Subgroups — C₃₁×#61382 #21518141

Order 7936.

1794 conjugacy classes of subgroups.

The subgroup lattice has 792 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3501	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3388	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3385	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3384	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3384	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3380	C₂	1	2	yes	maximal, complemented
8	Q₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3163	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3084	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3081	C₂	1	2	yes	maximal, complemented
13	C₃₁×#3076	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#2962	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#2946	C₂	1	2	yes	maximal
17	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
18	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
19	Q₈²×C₃₁	C₂²	1	2	yes	complemented
20	C₃₁×#557	C₂²	1	2	yes	complemented
21	C₃₁×#554	—	2	1	no	—
22	C₃₁×#554	—	2	2	no	—
23	C₃₁×#554	—	2	2	no	—
24	C₃₁×#554	C₂²	1	2	yes	—
25	C₃₁×#554	C₂²	1	4	yes	—
26	C₃₁×#554	C₂²	1	2	yes	complemented
27	D₄×Q₈×C₃₁	—	2	1	no	—
28	D₄×Q₈×C₃₁	—	2	2	no	—
29	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
30	D₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
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	1	no	—
36	C₃₁×#533	C₂²	1	1	yes	characteristic
37	C₃₁×#533	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
39	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
41	C₃₁×#520	C₂²	1	2	yes	—
42	C₃₁×#518	C₂²	1	2	yes	—
43	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
44	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#501	C₂²	1	4	yes	complemented
46	C₃₁×#501	C₂²	1	4	yes	complemented
47	C₃₁×#501	C₂²	1	4	yes	complemented
48	C₃₁×#500	C₂²	1	2	yes	complemented
49	C₃₁×#499	C₂²	1	4	yes	complemented
50	C₃₁×#498	C₂²	1	4	yes	complemented
51	C₃₁×#498	C₂²	1	2	yes	complemented
52	C₃₁×#479	C₂²	1	2	yes	complemented
53	C₃₁×#479	C₂²	1	4	yes	complemented
54	C₃₁×#478	C₂²	1	4	yes	—
55	C₃₁×#478	C₂²	1	4	yes	—
56	C₃₁×#478	C₂²	1	2	yes	complemented
57	C₃₁×#478	C₂²	1	4	yes	complemented
58	C₃₁×#477	C₂²	1	1	yes	characteristic, complemented
59	C₃₁×#476	C₂²	1	4	yes	—
60	C₃₁×#476	C₂²	1	2	yes	—
61	C₃₁×#476	C₂²	1	2	yes	—
62	C₃₁×#476	C₂²	1	4	yes	—
63	C₃₁×#476	C₂²	1	1	yes	characteristic, complemented
64	C₃₁×#476	C₂²	1	4	yes	complemented
65	C₃₁×#475	C₂²	1	1	yes	characteristic, complemented
66	C₃₁×#475	C₂²	1	4	yes	complemented
67	C₃₁×#474	C₂²	1	4	yes	—
68	C₃₁×#474	C₂²	1	1	yes	characteristic, complemented
69	C₃₁×#446	C₂²	1	2	yes	complemented
70	C₃₁×#446	C₂²	1	4	yes	complemented
71	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
72	C₃₁×#442	C₂²	1	2	yes	—
73	C₃₁×#442	C₂²	1	4	yes	—
74	C₃₁×#440	C₂²	1	4	yes	—
75	C₃₁×#440	C₂²	1	2	yes	—
76	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
77	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
78	C₄×D₈×C₃₁	C₂²	1	2	yes	—
79	C₄×D₈×C₃₁	C₂²	1	4	yes	—
80	C₃₁×#428	C₂²	1	2	yes	—
81	C₃₁×#428	C₂²	1	1	yes	characteristic
82	C₃₁×#428	C₂²	1	2	yes	—
83	C₃₁×#427	C₂²	1	2	yes	—
84	C₂×C₃₁×#106	C₂²	1	2	yes	—
85	C₃₁×#424	C₂²	1	2	yes	—
86	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
87	C₃₁×#418	C₂²	1	1	yes	characteristic
88	C₃₁×#417	C₂²	1	1	yes	characteristic
89	C₃₁×#416	C₂²	1	2	yes	—
90	C₂×C₃₁×#102	C₂²	1	2	yes	—
91	C₃₁×#405	C₂²	1	2	yes	—
92	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
93	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
94	C₂²×Q₈×C₃₁	—	2	2	no	—
95	C₂²×Q₈×C₃₁	—	2	2	no	—
96	C₂²×Q₈×C₃₁	—	2	1	no	—
97	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
98	C₂³×C₄×C₃₁	—	2	1	no	—
99	C₃₁×AGL(1,ℤ/8)	Q₈	1	8	yes	complemented
100	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
101	C₂×D₈×C₃₁	C₂³	1	2	yes	—
102	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
105	C₃₁×C₄⋊Q₈	—	2	2	no	—
106	C₃₁×C₄⋊Q₈	—	2	2	no	—
107	C₃₁×C₄⋊Q₈	—	2	2	no	—
108	C₃₁×C₄⋊Q₈	—	2	4	no	—
109	C₃₁×C₄⋊Q₈	—	2	2	no	—
110	C₃₁×C₄⋊Q₈	—	2	1	no	—
111	C₃₁×C₄⋊Q₈	—	2	1	no	—
112	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
113	C₃₁×C₄⋊Q₈	D₄	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	4	yes	—
117	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
118	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
119	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
120	C₃₁×#125	—	2	2	no	—
121	C₃₁×#125	—	2	4	no	—
122	C₃₁×#125	—	2	2	no	—
123	C₃₁×#125	D₄	1	2	yes	—
124	C₃₁×#125	C₂³	1	4	yes	—
125	C₃₁×#125	C₂³	1	4	yes	—
126	C₃₁×#125	C₂³	1	2	yes	—
127	C₃₁×C₂²⋊Q₈	—	2	2	no	—
128	C₃₁×C₂²⋊Q₈	—	2	2	no	—
129	C₃₁×C₂²⋊Q₈	—	2	4	no	—
130	C₃₁×C₂²⋊Q₈	—	2	1	no	—
131	C₃₁×C₂²⋊Q₈	—	2	1	no	—
132	C₃₁×C₂²⋊Q₈	—	2	2	no	—
133	C₃₁×C₂²⋊Q₈	—	2	4	no	—
134	C₃₁×C₂²⋊Q₈	—	2	2	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	—	2	4	no	—
141	C₃₁×C₂²⋊Q₈	—	2	4	no	—
142	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
143	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
144	C₄×Q₈×C₃₁	—	2	2	no	—
145	C₄×Q₈×C₃₁	—	2	2	no	—
146	C₄×Q₈×C₃₁	—	2	1	no	—
147	C₄×Q₈×C₃₁	—	2	1	no	—
148	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
149	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
150	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
151	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
152	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
153	C₄×D₄×C₃₁	—	2	1	no	—
154	C₄×D₄×C₃₁	—	2	1	no	—
155	C₄×D₄×C₃₁	—	2	2	no	—
156	C₄×D₄×C₃₁	—	2	2	no	—
157	C₄×D₄×C₃₁	—	2	4	no	—
158	C₄×D₄×C₃₁	—	2	1	no	—
159	C₄×D₄×C₃₁	—	2	1	no	—
160	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
161	C₄×D₄×C₃₁	C₂³	1	2	yes	—
162	C₄×D₄×C₃₁	C₂³	1	4	yes	—
163	C₄×D₄×C₃₁	C₂³	1	4	yes	—
164	C₄×D₄×C₃₁	C₂³	1	4	yes	—
165	C₄×D₄×C₃₁	C₂³	1	2	yes	—
166	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
167	C₃₁×#117	—	2	2	no	—
168	C₃₁×#117	—	2	1	no	—
169	C₃₁×#117	C₂³	1	2	yes	—
170	C₃₁×#117	C₂³	1	1	yes	characteristic
171	C₃₁×#117	C₂³	1	2	yes	—
172	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
175	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
176	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
178	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
179	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
180	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₂×C₄²×C₃₁	—	2	1	no	—
182	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
183	C₃₁×#107	C₂³	1	2	yes	—
184	C₃₁×#107	C₂³	1	2	yes	—
185	C₃₁×#107	C₂³	1	2	yes	—
186	C₃₁×#107	C₂³	1	4	yes	—
187	C₃₁×#107	C₂³	1	4	yes	—
188	C₃₁×#106	C₂³	1	4	yes	—
189	C₃₁×#106	C₂³	1	4	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	2	yes	—
194	C₃₁×#106	C₂³	1	4	yes	—
195	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
196	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
197	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
198	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
199	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
200	C₃₁×#103	C₂³	1	2	yes	—
201	C₃₁×#103	C₂³	1	2	yes	—
202	C₃₁×#103	C₂³	1	2	yes	—
203	C₃₁×#102	C₂³	1	4	yes	—
204	C₃₁×#102	C₂³	1	4	yes	—
205	C₃₁×#102	C₂³	1	2	yes	—
206	C₃₁×#102	C₂³	1	2	yes	—
207	C₃₁×#102	C₂³	1	2	yes	—
208	C₃₁×#102	C₂³	1	2	yes	—
209	C₃₁×#102	C₂³	1	2	yes	—
210	C₃₁×#97	C₂³	1	2	yes	—
211	C₃₁×#97	C₂³	1	4	yes	—
212	C₄×C₈×C₃₁	C₂³	1	2	yes	—
213	C₄×C₈×C₃₁	C₂³	1	4	yes	—
214	C₂⁴×C₃₁	—	2	1	no	—
215	#41×C₃₁	—	2	2	no	—
216	#41×C₃₁	C₂×Q₈	1	4	yes	complemented
217	C₂×Q₈×C₃₁	—	2	1	no	—
218	C₂×Q₈×C₃₁	—	2	2	no	—
219	C₂×Q₈×C₃₁	—	2	1	no	—
220	C₂×Q₈×C₃₁	—	2	4	no	—
221	C₂×Q₈×C₃₁	—	2	4	no	—
222	C₂×Q₈×C₃₁	—	2	2	no	—
223	C₂×Q₈×C₃₁	—	8	4	no	—
224	C₂×Q₈×C₃₁	—	8	2	no	—
225	C₂×Q₈×C₃₁	—	8	4	no	—
226	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
227	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
228	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
229	C₂×D₄×C₃₁	—	2	1	no	—
230	C₂×D₄×C₃₁	—	2	1	no	—
231	C₂×D₄×C₃₁	—	2	2	no	—
232	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	complemented
233	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
234	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
235	C₂²×C₄×C₃₁	—	2	1	no	—
236	C₂²×C₄×C₃₁	—	2	1	no	—
237	C₂²×C₄×C₃₁	—	2	2	no	—
238	C₂²×C₄×C₃₁	—	2	1	no	—
239	C₂²×C₄×C₃₁	—	2	1	no	—
240	C₂²×C₄×C₃₁	—	2	2	no	—
241	C₂²×C₄×C₃₁	—	2	4	no	—
242	C₂²×C₄×C₃₁	—	2	4	no	—
243	C₂²×C₄×C₃₁	—	2	2	no	—
244	C₂²×C₄×C₃₁	—	2	4	no	—
245	C₂²×C₄×C₃₁	—	8	2	no	—
246	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
247	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
248	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
249	SD₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
250	D₈×C₃₁	C₂×Q₈	1	8	yes	complemented
251	M₁₆×C₃₁	—	2	2	no	—
252	M₁₆×C₃₁	C₂×Q₈	1	4	yes	complemented
253	C₂×C₈×C₃₁	—	2	4	no	—
254	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
255	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
256	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
257	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
258	C₄⋊C₄×C₃₁	—	2	2	no	—
259	C₄⋊C₄×C₃₁	—	2	1	no	—
260	C₄⋊C₄×C₃₁	—	2	2	no	—
261	C₄⋊C₄×C₃₁	—	2	2	no	—
262	C₄⋊C₄×C₃₁	—	2	4	no	—
263	C₄⋊C₄×C₃₁	—	2	4	no	—
264	C₄⋊C₄×C₃₁	—	2	4	no	—
265	C₄⋊C₄×C₃₁	—	2	4	no	—
266	C₄⋊C₄×C₃₁	—	2	2	no	—
267	C₄⋊C₄×C₃₁	—	2	4	no	—
268	C₄⋊C₄×C₃₁	—	2	1	no	—
269	C₄⋊C₄×C₃₁	—	2	1	no	—
270	C₄⋊C₄×C₃₁	—	2	1	no	—
271	C₄⋊C₄×C₃₁	—	2	2	no	—
272	C₄⋊C₄×C₃₁	—	2	1	no	—
273	C₄⋊C₄×C₃₁	—	2	4	no	—
274	C₄⋊C₄×C₃₁	—	2	4	no	—
275	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
276	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
277	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
278	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
279	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
280	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
281	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
282	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
283	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
284	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
285	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
286	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
287	C₂²⋊C₄×C₃₁	—	2	2	no	—
288	C₂²⋊C₄×C₃₁	—	2	1	no	—
289	C₂²⋊C₄×C₃₁	—	2	4	no	—
290	C₂²⋊C₄×C₃₁	—	2	4	no	—
291	C₂²⋊C₄×C₃₁	—	2	2	no	—
292	C₂²⋊C₄×C₃₁	—	2	2	no	—
293	C₂²⋊C₄×C₃₁	—	2	1	no	—
294	C₂²⋊C₄×C₃₁	—	2	1	no	—
295	C₂²⋊C₄×C₃₁	—	2	4	no	—
296	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
297	C₄²×C₃₁	—	2	1	no	—
298	C₄²×C₃₁	—	2	1	no	—
299	C₄²×C₃₁	—	2	1	no	—
300	C₄²×C₃₁	—	2	4	no	—
301	C₄²×C₃₁	—	2	2	no	—
302	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
303	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
304	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
305	C₄²×C₃₁	C₂⁴	1	4	yes	—
306	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
307	#61382	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
308	C₂³×C₃₁	—	2	1	no	—
309	C₂³×C₃₁	—	2	2	no	—
310	C₂³×C₃₁	—	2	4	no	—
311	C₂³×C₃₁	—	8	1	no	—
312	C₂³×C₃₁	—	8	1	no	—
313	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
314	Q₈×C₃₁	—	2	4	no	—
315	Q₈×C₃₁	—	2	1	no	—
316	Q₈×C₃₁	—	4	2	no	—
317	Q₈×C₃₁	—	4	4	no	—
318	Q₈×C₃₁	—	4	2	no	—
319	Q₈×C₃₁	—	8	1	no	—
320	Q₈×C₃₁	—	8	2	no	—
321	Q₈×C₃₁	—	8	4	no	—
322	Q₈×C₃₁	—	8	4	no	—
323	Q₈×C₃₁	C₂²×Q₈	1	2	yes	—
324	D₄×C₃₁	—	2	4	no	—
325	D₄×C₃₁	—	2	4	no	—
326	D₄×C₃₁	—	2	2	no	—
327	D₄×C₃₁	—	2	1	no	—
328	D₄×C₃₁	C₂²×Q₈	1	2	yes	—
329	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
330	C₂×C₄×C₃₁	—	2	1	no	—
331	C₂×C₄×C₃₁	—	2	4	no	—
332	C₂×C₄×C₃₁	—	2	2	no	—
333	C₂×C₄×C₃₁	—	2	1	no	—
334	C₂×C₄×C₃₁	—	2	1	no	—
335	C₂×C₄×C₃₁	—	2	2	no	—
336	C₂×C₄×C₃₁	—	2	2	no	—
337	C₂×C₄×C₃₁	—	2	4	no	—
338	C₂×C₄×C₃₁	—	2	2	no	—
339	C₂×C₄×C₃₁	—	4	1	no	—
340	C₂×C₄×C₃₁	—	4	2	no	—
341	C₂×C₄×C₃₁	—	4	2	no	—
342	C₂×C₄×C₃₁	—	4	4	no	—
343	C₂×C₄×C₃₁	—	8	4	no	—
344	C₂×C₄×C₃₁	—	8	2	no	—
345	C₂×C₄×C₃₁	—	8	4	no	—
346	C₂×C₄×C₃₁	—	8	4	no	—
347	C₂×C₄×C₃₁	—	8	1	no	—
348	C₂×C₄×C₃₁	—	8	2	no	—
349	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
350	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
351	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
352	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
353	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
354	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
355	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
356	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
357	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
358	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
359	C₈×C₃₁	—	2	2	no	—
360	C₈×C₃₁	—	2	2	no	—
361	C₈×C₃₁	—	2	2	no	—
362	C₈×C₃₁	C₂²×Q₈	1	4	yes	—
363	#3503	C₂×C₃₁	1	1	yes	characteristic, complemented
364	#3501	C₂×C₃₁	1	1	yes	characteristic, complemented
365	#3388	C₂×C₃₁	1	4	yes	complemented
366	#3385	C₂×C₃₁	1	2	yes	complemented
367	#3384	C₂×C₃₁	1	2	yes	complemented
368	#3384	C₂×C₃₁	1	4	yes	complemented
369	#3380	C₂×C₃₁	1	2	yes	complemented
370	Q₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
371	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
372	#3163	C₂×C₃₁	1	2	yes	complemented
373	#3084	C₂×C₃₁	1	2	yes	complemented
374	#3081	C₂×C₃₁	1	2	yes	complemented
375	#3076	C₂×C₃₁	1	1	yes	characteristic, complemented
376	C₂×#476	C₂×C₃₁	1	1	yes	characteristic, complemented
377	#2962	C₂×C₃₁	1	1	yes	characteristic, complemented
378	#2946	C₂×C₃₁	1	2	yes	—
379	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic
380	C₂²×C₃₁	—	2	1	no	—
381	C₂²×C₃₁	—	2	1	no	—
382	C₂²×C₃₁	—	2	2	no	—
383	C₂²×C₃₁	—	2	4	no	—
384	C₂²×C₃₁	—	4	1	no	—
385	C₂²×C₃₁	—	4	2	no	—
386	C₂²×C₃₁	—	4	2	no	—
387	C₂²×C₃₁	—	4	1	no	—
388	C₂²×C₃₁	—	8	4	no	—
389	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
390	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
391	C₄×C₃₁	—	2	1	no	—
392	C₄×C₃₁	—	2	1	no	—
393	C₄×C₃₁	—	2	4	no	—
394	C₄×C₃₁	—	2	2	no	—
395	C₄×C₃₁	—	2	1	no	—
396	C₄×C₃₁	—	2	4	no	—
397	C₄×C₃₁	—	4	1	no	—
398	C₄×C₃₁	—	4	4	no	—
399	C₄×C₃₁	—	4	2	no	—
400	C₄×C₃₁	—	4	1	no	—
401	C₄×C₃₁	—	4	2	no	—
402	C₄×C₃₁	—	4	2	no	—
403	C₄×C₃₁	D₄×Q₈	1	2	yes	—
404	C₄×C₃₁	#577	1	2	yes	—
405	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
406	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
407	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
408	Q₈²	C₂²×C₃₁	1	2	yes	complemented
409	#557	C₂²×C₃₁	1	2	yes	complemented
410	#554	—	2	1	no	—
411	#554	—	2	2	no	—
412	#554	—	2	2	no	—
413	#554	C₂²×C₃₁	1	2	yes	—
414	#554	C₂²×C₃₁	1	4	yes	—
415	#554	C₂²×C₃₁	1	2	yes	complemented
416	D₄×Q₈	—	2	1	no	—
417	D₄×Q₈	—	2	2	no	—
418	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
419	D₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
420	#544	—	2	2	no	—
421	#544	C₂²×C₃₁	1	2	yes	—
422	#544	C₂²×C₃₁	1	2	yes	—
423	#543	—	2	2	no	—
424	#533	—	2	1	no	—
425	#533	C₂²×C₃₁	1	1	yes	characteristic
426	#533	C₂²×C₃₁	1	1	yes	characteristic
427	C₂×C₄⋊Q₈	—	2	1	no	—
428	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
429	C₂×C₂²⋊Q₈	—	2	2	no	—
430	#520	C₂²×C₃₁	1	2	yes	—
431	#518	C₂²×C₃₁	1	2	yes	—
432	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
433	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
434	#501	C₂²×C₃₁	1	4	yes	complemented
435	#501	C₂²×C₃₁	1	4	yes	complemented
436	#501	C₂²×C₃₁	1	4	yes	complemented
437	#500	C₂²×C₃₁	1	2	yes	complemented
438	#499	C₂²×C₃₁	1	4	yes	complemented
439	#498	C₂²×C₃₁	1	4	yes	complemented
440	#498	C₂²×C₃₁	1	2	yes	complemented
441	#479	C₂²×C₃₁	1	2	yes	complemented
442	#479	C₂²×C₃₁	1	4	yes	complemented
443	#478	C₂²×C₃₁	1	4	yes	—
444	#478	C₂²×C₃₁	1	4	yes	—
445	#478	C₂²×C₃₁	1	2	yes	complemented
446	#478	C₂²×C₃₁	1	4	yes	complemented
447	#477	C₂²×C₃₁	1	1	yes	characteristic, complemented
448	#476	C₂²×C₃₁	1	4	yes	—
449	#476	C₂²×C₃₁	1	2	yes	—
450	#476	C₂²×C₃₁	1	2	yes	—
451	#476	C₂²×C₃₁	1	4	yes	—
452	#476	C₂²×C₃₁	1	1	yes	characteristic, complemented
453	#476	C₂²×C₃₁	1	4	yes	complemented
454	#475	C₂²×C₃₁	1	1	yes	characteristic, complemented
455	#475	C₂²×C₃₁	1	4	yes	complemented
456	#474	C₂²×C₃₁	1	4	yes	—
457	#474	C₂²×C₃₁	1	1	yes	characteristic, complemented
458	#446	C₂²×C₃₁	1	2	yes	complemented
459	#446	C₂²×C₃₁	1	4	yes	complemented
460	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
461	#442	C₂²×C₃₁	1	2	yes	—
462	#442	C₂²×C₃₁	1	4	yes	—
463	#440	C₂²×C₃₁	1	4	yes	—
464	#440	C₂²×C₃₁	1	2	yes	—
465	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
466	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
467	C₄×D₈	C₂²×C₃₁	1	2	yes	—
468	C₄×D₈	C₂²×C₃₁	1	4	yes	—
469	#428	C₂²×C₃₁	1	2	yes	—
470	#428	C₂²×C₃₁	1	1	yes	characteristic
471	#428	C₂²×C₃₁	1	2	yes	—
472	#427	C₂²×C₃₁	1	2	yes	—
473	C₂×#106	C₂²×C₃₁	1	2	yes	—
474	#424	C₂²×C₃₁	1	2	yes	—
475	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
476	#418	C₂²×C₃₁	1	1	yes	characteristic
477	#417	C₂²×C₃₁	1	1	yes	characteristic
478	#416	C₂²×C₃₁	1	2	yes	—
479	C₂×#102	C₂²×C₃₁	1	2	yes	—
480	#405	C₂²×C₃₁	1	2	yes	—
481	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
482	C₂×C₃₁	—	2	2	no	—
483	C₂×C₃₁	—	4	2	no	—
484	C₂×C₃₁	—	4	4	no	—
485	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
486	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
487	C₂×C₃₁	#3585	1	1	yes	characteristic
488	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
489	C₂²×Q₈	—	2	2	no	—
490	C₂²×Q₈	—	2	2	no	—
491	C₂²×Q₈	—	2	1	no	—
492	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
493	C₂³×C₄	—	2	1	no	—
494	AGL(1,ℤ/8)	Q₈×C₃₁	1	8	yes	complemented
495	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
496	C₂×D₈	C₂³×C₃₁	1	2	yes	—
497	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
498	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
499	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
500	C₄⋊Q₈	—	2	2	no	—
501	C₄⋊Q₈	—	2	2	no	—
502	C₄⋊Q₈	—	2	2	no	—
503	C₄⋊Q₈	—	2	4	no	—
504	C₄⋊Q₈	—	2	2	no	—
505	C₄⋊Q₈	—	2	1	no	—
506	C₄⋊Q₈	—	2	1	no	—
507	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
508	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
509	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
510	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
511	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
512	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
513	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
514	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
515	#125	—	2	2	no	—
516	#125	—	2	4	no	—
517	#125	—	2	2	no	—
518	#125	D₄×C₃₁	1	2	yes	—
519	#125	C₂³×C₃₁	1	2	yes	—
520	#125	C₂³×C₃₁	1	4	yes	—
521	#125	C₂³×C₃₁	1	4	yes	—
522	C₂²⋊Q₈	—	2	2	no	—
523	C₂²⋊Q₈	—	2	2	no	—
524	C₂²⋊Q₈	—	2	4	no	—
525	C₂²⋊Q₈	—	2	1	no	—
526	C₂²⋊Q₈	—	2	1	no	—
527	C₂²⋊Q₈	—	2	2	no	—
528	C₂²⋊Q₈	—	2	4	no	—
529	C₂²⋊Q₈	—	2	2	no	—
530	C₂²⋊Q₈	—	2	4	no	—
531	C₂²⋊Q₈	—	2	4	no	—
532	C₂²⋊Q₈	—	2	4	no	—
533	C₂²⋊Q₈	—	2	4	no	—
534	C₂²⋊Q₈	—	2	4	no	—
535	C₂²⋊Q₈	—	2	4	no	—
536	C₂²⋊Q₈	—	2	4	no	—
537	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
538	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
539	C₄×Q₈	—	2	2	no	—
540	C₄×Q₈	—	2	2	no	—
541	C₄×Q₈	—	2	1	no	—
542	C₄×Q₈	—	2	1	no	—
543	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
544	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
545	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
546	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
547	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
548	C₄×D₄	—	2	1	no	—
549	C₄×D₄	—	2	1	no	—
550	C₄×D₄	—	2	2	no	—
551	C₄×D₄	—	2	2	no	—
552	C₄×D₄	—	2	4	no	—
553	C₄×D₄	—	2	1	no	—
554	C₄×D₄	—	2	1	no	—
555	C₄×D₄	C₂³×C₃₁	1	2	yes	—
556	C₄×D₄	C₂³×C₃₁	1	2	yes	—
557	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
558	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
559	C₄×D₄	C₂³×C₃₁	1	4	yes	—
560	C₄×D₄	C₂³×C₃₁	1	4	yes	—
561	C₄×D₄	C₂³×C₃₁	1	4	yes	—
562	#117	—	2	2	no	—
563	#117	—	2	1	no	—
564	#117	C₂³×C₃₁	1	1	yes	characteristic
565	#117	C₂³×C₃₁	1	2	yes	—
566	#117	C₂³×C₃₁	1	2	yes	—
567	C₂×C₄⋊C₄	—	2	1	no	—
568	C₂×C₄⋊C₄	—	2	2	no	—
569	C₂×C₄⋊C₄	—	2	4	no	—
570	C₂×C₄⋊C₄	—	2	2	no	—
571	C₂×C₄⋊C₄	—	2	2	no	—
572	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
573	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
574	C₂×C₂²⋊C₄	—	2	1	no	—
575	C₂×C₂²⋊C₄	—	2	4	no	—
576	C₂×C₄²	—	2	1	no	—
577	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
578	#107	C₂³×C₃₁	1	2	yes	—
579	#107	C₂³×C₃₁	1	2	yes	—
580	#107	C₂³×C₃₁	1	2	yes	—
581	#107	C₂³×C₃₁	1	4	yes	—
582	#107	C₂³×C₃₁	1	4	yes	—
583	#106	C₂³×C₃₁	1	4	yes	—
584	#106	C₂³×C₃₁	1	4	yes	—
585	#106	C₂³×C₃₁	1	2	yes	—
586	#106	C₂³×C₃₁	1	2	yes	—
587	#106	C₂³×C₃₁	1	4	yes	—
588	#106	C₂³×C₃₁	1	2	yes	—
589	#106	C₂³×C₃₁	1	4	yes	—
590	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
591	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
592	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
593	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
594	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
595	#103	C₂³×C₃₁	1	2	yes	—
596	#103	C₂³×C₃₁	1	2	yes	—
597	#103	C₂³×C₃₁	1	2	yes	—
598	#102	C₂³×C₃₁	1	4	yes	—
599	#102	C₂³×C₃₁	1	4	yes	—
600	#102	C₂³×C₃₁	1	2	yes	—
601	#102	C₂³×C₃₁	1	2	yes	—
602	#102	C₂³×C₃₁	1	2	yes	—
603	#102	C₂³×C₃₁	1	2	yes	—
604	#102	C₂³×C₃₁	1	2	yes	—
605	#97	C₂³×C₃₁	1	2	yes	—
606	#97	C₂³×C₃₁	1	4	yes	—
607	C₄×C₈	C₂³×C₃₁	1	2	yes	—
608	C₄×C₈	C₂³×C₃₁	1	4	yes	—
609	C₃₁	#61382	1	1	yes	characteristic, direct product factor, Sylow
610	C₂⁴	—	2	1	no	—
611	#41	—	2	2	no	—
612	#41	C₂×Q₈×C₃₁	1	4	yes	complemented
613	C₂×Q₈	—	2	1	no	—
614	C₂×Q₈	—	2	2	no	—
615	C₂×Q₈	—	2	1	no	—
616	C₂×Q₈	—	2	4	no	—
617	C₂×Q₈	—	2	4	no	—
618	C₂×Q₈	—	2	2	no	—
619	C₂×Q₈	—	8	4	no	—
620	C₂×Q₈	—	8	2	no	—
621	C₂×Q₈	—	8	4	no	—
622	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
623	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
624	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
625	C₂×D₄	—	2	1	no	—
626	C₂×D₄	—	2	1	no	—
627	C₂×D₄	—	2	2	no	—
628	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	complemented
629	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
630	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
631	C₂²×C₄	—	2	1	no	—
632	C₂²×C₄	—	2	1	no	—
633	C₂²×C₄	—	2	2	no	—
634	C₂²×C₄	—	2	1	no	—
635	C₂²×C₄	—	2	1	no	—
636	C₂²×C₄	—	2	2	no	—
637	C₂²×C₄	—	2	4	no	—
638	C₂²×C₄	—	2	4	no	—
639	C₂²×C₄	—	2	2	no	—
640	C₂²×C₄	—	2	4	no	—
641	C₂²×C₄	—	8	2	no	—
642	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
643	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
644	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
645	SD₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
646	D₈	C₂×Q₈×C₃₁	1	8	yes	complemented
647	M₁₆	—	2	2	no	—
648	M₁₆	C₂×Q₈×C₃₁	1	4	yes	complemented
649	C₂×C₈	—	2	4	no	—
650	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
651	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
652	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
653	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
654	C₄⋊C₄	—	2	2	no	—
655	C₄⋊C₄	—	2	1	no	—
656	C₄⋊C₄	—	2	2	no	—
657	C₄⋊C₄	—	2	2	no	—
658	C₄⋊C₄	—	2	4	no	—
659	C₄⋊C₄	—	2	4	no	—
660	C₄⋊C₄	—	2	4	no	—
661	C₄⋊C₄	—	2	4	no	—
662	C₄⋊C₄	—	2	2	no	—
663	C₄⋊C₄	—	2	4	no	—
664	C₄⋊C₄	—	2	1	no	—
665	C₄⋊C₄	—	2	1	no	—
666	C₄⋊C₄	—	2	1	no	—
667	C₄⋊C₄	—	2	2	no	—
668	C₄⋊C₄	—	2	1	no	—
669	C₄⋊C₄	—	2	4	no	—
670	C₄⋊C₄	—	2	4	no	—
671	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
672	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
673	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
674	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
675	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
676	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
677	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
678	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
679	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
680	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
681	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
682	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
683	C₂²⋊C₄	—	2	2	no	—
684	C₂²⋊C₄	—	2	1	no	—
685	C₂²⋊C₄	—	2	4	no	—
686	C₂²⋊C₄	—	2	4	no	—
687	C₂²⋊C₄	—	2	2	no	—
688	C₂²⋊C₄	—	2	2	no	—
689	C₂²⋊C₄	—	2	1	no	—
690	C₂²⋊C₄	—	2	1	no	—
691	C₂²⋊C₄	—	2	4	no	—
692	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
693	C₄²	—	2	1	no	—
694	C₄²	—	2	1	no	—
695	C₄²	—	2	1	no	—
696	C₄²	—	2	4	no	—
697	C₄²	—	2	2	no	—
698	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
699	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
700	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
701	C₄²	C₂⁴×C₃₁	1	4	yes	—
702	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
703	C₂³	—	2	1	no	—
704	C₂³	—	2	2	no	—
705	C₂³	—	2	4	no	—
706	C₂³	—	8	1	no	—
707	C₂³	—	8	1	no	—
708	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
709	Q₈	—	2	4	no	—
710	Q₈	—	2	1	no	—
711	Q₈	—	4	2	no	—
712	Q₈	—	4	4	no	—
713	Q₈	—	4	2	no	—
714	Q₈	—	8	1	no	—
715	Q₈	—	8	4	no	—
716	Q₈	—	8	2	no	—
717	Q₈	—	8	4	no	—
718	Q₈	C₂²×Q₈×C₃₁	1	2	yes	—
719	D₄	—	2	4	no	—
720	D₄	—	2	4	no	—
721	D₄	—	2	2	no	—
722	D₄	—	2	1	no	—
723	D₄	C₂²×Q₈×C₃₁	1	2	yes	—
724	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
725	C₂×C₄	—	2	1	no	—
726	C₂×C₄	—	2	4	no	—
727	C₂×C₄	—	2	2	no	—
728	C₂×C₄	—	2	1	no	—
729	C₂×C₄	—	2	1	no	—
730	C₂×C₄	—	2	2	no	—
731	C₂×C₄	—	2	2	no	—
732	C₂×C₄	—	2	4	no	—
733	C₂×C₄	—	2	2	no	—
734	C₂×C₄	—	4	1	no	—
735	C₂×C₄	—	4	2	no	—
736	C₂×C₄	—	4	2	no	—
737	C₂×C₄	—	4	4	no	—
738	C₂×C₄	—	8	4	no	—
739	C₂×C₄	—	8	2	no	—
740	C₂×C₄	—	8	4	no	—
741	C₂×C₄	—	8	4	no	—
742	C₂×C₄	—	8	1	no	—
743	C₂×C₄	—	8	2	no	—
744	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
745	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
746	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
747	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
748	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
749	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
750	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
751	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
752	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
753	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
754	C₈	—	2	2	no	—
755	C₈	—	2	2	no	—
756	C₈	—	2	2	no	—
757	C₈	C₂²×Q₈×C₃₁	1	4	yes	—
758	C₂²	—	2	1	no	—
759	C₂²	—	2	1	no	—
760	C₂²	—	2	2	no	—
761	C₂²	—	2	4	no	—
762	C₂²	—	4	2	no	—
763	C₂²	—	4	1	no	—
764	C₂²	—	4	1	no	—
765	C₂²	—	4	2	no	—
766	C₂²	—	8	4	no	—
767	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
768	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
769	C₄	—	2	2	no	—
770	C₄	—	2	1	no	—
771	C₄	—	2	4	no	—
772	C₄	—	2	1	no	—
773	C₄	—	2	1	no	—
774	C₄	—	2	4	no	—
775	C₄	—	4	2	no	—
776	C₄	—	4	1	no	—
777	C₄	—	4	2	no	—
778	C₄	—	4	1	no	—
779	C₄	—	4	4	no	—
780	C₄	—	4	2	no	—
781	C₄	D₄×Q₈×C₃₁	1	2	yes	—
782	C₄	C₃₁×#577	1	2	yes	—
783	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
784	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
785	C₂	—	2	2	no	—
786	C₂	—	4	4	no	—
787	C₂	—	4	2	no	—
788	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
789	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
790	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	C₂×C₃₁×AGL(1,ℤ/8)	C₂²
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×#3503	C₂	semidirect
2. C₃₁×#3501	C₂	semidirect
3. C₃₁×#3388	C₂	semidirect
4. C₃₁×#3385	C₂	semidirect
5. C₃₁×#3384	C₂	semidirect
6. C₃₁×#3384	C₂	semidirect
7. C₃₁×#3380	C₂	semidirect
8. Q₈×SD₁₆×C₃₁	C₂	semidirect
9. C₃₁×#3165	C₂	semidirect
10. C₃₁×#3163	C₂	semidirect
11. C₃₁×#3084	C₂	semidirect
12. C₃₁×#3081	C₂	semidirect
13. C₃₁×#3076	C₂	semidirect
14. C₂×C₃₁×#476	C₂	semidirect
15. C₃₁×#2962	C₂	semidirect
19. Q₈²×C₃₁	C₂²	semidirect
20. C₃₁×#557	C₂²	semidirect
26. C₃₁×#554	C₂²	semidirect
30. D₄×Q₈×C₃₁	C₂²	semidirect
45. C₃₁×#501	C₂²	semidirect
46. C₃₁×#501	C₂²	semidirect
47. C₃₁×#501	C₂²	semidirect
48. C₃₁×#500	C₂²	semidirect
49. C₃₁×#499	C₂²	semidirect
50. C₃₁×#498	C₂²	semidirect
51. C₃₁×#498	C₂²	semidirect
52. C₃₁×#479	C₂²	semidirect
53. C₃₁×#479	C₂²	semidirect
56. C₃₁×#478	C₂²	semidirect
57. C₃₁×#478	C₂²	semidirect
58. C₃₁×#477	C₂²	semidirect
63. C₃₁×#476	C₂²	semidirect
64. C₃₁×#476	C₂²	semidirect
65. C₃₁×#475	C₂²	semidirect
66. C₃₁×#475	C₂²	semidirect
68. C₃₁×#474	C₂²	semidirect
69. C₃₁×#446	C₂²	semidirect
70. C₃₁×#446	C₂²	semidirect
71. C₈×Q₈×C₃₁	C₂²	semidirect
99. C₃₁×AGL(1,ℤ/8)	Q₈	semidirect
216. #41×C₃₁	C₂×Q₈	semidirect
232. C₂×D₄×C₃₁	C₂×Q₈	semidirect
249. SD₁₆×C₃₁	C₂×Q₈	semidirect
250. D₈×C₃₁	C₂×Q₈	semidirect
252. M₁₆×C₃₁	C₂×Q₈	semidirect
307. #61382	C₃₁	direct (N × H)
363. #3503	C₂×C₃₁	semidirect
364. #3501	C₂×C₃₁	semidirect
365. #3388	C₂×C₃₁	semidirect
366. #3385	C₂×C₃₁	semidirect
367. #3384	C₂×C₃₁	semidirect
368. #3384	C₂×C₃₁	semidirect
369. #3380	C₂×C₃₁	semidirect
370. Q₈×SD₁₆	C₂×C₃₁	semidirect
371. #3165	C₂×C₃₁	semidirect
372. #3163	C₂×C₃₁	semidirect
373. #3084	C₂×C₃₁	semidirect
374. #3081	C₂×C₃₁	semidirect
375. #3076	C₂×C₃₁	semidirect
376. C₂×#476	C₂×C₃₁	semidirect
377. #2962	C₂×C₃₁	semidirect
408. Q₈²	C₂²×C₃₁	semidirect
409. #557	C₂²×C₃₁	semidirect
415. #554	C₂²×C₃₁	semidirect
419. D₄×Q₈	C₂²×C₃₁	semidirect
434. #501	C₂²×C₃₁	semidirect
435. #501	C₂²×C₃₁	semidirect
436. #501	C₂²×C₃₁	semidirect
437. #500	C₂²×C₃₁	semidirect
438. #499	C₂²×C₃₁	semidirect
439. #498	C₂²×C₃₁	semidirect
440. #498	C₂²×C₃₁	semidirect
441. #479	C₂²×C₃₁	semidirect
442. #479	C₂²×C₃₁	semidirect
445. #478	C₂²×C₃₁	semidirect
446. #478	C₂²×C₃₁	semidirect
447. #477	C₂²×C₃₁	semidirect
452. #476	C₂²×C₃₁	semidirect
453. #476	C₂²×C₃₁	semidirect
454. #475	C₂²×C₃₁	semidirect
455. #475	C₂²×C₃₁	semidirect
457. #474	C₂²×C₃₁	semidirect
458. #446	C₂²×C₃₁	semidirect
459. #446	C₂²×C₃₁	semidirect
460. C₈×Q₈	C₂²×C₃₁	semidirect
494. AGL(1,ℤ/8)	Q₈×C₃₁	semidirect
609. C₃₁	#61382	direct (N × H)
612. #41	C₂×Q₈×C₃₁	semidirect
628. C₂×D₄	C₂×Q₈×C₃₁	semidirect
645. SD₁₆	C₂×Q₈×C₃₁	semidirect
646. D₈	C₂×Q₈×C₃₁	semidirect
648. M₁₆	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×#3503	2
C₃₁×#3501	2
C₃₁×#3388	2
C₃₁×#3385	2
C₃₁×#3384	2
C₃₁×#3380	2
Q₈×SD₁₆×C₃₁	2
C₃₁×#3165	2
C₃₁×#3163	2
C₃₁×#3084	2
C₃₁×#3081	2
C₃₁×#3076	2
C₂×C₃₁×#476	2
C₃₁×#2962	2
C₃₁×#2946	2
C₄×C₃₁×AGL(1,ℤ/8)	2
#61382	31

Subgroups — C31×#61382 #21518141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61382 #21518141