁×#554	—	2	2	no	—
27	C₃₁×#554	C₂²	1	2	yes	—
28	C₃₁×#554	C₂²	1	4	yes	—
29	D₄×Q₈×C₃₁	—	2	1	no	—
30	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
31	C₃₁×#544	—	2	2	no	—
32	C₃₁×#544	C₂²	1	2	yes	—
33	C₃₁×#544	C₂²	1	2	yes	—
34	C₃₁×#543	—	2	2	no	—
35	C₃₁×#533	—	2	2	no	—
36	C₃₁×#533	C₂²	1	1	yes	characteristic
37	C₃₁×#533	C₂²	1	1	yes	characteristic
38	C₃₁×#533	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
40	C₃₁×#519	C₂²	1	2	yes	—
41	C₃₁×#518	C₂²	1	2	yes	—
42	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₃₁×#501	C₂²	1	4	yes	complemented
45	C₃₁×#501	C₂²	1	4	yes	complemented
46	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
47	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
48	C₃₁×#499	C₂²	1	2	yes	complemented
49	C₃₁×#499	C₂²	1	4	yes	complemented
50	C₃₁×#499	C₂²	1	2	yes	complemented
51	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
52	C₃₁×#498	C₂²	1	4	yes	complemented
53	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
54	C₃₁×#479	C₂²	1	4	yes	complemented
55	C₃₁×#479	C₂²	1	4	yes	complemented
56	C₃₁×#479	C₂²	1	4	yes	complemented
57	C₃₁×#478	C₂²	1	4	yes	—
58	C₃₁×#478	C₂²	1	4	yes	—
59	C₃₁×#478	C₂²	1	4	yes	—
60	C₃₁×#477	C₂²	1	2	yes	complemented
61	C₃₁×#477	C₂²	1	4	yes	complemented
62	C₃₁×#476	C₂²	1	2	yes	—
63	C₃₁×#476	C₂²	1	4	yes	—
64	C₃₁×#475	C₂²	1	2	yes	complemented
65	C₃₁×#475	C₂²	1	4	yes	complemented
66	C₃₁×#474	C₂²	1	2	yes	—
67	C₃₁×#474	C₂²	1	4	yes	—
68	C₃₁×#446	C₂²	1	2	yes	complemented
69	C₃₁×#446	C₂²	1	2	yes	complemented
70	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
71	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
72	C₃₁×#440	C₂²	1	4	yes	—
73	C₃₁×#440	C₂²	1	4	yes	—
74	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
75	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
76	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
77	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
78	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
79	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
80	C₃₁×#428	C₂²	1	2	yes	—
81	C₃₁×#428	C₂²	1	2	yes	—
82	C₃₁×#427	C₂²	1	2	yes	—
83	C₃₁×#427	C₂²	1	2	yes	—
84	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
85	C₃₁×#424	C₂²	1	2	yes	—
86	C₃₁×#423	C₂²	1	1	yes	characteristic
87	C₃₁×#419	C₂²	1	2	yes	—
88	C₃₁×#418	C₂²	1	2	yes	—
89	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
90	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
91	C₃₁×#405	C₂²	1	2	yes	—
92	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
93	C₂²×Q₈×C₃₁	—	2	2	no	—
94	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
96	C₂³×C₄×C₃₁	—	2	1	no	—
97	C₂×SD₁₆×C₃₁	Q₈	1	8	yes	complemented
98	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
99	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
100	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
101	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
103	C₃₁×C₄⋊Q₈	—	2	2	no	—
104	C₃₁×C₄⋊Q₈	—	2	4	no	—
105	C₃₁×C₄⋊Q₈	—	2	2	no	—
106	C₃₁×C₄⋊Q₈	—	2	4	no	—
107	C₃₁×C₄⋊Q₈	—	2	2	no	—
108	C₃₁×C₄⋊Q₈	—	2	2	no	—
109	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
110	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
111	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
113	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
114	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
115	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
116	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
117	C₃₁×#125	—	2	4	no	—
118	C₃₁×#125	—	2	4	no	—
119	C₃₁×#125	—	2	4	no	—
120	C₃₁×#125	—	2	2	no	—
121	C₃₁×#125	—	2	2	no	—
122	C₃₁×#125	D₄	1	2	yes	—
123	C₃₁×#125	C₂³	1	2	yes	—
124	C₃₁×#125	C₂³	1	4	yes	—
125	C₃₁×#125	C₂³	1	2	yes	—
126	C₃₁×#125	C₂³	1	4	yes	—
127	C₃₁×C₂²⋊Q₈	—	2	2	no	—
128	C₃₁×C₂²⋊Q₈	—	2	4	no	—
129	C₃₁×C₂²⋊Q₈	—	2	1	no	—
130	C₃₁×C₂²⋊Q₈	—	2	1	no	—
131	C₃₁×C₂²⋊Q₈	—	2	1	no	—
132	C₃₁×C₂²⋊Q₈	—	2	1	no	—
133	C₃₁×C₂²⋊Q₈	—	2	2	no	—
134	C₃₁×C₂²⋊Q₈	—	2	4	no	—
135	C₃₁×C₂²⋊Q₈	—	2	4	no	—
136	C₃₁×C₂²⋊Q₈	—	2	4	no	—
137	C₃₁×C₂²⋊Q₈	—	2	4	no	—
138	C₃₁×C₂²⋊Q₈	—	2	4	no	—
139	C₃₁×C₂²⋊Q₈	—	2	4	no	—
140	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
141	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
142	C₄×Q₈×C₃₁	—	2	2	no	—
143	C₄×Q₈×C₃₁	—	2	2	no	—
144	C₄×Q₈×C₃₁	—	2	2	no	—
145	C₄×Q₈×C₃₁	—	2	4	no	—
146	C₄×Q₈×C₃₁	—	2	2	no	—
147	C₄×Q₈×C₃₁	—	2	1	no	—
148	C₄×Q₈×C₃₁	—	2	1	no	—
149	C₄×Q₈×C₃₁	—	2	1	no	—
150	C₄×Q₈×C₃₁	—	2	1	no	—
151	C₄×Q₈×C₃₁	—	2	1	no	—
152	C₄×Q₈×C₃₁	—	2	1	no	—
153	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
154	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
155	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
156	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
157	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
158	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
159	C₄×D₄×C₃₁	—	2	2	no	—
160	C₄×D₄×C₃₁	—	2	2	no	—
161	C₄×D₄×C₃₁	—	2	1	no	—
162	C₄×D₄×C₃₁	—	2	1	no	—
163	C₄×D₄×C₃₁	C₂³	1	2	yes	—
164	C₄×D₄×C₃₁	C₂³	1	2	yes	—
165	C₄×D₄×C₃₁	C₂³	1	4	yes	—
166	C₄×D₄×C₃₁	C₂³	1	4	yes	—
167	C₃₁×#117	—	2	4	no	—
168	C₃₁×#117	C₂³	1	2	yes	—
169	C₃₁×#117	C₂³	1	2	yes	—
170	C₃₁×#117	C₂³	1	2	yes	—
171	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
175	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
176	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
177	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
178	C₂×C₄²×C₃₁	—	2	1	no	—
179	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
180	C₃₁×#107	C₂³	1	2	yes	—
181	C₃₁×#107	C₂³	1	2	yes	—
182	C₃₁×#107	C₂³	1	2	yes	—
183	C₃₁×#107	C₂³	1	2	yes	—
184	C₃₁×#107	C₂³	1	4	yes	—
185	C₃₁×#107	C₂³	1	4	yes	—
186	C₃₁×#106	C₂³	1	2	yes	—
187	C₃₁×#106	C₂³	1	2	yes	—
188	C₃₁×#106	C₂³	1	2	yes	—
189	C₃₁×#106	C₂³	1	2	yes	—
190	C₃₁×#106	C₂³	1	2	yes	—
191	C₃₁×#106	C₂³	1	2	yes	—
192	C₃₁×#106	C₂³	1	4	yes	—
193	C₃₁×#106	C₂³	1	4	yes	—
194	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
195	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
196	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
197	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
198	C₃₁×#103	C₂³	1	2	yes	—
199	C₃₁×#103	C₂³	1	2	yes	—
200	C₃₁×#103	C₂³	1	4	yes	—
201	C₃₁×#103	C₂³	1	4	yes	—
202	C₃₁×#102	C₂³	1	2	yes	—
203	C₃₁×#102	C₂³	1	2	yes	—
204	C₃₁×#102	C₂³	1	4	yes	—
205	C₃₁×#102	C₂³	1	4	yes	—
206	C₃₁×#97	C₂³	1	4	yes	—
207	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
208	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
209	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
210	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
211	C₄×C₈×C₃₁	C₂³	1	4	yes	—
212	C₂⁴×C₃₁	—	2	1	no	—
213	C₂×Q₈×C₃₁	—	2	1	no	—
214	C₂×Q₈×C₃₁	—	2	1	no	—
215	C₂×Q₈×C₃₁	—	2	1	no	—
216	C₂×Q₈×C₃₁	—	2	1	no	—
217	C₂×Q₈×C₃₁	—	2	1	no	—
218	C₂×Q₈×C₃₁	—	2	4	no	—
219	C₂×Q₈×C₃₁	—	2	4	no	—
220	C₂×Q₈×C₃₁	—	2	2	no	—
221	C₂×Q₈×C₃₁	—	8	4	no	—
222	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
223	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
224	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	complemented
225	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
226	C₂×D₄×C₃₁	—	2	1	no	—
227	C₂×D₄×C₃₁	—	2	2	no	—
228	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
229	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
230	C₂²×C₄×C₃₁	—	2	1	no	—
231	C₂²×C₄×C₃₁	—	2	1	no	—
232	C₂²×C₄×C₃₁	—	2	1	no	—
233	C₂²×C₄×C₃₁	—	2	2	no	—
234	C₂²×C₄×C₃₁	—	2	4	no	—
235	C₂²×C₄×C₃₁	—	2	4	no	—
236	C₂²×C₄×C₃₁	—	2	2	no	—
237	C₂²×C₄×C₃₁	—	8	2	no	—
238	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
239	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
240	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
241	SD₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
242	SD₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
243	M₁₆×C₃₁	—	2	4	no	—
244	C₂×C₈×C₃₁	—	2	2	no	—
245	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
246	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
247	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
248	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
249	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
250	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
251	C₄⋊C₄×C₃₁	—	2	2	no	—
252	C₄⋊C₄×C₃₁	—	2	1	no	—
253	C₄⋊C₄×C₃₁	—	2	2	no	—
254	C₄⋊C₄×C₃₁	—	2	2	no	—
255	C₄⋊C₄×C₃₁	—	2	4	no	—
256	C₄⋊C₄×C₃₁	—	2	4	no	—
257	C₄⋊C₄×C₃₁	—	2	4	no	—
258	C₄⋊C₄×C₃₁	—	2	1	no	—
259	C₄⋊C₄×C₃₁	—	2	1	no	—
260	C₄⋊C₄×C₃₁	—	2	4	no	—
261	C₄⋊C₄×C₃₁	—	2	4	no	—
262	C₄⋊C₄×C₃₁	—	2	2	no	—
263	C₄⋊C₄×C₃₁	—	2	1	no	—
264	C₄⋊C₄×C₃₁	—	2	2	no	—
265	C₄⋊C₄×C₃₁	—	2	1	no	—
266	C₄⋊C₄×C₃₁	—	2	1	no	—
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	4	no	—
271	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
272	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
273	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
274	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
275	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
276	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
277	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
278	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
279	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
280	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
281	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
282	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
283	C₂²⋊C₄×C₃₁	—	2	2	no	—
284	C₂²⋊C₄×C₃₁	—	2	2	no	—
285	C₂²⋊C₄×C₃₁	—	2	4	no	—
286	C₂²⋊C₄×C₃₁	—	2	4	no	—
287	C₂²⋊C₄×C₃₁	—	2	2	no	—
288	C₂²⋊C₄×C₃₁	—	2	2	no	—
289	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
290	C₄²×C₃₁	—	2	2	no	—
291	C₄²×C₃₁	—	2	2	no	—
292	C₄²×C₃₁	—	2	4	no	—
293	C₄²×C₃₁	—	2	4	no	—
294	C₄²×C₃₁	—	2	2	no	—
295	C₄²×C₃₁	—	2	2	no	—
296	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
297	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
298	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
299	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
300	C₄²×C₃₁	C₂⁴	1	4	yes	—
301	#61371	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₂³×C₃₁	—	2	2	no	—
303	C₂³×C₃₁	—	2	4	no	—
304	C₂³×C₃₁	—	8	1	no	—
305	C₂³×C₃₁	—	8	1	no	—
306	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
307	Q₈×C₃₁	—	2	4	no	—
308	Q₈×C₃₁	—	2	4	no	—
309	Q₈×C₃₁	—	2	2	no	—
310	Q₈×C₃₁	—	4	2	no	—
311	Q₈×C₃₁	—	4	2	no	—
312	Q₈×C₃₁	—	4	2	no	—
313	Q₈×C₃₁	—	8	1	no	—
314	Q₈×C₃₁	—	8	1	no	—
315	Q₈×C₃₁	—	8	4	no	—
316	Q₈×C₃₁	—	8	4	no	—
317	Q₈×C₃₁	C₂²×Q₈	1	4	yes	—
318	D₄×C₃₁	—	2	4	no	—
319	D₄×C₃₁	—	2	2	no	—
320	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
321	C₂×C₄×C₃₁	—	2	4	no	—
322	C₂×C₄×C₃₁	—	2	2	no	—
323	C₂×C₄×C₃₁	—	2	2	no	—
324	C₂×C₄×C₃₁	—	2	2	no	—
325	C₂×C₄×C₃₁	—	2	2	no	—
326	C₂×C₄×C₃₁	—	2	2	no	—
327	C₂×C₄×C₃₁	—	2	2	no	—
328	C₂×C₄×C₃₁	—	2	2	no	—
329	C₂×C₄×C₃₁	—	2	2	no	—
330	C₂×C₄×C₃₁	—	4	1	no	—
331	C₂×C₄×C₃₁	—	4	4	no	—
332	C₂×C₄×C₃₁	—	8	4	no	—
333	C₂×C₄×C₃₁	—	8	2	no	—
334	C₂×C₄×C₃₁	—	8	4	no	—
335	C₂×C₄×C₃₁	—	8	1	no	—
336	C₂×C₄×C₃₁	—	8	1	no	—
337	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
338	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
339	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
340	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
341	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
342	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
343	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
344	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
345	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
346	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
347	C₈×C₃₁	—	2	1	no	—
348	C₈×C₃₁	—	2	1	no	—
349	C₈×C₃₁	—	2	4	no	—
350	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
351	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
352	#3502	C₂×C₃₁	1	1	yes	characteristic, complemented
353	#3501	C₂×C₃₁	1	1	yes	characteristic
354	#3388	C₂×C₃₁	1	4	yes	complemented
355	#3386	C₂×C₃₁	1	2	yes	complemented
356	#3385	C₂×C₃₁	1	2	yes	complemented
357	#3381	C₂×C₃₁	1	2	yes	complemented
358	#3381	C₂×C₃₁	1	4	yes	complemented
359	Q₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
360	#3163	C₂×C₃₁	1	2	yes	complemented
361	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
362	#3082	C₂×C₃₁	1	2	yes	complemented
363	#3081	C₂×C₃₁	1	2	yes	—
364	#3078	C₂×C₃₁	1	1	yes	characteristic, complemented
365	#3077	C₂×C₃₁	1	1	yes	characteristic
366	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
367	#2949	C₂×C₃₁	1	2	yes	—
368	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
369	C₂²×C₃₁	—	2	1	no	—
370	C₂²×C₃₁	—	2	1	no	—
371	C₂²×C₃₁	—	2	4	no	—
372	C₂²×C₃₁	—	4	2	no	—
373	C₂²×C₃₁	—	4	2	no	—
374	C₂²×C₃₁	—	8	4	no	—
375	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
376	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
377	C₄×C₃₁	—	2	1	no	—
378	C₄×C₃₁	—	2	1	no	—
379	C₄×C₃₁	—	2	4	no	—
380	C₄×C₃₁	—	2	4	no	—
381	C₄×C₃₁	—	2	1	no	—
382	C₄×C₃₁	—	2	4	no	—
383	C₄×C₃₁	—	4	2	no	—
384	C₄×C₃₁	—	4	2	no	—
385	C₄×C₃₁	—	4	2	no	—
386	C₄×C₃₁	—	4	2	no	—
387	C₄×C₃₁	—	4	2	no	—
388	C₄×C₃₁	—	4	2	no	—
389	C₄×C₃₁	D₄×Q₈	1	2	yes	—
390	C₄×C₃₁	#577	1	2	yes	—
391	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
392	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
393	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
394	Q₈²	—	2	1	no	—
395	Q₈²	C₂²×C₃₁	1	2	yes	complemented
396	#557	—	2	1	no	—
397	#557	—	2	2	no	—
398	#557	C₂²×C₃₁	1	2	yes	complemented
399	#557	C₂²×C₃₁	1	4	yes	complemented
400	#554	—	2	1	no	—
401	#554	—	2	2	no	—
402	#554	C₂²×C₃₁	1	2	yes	—
403	#554	C₂²×C₃₁	1	4	yes	—
404	D₄×Q₈	—	2	1	no	—
405	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
406	#544	—	2	2	no	—
407	#544	C₂²×C₃₁	1	2	yes	—
408	#544	C₂²×C₃₁	1	2	yes	—
409	#543	—	2	2	no	—
410	#533	—	2	2	no	—
411	#533	C₂²×C₃₁	1	1	yes	characteristic
412	#533	C₂²×C₃₁	1	1	yes	characteristic
413	#533	C₂²×C₃₁	1	1	yes	characteristic
414	C₂×C₂²⋊Q₈	—	2	2	no	—
415	#519	C₂²×C₃₁	1	2	yes	—
416	#518	C₂²×C₃₁	1	2	yes	—
417	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
418	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
419	#501	C₂²×C₃₁	1	4	yes	complemented
420	#501	C₂²×C₃₁	1	4	yes	complemented
421	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
422	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
423	#499	C₂²×C₃₁	1	2	yes	complemented
424	#499	C₂²×C₃₁	1	4	yes	complemented
425	#499	C₂²×C₃₁	1	2	yes	complemented
426	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
427	#498	C₂²×C₃₁	1	4	yes	complemented
428	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
429	#479	C₂²×C₃₁	1	4	yes	complemented
430	#479	C₂²×C₃₁	1	4	yes	complemented
431	#479	C₂²×C₃₁	1	4	yes	complemented
432	#478	C₂²×C₃₁	1	4	yes	—
433	#478	C₂²×C₃₁	1	4	yes	—
434	#478	C₂²×C₃₁	1	4	yes	—
435	#477	C₂²×C₃₁	1	2	yes	complemented
436	#477	C₂²×C₃₁	1	4	yes	complemented
437	#476	C₂²×C₃₁	1	2	yes	—
438	#476	C₂²×C₃₁	1	4	yes	—
439	#475	C₂²×C₃₁	1	2	yes	complemented
440	#475	C₂²×C₃₁	1	4	yes	complemented
441	#474	C₂²×C₃₁	1	2	yes	—
442	#474	C₂²×C₃₁	1	4	yes	—
443	#446	C₂²×C₃₁	1	2	yes	complemented
444	#446	C₂²×C₃₁	1	2	yes	complemented
445	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
446	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
447	#440	C₂²×C₃₁	1	4	yes	—
448	#440	C₂²×C₃₁	1	4	yes	—
449	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
450	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
451	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
452	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
453	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
454	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
455	#428	C₂²×C₃₁	1	2	yes	—
456	#428	C₂²×C₃₁	1	2	yes	—
457	#427	C₂²×C₃₁	1	2	yes	—
458	#427	C₂²×C₃₁	1	2	yes	—
459	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
460	#424	C₂²×C₃₁	1	2	yes	—
461	#423	C₂²×C₃₁	1	1	yes	characteristic
462	#419	C₂²×C₃₁	1	2	yes	—
463	#418	C₂²×C₃₁	1	2	yes	—
464	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
465	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
466	#405	C₂²×C₃₁	1	2	yes	—
467	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
468	C₂×C₃₁	—	2	2	no	—
469	C₂×C₃₁	—	4	4	no	—
470	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
471	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
472	C₂×C₃₁	#3584	1	1	yes	characteristic
473	C₂²×Q₈	—	2	2	no	—
474	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
475	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
476	C₂³×C₄	—	2	1	no	—
477	C₂×SD₁₆	Q₈×C₃₁	1	8	yes	complemented
478	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
479	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
480	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
481	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
482	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
483	C₄⋊Q₈	—	2	2	no	—
484	C₄⋊Q₈	—	2	4	no	—
485	C₄⋊Q₈	—	2	2	no	—
486	C₄⋊Q₈	—	2	4	no	—
487	C₄⋊Q₈	—	2	2	no	—
488	C₄⋊Q₈	—	2	2	no	—
489	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
490	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
491	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
492	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
493	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
494	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
495	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
496	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
497	#125	—	2	4	no	—
498	#125	—	2	4	no	—
499	#125	—	2	4	no	—
500	#125	—	2	2	no	—
501	#125	—	2	2	no	—
502	#125	D₄×C₃₁	1	2	yes	—
503	#125	C₂³×C₃₁	1	2	yes	—
504	#125	C₂³×C₃₁	1	2	yes	—
505	#125	C₂³×C₃₁	1	4	yes	—
506	#125	C₂³×C₃₁	1	4	yes	—
507	C₂²⋊Q₈	—	2	2	no	—
508	C₂²⋊Q₈	—	2	4	no	—
509	C₂²⋊Q₈	—	2	1	no	—
510	C₂²⋊Q₈	—	2	1	no	—
511	C₂²⋊Q₈	—	2	1	no	—
512	C₂²⋊Q₈	—	2	1	no	—
513	C₂²⋊Q₈	—	2	2	no	—
514	C₂²⋊Q₈	—	2	4	no	—
515	C₂²⋊Q₈	—	2	4	no	—
516	C₂²⋊Q₈	—	2	4	no	—
517	C₂²⋊Q₈	—	2	4	no	—
518	C₂²⋊Q₈	—	2	4	no	—
519	C₂²⋊Q₈	—	2	4	no	—
520	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
521	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
522	C₄×Q₈	—	2	2	no	—
523	C₄×Q₈	—	2	2	no	—
524	C₄×Q₈	—	2	2	no	—
525	C₄×Q₈	—	2	4	no	—
526	C₄×Q₈	—	2	2	no	—
527	C₄×Q₈	—	2	1	no	—
528	C₄×Q₈	—	2	1	no	—
529	C₄×Q₈	—	2	1	no	—
530	C₄×Q₈	—	2	1	no	—
531	C₄×Q₈	—	2	1	no	—
532	C₄×Q₈	—	2	1	no	—
533	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
534	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
535	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
536	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
537	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
538	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
539	C₄×D₄	—	2	2	no	—
540	C₄×D₄	—	2	2	no	—
541	C₄×D₄	—	2	1	no	—
542	C₄×D₄	—	2	1	no	—
543	C₄×D₄	C₂³×C₃₁	1	2	yes	—
544	C₄×D₄	C₂³×C₃₁	1	2	yes	—
545	C₄×D₄	C₂³×C₃₁	1	4	yes	—
546	C₄×D₄	C₂³×C₃₁	1	4	yes	—
547	#117	—	2	4	no	—
548	#117	C₂³×C₃₁	1	2	yes	—
549	#117	C₂³×C₃₁	1	2	yes	—
550	#117	C₂³×C₃₁	1	2	yes	—
551	C₂×C₄⋊C₄	—	2	1	no	—
552	C₂×C₄⋊C₄	—	2	4	no	—
553	C₂×C₄⋊C₄	—	2	2	no	—
554	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
555	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
556	C₂×C₂²⋊C₄	—	2	1	no	—
557	C₂×C₂²⋊C₄	—	2	4	no	—
558	C₂×C₄²	—	2	1	no	—
559	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
560	#107	C₂³×C₃₁	1	2	yes	—
561	#107	C₂³×C₃₁	1	2	yes	—
562	#107	C₂³×C₃₁	1	2	yes	—
563	#107	C₂³×C₃₁	1	2	yes	—
564	#107	C₂³×C₃₁	1	4	yes	—
565	#107	C₂³×C₃₁	1	4	yes	—
566	#106	C₂³×C₃₁	1	2	yes	—
567	#106	C₂³×C₃₁	1	2	yes	—
568	#106	C₂³×C₃₁	1	2	yes	—
569	#106	C₂³×C₃₁	1	2	yes	—
570	#106	C₂³×C₃₁	1	2	yes	—
571	#106	C₂³×C₃₁	1	2	yes	—
572	#106	C₂³×C₃₁	1	4	yes	—
573	#106	C₂³×C₃₁	1	4	yes	—
574	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
575	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
576	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
577	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
578	#103	C₂³×C₃₁	1	2	yes	—
579	#103	C₂³×C₃₁	1	2	yes	—
580	#103	C₂³×C₃₁	1	4	yes	—
581	#103	C₂³×C₃₁	1	4	yes	—
582	#102	C₂³×C₃₁	1	2	yes	—
583	#102	C₂³×C₃₁	1	2	yes	—
584	#102	C₂³×C₃₁	1	4	yes	—
585	#102	C₂³×C₃₁	1	4	yes	—
586	#97	C₂³×C₃₁	1	4	yes	—
587	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
588	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
589	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
590	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
591	C₄×C₈	C₂³×C₃₁	1	4	yes	—
592	C₃₁	#61371	1	1	yes	characteristic, direct product factor, Sylow
593	C₂⁴	—	2	1	no	—
594	C₂×Q₈	—	2	1	no	—
595	C₂×Q₈	—	2	1	no	—
596	C₂×Q₈	—	2	1	no	—
597	C₂×Q₈	—	2	1	no	—
598	C₂×Q₈	—	2	1	no	—
599	C₂×Q₈	—	2	4	no	—
600	C₂×Q₈	—	2	4	no	—
601	C₂×Q₈	—	2	2	no	—
602	C₂×Q₈	—	8	4	no	—
603	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
604	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
605	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	complemented
606	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
607	C₂×D₄	—	2	1	no	—
608	C₂×D₄	—	2	2	no	—
609	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
610	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
611	C₂²×C₄	—	2	1	no	—
612	C₂²×C₄	—	2	1	no	—
613	C₂²×C₄	—	2	1	no	—
614	C₂²×C₄	—	2	2	no	—
615	C₂²×C₄	—	2	4	no	—
616	C₂²×C₄	—	2	4	no	—
617	C₂²×C₄	—	2	2	no	—
618	C₂²×C₄	—	8	2	no	—
619	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
620	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
621	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
622	SD₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
623	SD₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
624	M₁₆	—	2	4	no	—
625	C₂×C₈	—	2	2	no	—
626	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
627	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
628	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
629	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
630	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
631	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
632	C₄⋊C₄	—	2	2	no	—
633	C₄⋊C₄	—	2	1	no	—
634	C₄⋊C₄	—	2	2	no	—
635	C₄⋊C₄	—	2	2	no	—
636	C₄⋊C₄	—	2	4	no	—
637	C₄⋊C₄	—	2	4	no	—
638	C₄⋊C₄	—	2	4	no	—
639	C₄⋊C₄	—	2	1	no	—
640	C₄⋊C₄	—	2	1	no	—
641	C₄⋊C₄	—	2	4	no	—
642	C₄⋊C₄	—	2	4	no	—
643	C₄⋊C₄	—	2	2	no	—
644	C₄⋊C₄	—	2	1	no	—
645	C₄⋊C₄	—	2	2	no	—
646	C₄⋊C₄	—	2	1	no	—
647	C₄⋊C₄	—	2	1	no	—
648	C₄⋊C₄	—	2	4	no	—
649	C₄⋊C₄	—	2	4	no	—
650	C₄⋊C₄	—	2	2	no	—
651	C₄⋊C₄	—	2	4	no	—
652	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
653	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
654	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
655	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
656	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
657	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
658	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
659	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
660	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
661	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
662	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
663	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
664	C₂²⋊C₄	—	2	2	no	—
665	C₂²⋊C₄	—	2	2	no	—
666	C₂²⋊C₄	—	2	4	no	—
667	C₂²⋊C₄	—	2	4	no	—
668	C₂²⋊C₄	—	2	2	no	—
669	C₂²⋊C₄	—	2	2	no	—
670	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
671	C₄²	—	2	2	no	—
672	C₄²	—	2	2	no	—
673	C₄²	—	2	4	no	—
674	C₄²	—	2	4	no	—
675	C₄²	—	2	2	no	—
676	C₄²	—	2	2	no	—
677	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
678	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
679	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
680	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
681	C₄²	C₂⁴×C₃₁	1	4	yes	—
682	C₂³	—	2	2	no	—
683	C₂³	—	2	4	no	—
684	C₂³	—	8	1	no	—
685	C₂³	—	8	1	no	—
686	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
687	Q₈	—	2	4	no	—
688	Q₈	—	2	4	no	—
689	Q₈	—	2	2	no	—
690	Q₈	—	4	2	no	—
691	Q₈	—	4	2	no	—
692	Q₈	—	4	2	no	—
693	Q₈	—	8	1	no	—
694	Q₈	—	8	1	no	—
695	Q₈	—	8	4	no	—
696	Q₈	—	8	4	no	—
697	Q₈	C₂²×Q₈×C₃₁	1	4	yes	—
698	D₄	—	2	4	no	—
699	D₄	—	2	2	no	—
700	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
701	C₂×C₄	—	2	4	no	—
702	C₂×C₄	—	2	2	no	—
703	C₂×C₄	—	2	2	no	—
704	C₂×C₄	—	2	2	no	—
705	C₂×C₄	—	2	2	no	—
706	C₂×C₄	—	2	2	no	—
707	C₂×C₄	—	2	2	no	—
708	C₂×C₄	—	2	2	no	—
709	C₂×C₄	—	2	2	no	—
710	C₂×C₄	—	4	1	no	—
711	C₂×C₄	—	4	4	no	—
712	C₂×C₄	—	8	4	no	—
713	C₂×C₄	—	8	2	no	—
714	C₂×C₄	—	8	4	no	—
715	C₂×C₄	—	8	1	no	—
716	C₂×C₄	—	8	1	no	—
717	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
718	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
719	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
720	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
721	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
722	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
723	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
724	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
725	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
726	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
727	C₈	—	2	1	no	—
728	C₈	—	2	1	no	—
729	C₈	—	2	4	no	—
730	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
731	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
732	C₂²	—	2	1	no	—
733	C₂²	—	2	1	no	—
734	C₂²	—	2	4	no	—
735	C₂²	—	4	2	no	—
736	C₂²	—	4	2	no	—
737	C₂²	—	8	4	no	—
738	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
739	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
740	C₄	—	2	4	no	—
741	C₄	—	2	1	no	—
742	C₄	—	2	4	no	—
743	C₄	—	2	1	no	—
744	C₄	—	2	1	no	—
745	C₄	—	2	4	no	—
746	C₄	—	4	2	no	—
747	C₄	—	4	2	no	—
748	C₄	—	4	2	no	—
749	C₄	—	4	2	no	—
750	C₄	—	4	2	no	—
751	C₄	—	4	2	no	—
752	C₄	D₄×Q₈×C₃₁	1	2	yes	—
753	C₄	C₃₁×#577	1	2	yes	—
754	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
755	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
756	C₂	—	2	2	no	—
757	C₂	—	4	4	no	—
758	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
759	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
760	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3502	C₂	semidirect
3. C₃₁×#3388	C₂	semidirect
4. C₃₁×#3386	C₂	semidirect
5. C₃₁×#3385	C₂	semidirect
6. C₃₁×#3381	C₂	semidirect
7. C₃₁×#3381	C₂	semidirect
8. Q₈×SD₁₆×C₃₁	C₂	semidirect
9. C₃₁×#3163	C₂	semidirect
10. C₃₁×#3160	C₂	semidirect
11. C₃₁×#3082	C₂	semidirect
13. C₃₁×#3078	C₂	semidirect
15. C₃₁×#2960	C₂	semidirect
20. Q₈²×C₃₁	C₂²	semidirect
23. C₃₁×#557	C₂²	semidirect
24. C₃₁×#557	C₂²	semidirect
44. C₃₁×#501	C₂²	semidirect
45. C₃₁×#501	C₂²	semidirect
46. C₃₁×#500	C₂²	semidirect
47. C₃₁×#500	C₂²	semidirect
48. C₃₁×#499	C₂²	semidirect
49. C₃₁×#499	C₂²	semidirect
50. C₃₁×#499	C₂²	semidirect
51. C₃₁×#498	C₂²	semidirect
52. C₃₁×#498	C₂²	semidirect
53. C₃₁×#498	C₂²	semidirect
54. C₃₁×#479	C₂²	semidirect
55. C₃₁×#479	C₂²	semidirect
56. C₃₁×#479	C₂²	semidirect
60. C₃₁×#477	C₂²	semidirect
61. C₃₁×#477	C₂²	semidirect
64. C₃₁×#475	C₂²	semidirect
65. C₃₁×#475	C₂²	semidirect
68. C₃₁×#446	C₂²	semidirect
69. C₃₁×#446	C₂²	semidirect
70. C₈×Q₈×C₃₁	C₂²	semidirect
71. C₈×Q₈×C₃₁	C₂²	semidirect
97. C₂×SD₁₆×C₃₁	Q₈	semidirect
224. C₂×Q₈×C₃₁	C₂×Q₈	semidirect
241. SD₁₆×C₃₁	C₂×Q₈	semidirect
242. SD₁₆×C₃₁	C₂×Q₈	semidirect
245. C₂×C₈×C₃₁	C₂×Q₈	semidirect
301. #61371	C₃₁	direct (N × H)
352. #3502	C₂×C₃₁	semidirect
354. #3388	C₂×C₃₁	semidirect
355. #3386	C₂×C₃₁	semidirect
356. #3385	C₂×C₃₁	semidirect
357. #3381	C₂×C₃₁	semidirect
358. #3381	C₂×C₃₁	semidirect
359. Q₈×SD₁₆	C₂×C₃₁	semidirect
360. #3163	C₂×C₃₁	semidirect
361. #3160	C₂×C₃₁	semidirect
362. #3082	C₂×C₃₁	semidirect
364. #3078	C₂×C₃₁	semidirect
366. #2960	C₂×C₃₁	semidirect
395. Q₈²	C₂²×C₃₁	semidirect
398. #557	C₂²×C₃₁	semidirect
399. #557	C₂²×C₃₁	semidirect
419. #501	C₂²×C₃₁	semidirect
420. #501	C₂²×C₃₁	semidirect
421. #500	C₂²×C₃₁	semidirect
422. #500	C₂²×C₃₁	semidirect
423. #499	C₂²×C₃₁	semidirect
424. #499	C₂²×C₃₁	semidirect
425. #499	C₂²×C₃₁	semidirect
426. #498	C₂²×C₃₁	semidirect
427. #498	C₂²×C₃₁	semidirect
428. #498	C₂²×C₃₁	semidirect
429. #479	C₂²×C₃₁	semidirect
430. #479	C₂²×C₃₁	semidirect
431. #479	C₂²×C₃₁	semidirect
435. #477	C₂²×C₃₁	semidirect
436. #477	C₂²×C₃₁	semidirect
439. #475	C₂²×C₃₁	semidirect
440. #475	C₂²×C₃₁	semidirect
443. #446	C₂²×C₃₁	semidirect
444. #446	C₂²×C₃₁	semidirect
445. C₈×Q₈	C₂²×C₃₁	semidirect
446. C₈×Q₈	C₂²×C₃₁	semidirect
477. C₂×SD₁₆	Q₈×C₃₁	semidirect
592. C₃₁	#61371	direct (N × H)
605. C₂×Q₈	C₂×Q₈×C₃₁	semidirect
622. SD₁₆	C₂×Q₈×C₃₁	semidirect
623. SD₁₆	C₂×Q₈×C₃₁	semidirect
626. C₂×C₈	C₂×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3502	2
C₃₁×#3501	2
C₃₁×#3388	2
C₃₁×#3386	2
C₃₁×#3385	2
C₃₁×#3381	2
Q₈×SD₁₆×C₃₁	2
C₃₁×#3163	2
C₃₁×#3160	2
C₃₁×#3082	2
C₃₁×#3081	2
C₃₁×#3078	2
C₃₁×#3077	2
C₃₁×#2960	2
C₃₁×#2949	2
C₂×C₄×SD₁₆×C₃₁	2
#61371	31

Subgroups — C31×#61371 #21518130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61371 #21518130