Subgroups — C₃₁×#34330 #21491089

Order 7936.

2772 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2366	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2364	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2363	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2357	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2339	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2333	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2328	C₂	1	2	yes	maximal, complemented
12	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₃₁×#121	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2297	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#2275	C₂	1	2	yes	maximal, complemented
17	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#550	C₄	1	16	yes	complemented
20	C₂×C₃₁×#127	C₂²	1	2	yes	—
21	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#121	C₂²	1	4	yes	—
23	C₂×C₃₁×#121	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
33	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
35	C₂×C₃₁×#117	C₂²	1	2	yes	—
36	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
37	C₃₁×#391	C₂²	1	1	yes	characteristic
38	C₃₁×#391	C₂²	1	2	yes	—
39	C₃₁×#391	C₂²	1	1	yes	characteristic
40	C₃₁×#390	C₂²	1	2	yes	—
41	C₃₁×#390	C₂²	1	2	yes	—
42	C₃₁×#390	C₂²	1	2	yes	—
43	C₃₁×#390	C₂²	1	4	yes	—
44	C₃₁×#390	C₂²	1	2	yes	—
45	C₃₁×#390	C₂²	1	2	yes	—
46	C₃₁×#390	C₂²	1	2	yes	—
47	C₃₁×#389	C₂²	1	1	yes	characteristic
48	C₃₁×#389	C₂²	1	1	yes	characteristic
49	C₃₁×#388	C₂²	1	4	yes	—
50	C₃₁×#388	C₂²	1	4	yes	—
51	C₃₁×#388	C₂²	1	4	yes	—
52	C₃₁×#387	C₂²	1	2	yes	—
53	C₃₁×#387	C₂²	1	4	yes	—
54	C₃₁×#387	C₂²	1	2	yes	—
55	C₃₁×#386	C₂²	1	4	yes	—
56	C₃₁×#386	C₂²	1	4	yes	—
57	C₃₁×#386	C₂²	1	4	yes	—
58	C₃₁×#386	C₂²	1	4	yes	—
59	C₃₁×#386	C₂²	1	4	yes	—
60	C₃₁×#386	C₂²	1	4	yes	—
61	C₃₁×#386	C₂²	1	4	yes	—
62	C₃₁×#384	C₂²	1	1	yes	characteristic
63	C₃₁×#382	C₂²	1	2	yes	—
64	C₃₁×#381	C₂²	1	4	yes	—
65	C₃₁×#380	C₂²	1	2	yes	—
66	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
67	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
68	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
70	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
72	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
74	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
75	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
76	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
77	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
78	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
79	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
80	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
81	C₃₁×#376	C₂²	1	2	yes	—
82	C₂×C₃₁×#95	C₂²	1	4	yes	—
83	C₄³×C₃₁	C₂²	1	1	yes	characteristic
84	C₂×#41×C₃₁	—	4	4	no	—
85	C₂×#41×C₃₁	—	4	4	no	—
86	C₂×#41×C₃₁	C₂×C₄	1	16	yes	complemented
87	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
90	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
91	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
92	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
93	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
94	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
95	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
96	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
97	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
98	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
99	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
100	C₃₁×#127	C₂×C₄	1	8	yes	complemented
101	C₃₁×#127	C₂×C₄	1	16	yes	complemented
102	C₃₁×#121	C₂×C₄	1	16	yes	complemented
103	C₃₁×#121	C₂×C₄	1	32	yes	complemented
104	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
105	C₄×D₄×C₃₁	—	2	4	no	—
106	C₄×D₄×C₃₁	—	2	8	no	—
107	C₄×D₄×C₃₁	—	2	4	no	—
108	C₄×D₄×C₃₁	—	2	8	no	—
109	C₄×D₄×C₃₁	—	4	4	no	—
110	C₄×D₄×C₃₁	—	4	4	no	—
111	C₄×D₄×C₃₁	—	4	4	no	—
112	C₄×D₄×C₃₁	—	4	8	no	—
113	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
114	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
115	C₃₁×#117	—	4	4	no	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
123	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
126	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
128	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
129	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
132	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
133	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
135	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
136	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
137	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
138	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
139	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
140	C₂×C₄²×C₃₁	—	4	4	no	—
141	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
142	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
143	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
145	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
146	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
147	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
148	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
149	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
150	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
151	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
152	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
153	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
154	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
155	C₃₁×#95	C₂³	1	2	yes	—
156	C₃₁×#95	C₂³	1	2	yes	—
157	C₃₁×#95	C₂³	1	4	yes	—
158	C₃₁×#95	C₂³	1	4	yes	—
159	C₃₁×#95	C₂³	1	2	yes	—
160	C₃₁×#95	C₂³	1	2	yes	—
161	C₃₁×#95	C₂³	1	2	yes	—
162	C₃₁×#95	C₂³	1	2	yes	—
163	C₃₁×#95	C₂³	1	2	yes	—
164	C₃₁×#95	C₂³	1	2	yes	—
165	C₃₁×#95	C₂³	1	4	yes	—
166	C₃₁×#95	C₂³	1	2	yes	—
167	C₃₁×#95	C₂³	1	4	yes	—
168	C₃₁×#95	C₂³	1	2	yes	—
169	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
170	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
171	#41×C₃₁	—	4	32	no	—
172	C₂×Q₈×C₃₁	—	2	2	no	—
173	C₂×Q₈×C₃₁	—	2	2	no	—
174	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
175	C₂×D₄×C₃₁	—	2	2	no	—
176	C₂×D₄×C₃₁	—	2	4	no	—
177	C₂×D₄×C₃₁	—	2	2	no	—
178	C₂×D₄×C₃₁	—	2	4	no	—
179	C₂×D₄×C₃₁	—	2	4	no	—
180	C₂×D₄×C₃₁	—	2	4	no	—
181	C₂×D₄×C₃₁	—	4	2	no	—
182	C₂×D₄×C₃₁	—	4	4	no	—
183	C₂×D₄×C₃₁	—	4	2	no	—
184	C₂×D₄×C₃₁	—	4	4	no	—
185	C₂×D₄×C₃₁	—	4	4	no	—
186	C₂×D₄×C₃₁	—	4	4	no	—
187	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
188	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
189	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
190	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
191	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
192	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
193	C₂²×C₄×C₃₁	—	2	4	no	—
194	C₂²×C₄×C₃₁	—	2	4	no	—
195	C₂²×C₄×C₃₁	—	2	4	no	—
196	C₂²×C₄×C₃₁	—	2	4	no	—
197	C₂²×C₄×C₃₁	—	2	4	no	—
198	C₂²×C₄×C₃₁	—	2	8	no	—
199	C₂²×C₄×C₃₁	—	2	8	no	—
200	C₂²×C₄×C₃₁	—	4	2	no	—
201	C₂²×C₄×C₃₁	—	4	2	no	—
202	C₂²×C₄×C₃₁	—	4	2	no	—
203	C₂²×C₄×C₃₁	—	4	4	no	—
204	C₂²×C₄×C₃₁	—	4	4	no	—
205	C₂²×C₄×C₃₁	—	4	4	no	—
206	C₂²×C₄×C₃₁	—	4	4	no	—
207	C₂²×C₄×C₃₁	—	4	4	no	—
208	C₂²×C₄×C₃₁	—	4	4	no	—
209	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
210	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
211	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
214	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
216	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
217	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
218	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
219	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
220	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
222	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
223	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
224	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
225	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
226	C₄⋊C₄×C₃₁	—	2	2	no	—
227	C₄⋊C₄×C₃₁	—	2	4	no	—
228	C₄⋊C₄×C₃₁	—	2	2	no	—
229	C₄⋊C₄×C₃₁	—	2	4	no	—
230	C₄⋊C₄×C₃₁	—	2	4	no	—
231	C₄⋊C₄×C₃₁	—	2	2	no	—
232	C₄⋊C₄×C₃₁	—	4	2	no	—
233	C₄⋊C₄×C₃₁	—	4	2	no	—
234	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
235	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
236	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
237	C₄⋊C₄×C₃₁	#41	1	4	yes	complemented
238	C₂²⋊C₄×C₃₁	—	2	4	no	—
239	C₂²⋊C₄×C₃₁	—	2	8	no	—
240	C₂²⋊C₄×C₃₁	—	2	8	no	—
241	C₂²⋊C₄×C₃₁	—	2	4	no	—
242	C₂²⋊C₄×C₃₁	—	2	8	no	—
243	C₂²⋊C₄×C₃₁	—	4	4	no	—
244	C₂²⋊C₄×C₃₁	—	4	4	no	—
245	C₂²⋊C₄×C₃₁	—	4	4	no	—
246	C₂²⋊C₄×C₃₁	—	4	4	no	—
247	C₂²⋊C₄×C₃₁	—	4	4	no	—
248	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
249	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
250	C₄²×C₃₁	—	2	2	no	—
251	C₄²×C₃₁	—	2	4	no	—
252	C₄²×C₃₁	—	2	2	no	—
253	C₄²×C₃₁	—	2	4	no	—
254	C₄²×C₃₁	—	2	2	no	—
255	C₄²×C₃₁	—	2	2	no	—
256	C₄²×C₃₁	—	4	2	no	—
257	C₄²×C₃₁	—	4	2	no	—
258	C₄²×C₃₁	—	4	1	no	—
259	C₄²×C₃₁	—	4	4	no	—
260	C₄²×C₃₁	—	4	4	no	—
261	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
262	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
263	C₄²×C₃₁	#41	1	4	yes	complemented
264	#34330	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₂³×C₃₁	—	2	2	no	—
266	C₂³×C₃₁	—	2	4	no	—
267	C₂³×C₃₁	—	2	2	no	—
268	C₂³×C₃₁	—	2	4	no	—
269	C₂³×C₃₁	—	2	4	no	—
270	C₂³×C₃₁	—	2	2	no	—
271	C₂³×C₃₁	—	2	2	no	—
272	C₂³×C₃₁	—	2	4	no	—
273	C₂³×C₃₁	—	2	4	no	—
274	C₂³×C₃₁	—	2	2	no	—
275	C₂³×C₃₁	—	2	4	no	—
276	C₂³×C₃₁	—	2	2	no	—
277	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
278	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
279	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
280	Q₈×C₃₁	—	2	8	no	—
281	D₄×C₃₁	—	2	8	no	—
282	D₄×C₃₁	—	2	16	no	—
283	D₄×C₃₁	—	2	16	no	—
284	D₄×C₃₁	—	4	8	no	—
285	D₄×C₃₁	—	4	16	no	—
286	D₄×C₃₁	—	4	16	no	—
287	C₂×C₄×C₃₁	—	2	1	no	—
288	C₂×C₄×C₃₁	—	2	1	no	—
289	C₂×C₄×C₃₁	—	2	1	no	—
290	C₂×C₄×C₃₁	—	2	2	no	—
291	C₂×C₄×C₃₁	—	2	1	no	—
292	C₂×C₄×C₃₁	—	2	1	no	—
293	C₂×C₄×C₃₁	—	2	1	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	2	2	no	—
296	C₂×C₄×C₃₁	—	2	2	no	—
297	C₂×C₄×C₃₁	—	2	2	no	—
298	C₂×C₄×C₃₁	—	2	2	no	—
299	C₂×C₄×C₃₁	—	2	2	no	—
300	C₂×C₄×C₃₁	—	2	1	no	—
301	C₂×C₄×C₃₁	—	2	1	no	—
302	C₂×C₄×C₃₁	—	2	2	no	—
303	C₂×C₄×C₃₁	—	2	16	no	—
304	C₂×C₄×C₃₁	—	2	1	no	—
305	C₂×C₄×C₃₁	—	2	2	no	—
306	C₂×C₄×C₃₁	—	2	2	no	—
307	C₂×C₄×C₃₁	—	2	2	no	—
308	C₂×C₄×C₃₁	—	2	4	no	—
309	C₂×C₄×C₃₁	—	2	2	no	—
310	C₂×C₄×C₃₁	—	2	1	no	—
311	C₂×C₄×C₃₁	—	2	2	no	—
312	C₂×C₄×C₃₁	—	2	4	no	—
313	C₂×C₄×C₃₁	—	2	4	no	—
314	C₂×C₄×C₃₁	—	2	2	no	—
315	C₂×C₄×C₃₁	—	2	2	no	—
316	C₂×C₄×C₃₁	—	2	1	no	—
317	C₂×C₄×C₃₁	—	2	2	no	—
318	C₂×C₄×C₃₁	—	2	2	no	—
319	C₂×C₄×C₃₁	—	4	16	no	—
320	C₂×C₄×C₃₁	—	4	8	no	—
321	C₂×C₄×C₃₁	—	4	8	no	—
322	C₂×C₄×C₃₁	—	8	8	no	—
323	C₂×C₄×C₃₁	—	8	8	no	—
324	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
325	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
326	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
327	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
328	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
329	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
330	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
331	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
332	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
333	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
334	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
335	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
336	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
337	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
338	C₂×#550	C₂×C₃₁	1	1	yes	characteristic
339	#2366	C₂×C₃₁	1	2	yes	complemented
340	#2366	C₂×C₃₁	1	1	yes	characteristic, complemented
341	#2364	C₂×C₃₁	1	4	yes	complemented
342	#2363	C₂×C₃₁	1	2	yes	complemented
343	#2357	C₂×C₃₁	1	2	yes	complemented
344	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
345	#2354	C₂×C₃₁	1	1	yes	characteristic, complemented
346	#2339	C₂×C₃₁	1	2	yes	complemented
347	#2333	C₂×C₃₁	1	4	yes	complemented
348	#2328	C₂×C₃₁	1	2	yes	complemented
349	C₄×#127	C₂×C₃₁	1	1	yes	characteristic, complemented
350	C₄×#121	C₂×C₃₁	1	2	yes	complemented
351	#2297	C₂×C₃₁	1	2	yes	complemented
352	#2295	C₂×C₃₁	1	1	yes	characteristic, complemented
353	#2275	C₂×C₃₁	1	2	yes	complemented
354	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
355	C₂²×C₃₁	—	2	4	no	—
356	C₂²×C₃₁	—	2	8	no	—
357	C₂²×C₃₁	—	2	4	no	—
358	C₂²×C₃₁	—	2	8	no	—
359	C₂²×C₃₁	—	2	8	no	—
360	C₂²×C₃₁	—	2	4	no	—
361	C₂²×C₃₁	—	4	2	no	—
362	C₂²×C₃₁	—	4	4	no	—
363	C₂²×C₃₁	—	4	4	no	—
364	C₂²×C₃₁	—	4	2	no	—
365	C₂²×C₃₁	—	4	2	no	—
366	C₂²×C₃₁	—	4	4	no	—
367	C₂²×C₃₁	—	4	2	no	—
368	C₂²×C₃₁	—	4	4	no	—
369	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
370	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
371	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
372	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
373	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
374	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
375	C₂²×C₃₁	#585	1	1	yes	characteristic
376	C₄×C₃₁	—	2	4	no	—
377	C₄×C₃₁	—	2	4	no	—
378	C₄×C₃₁	—	2	2	no	—
379	C₄×C₃₁	—	2	4	no	—
380	C₄×C₃₁	—	2	2	no	—
381	C₄×C₃₁	—	2	2	no	—
382	C₄×C₃₁	—	2	2	no	—
383	C₄×C₃₁	—	2	4	no	—
384	C₄×C₃₁	—	4	2	no	—
385	C₄×C₃₁	—	4	4	no	—
386	C₄×C₃₁	—	4	1	no	—
387	C₄×C₃₁	—	4	2	no	—
388	C₄×C₃₁	—	4	2	no	—
389	C₄×C₃₁	C₄×#41	1	4	yes	—
390	C₄×C₃₁	#518	1	4	yes	—
391	C₂²×#41	C₂²×C₃₁	1	2	yes	—
392	#550	C₄×C₃₁	1	16	yes	complemented
393	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
394	C₂×#127	C₂²×C₃₁	1	2	yes	—
395	C₂×#121	C₂²×C₃₁	1	4	yes	—
396	C₂×#121	C₂²×C₃₁	1	2	yes	—
397	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
398	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
399	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
400	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
401	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
402	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
403	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
404	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
405	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
406	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
407	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
408	C₂×#117	C₂²×C₃₁	1	2	yes	—
409	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
410	#391	C₂²×C₃₁	1	1	yes	characteristic
411	#391	C₂²×C₃₁	1	2	yes	—
412	#391	C₂²×C₃₁	1	1	yes	characteristic
413	#390	C₂²×C₃₁	1	2	yes	—
414	#390	C₂²×C₃₁	1	2	yes	—
415	#390	C₂²×C₃₁	1	2	yes	—
416	#390	C₂²×C₃₁	1	2	yes	—
417	#390	C₂²×C₃₁	1	2	yes	—
418	#390	C₂²×C₃₁	1	4	yes	—
419	#390	C₂²×C₃₁	1	2	yes	—
420	#389	C₂²×C₃₁	1	1	yes	characteristic
421	#389	C₂²×C₃₁	1	1	yes	characteristic
422	#388	C₂²×C₃₁	1	4	yes	—
423	#388	C₂²×C₃₁	1	4	yes	—
424	#388	C₂²×C₃₁	1	4	yes	—
425	#387	C₂²×C₃₁	1	2	yes	—
426	#387	C₂²×C₃₁	1	4	yes	—
427	#387	C₂²×C₃₁	1	2	yes	—
428	#386	C₂²×C₃₁	1	4	yes	—
429	#386	C₂²×C₃₁	1	4	yes	—
430	#386	C₂²×C₃₁	1	4	yes	—
431	#386	C₂²×C₃₁	1	4	yes	—
432	#386	C₂²×C₃₁	1	4	yes	—
433	#386	C₂²×C₃₁	1	4	yes	—
434	#386	C₂²×C₃₁	1	4	yes	—
435	#384	C₂²×C₃₁	1	1	yes	characteristic
436	#382	C₂²×C₃₁	1	2	yes	—
437	#381	C₂²×C₃₁	1	4	yes	—
438	#380	C₂²×C₃₁	1	2	yes	—
439	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
440	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
441	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
442	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
443	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
444	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
445	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
446	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
447	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
448	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
449	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
450	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
451	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
452	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
453	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
454	#376	C₂²×C₃₁	1	2	yes	—
455	C₂×#95	C₂²×C₃₁	1	4	yes	—
456	C₄³	C₂²×C₃₁	1	1	yes	characteristic
457	C₂×C₃₁	—	4	4	no	—
458	C₂×C₃₁	—	4	8	no	—
459	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
460	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
461	C₂×C₃₁	C₄×H(5,𝔽₂)	1	1	yes	characteristic
462	C₂×C₃₁	#3506	1	1	yes	characteristic
463	C₂×C₃₁	#3508	1	1	yes	characteristic
464	C₂×C₃₁	#3517	1	1	yes	characteristic
465	C₂×C₃₁	#3519	1	1	yes	characteristic
466	C₂×#41	—	4	4	no	—
467	C₂×#41	—	4	4	no	—
468	C₂×#41	C₂×C₄×C₃₁	1	16	yes	complemented
469	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
470	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
471	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
472	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
473	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
474	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
475	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
476	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
477	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
478	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
479	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
480	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
481	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
482	#127	C₂×C₄×C₃₁	1	8	yes	complemented
483	#127	C₂×C₄×C₃₁	1	16	yes	complemented
484	#121	C₂×C₄×C₃₁	1	16	yes	complemented
485	#121	C₂×C₄×C₃₁	1	32	yes	complemented
486	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
487	C₄×D₄	—	2	4	no	—
488	C₄×D₄	—	2	8	no	—
489	C₄×D₄	—	2	4	no	—
490	C₄×D₄	—	2	8	no	—
491	C₄×D₄	—	4	4	no	—
492	C₄×D₄	—	4	4	no	—
493	C₄×D₄	—	4	4	no	—
494	C₄×D₄	—	4	8	no	—
495	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
496	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
497	#117	—	4	4	no	—
498	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
499	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
500	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
501	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
502	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
503	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
504	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
505	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
506	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
507	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
508	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
509	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
510	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
511	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
512	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
513	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
514	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
515	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
516	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
517	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
518	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
519	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
520	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
521	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
522	C₂×C₄²	—	4	4	no	—
523	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
524	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
525	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
526	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
527	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
528	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
529	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
530	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
531	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
532	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
533	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
534	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
535	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
536	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
537	#95	C₂³×C₃₁	1	2	yes	—
538	#95	C₂³×C₃₁	1	2	yes	—
539	#95	C₂³×C₃₁	1	4	yes	—
540	#95	C₂³×C₃₁	1	4	yes	—
541	#95	C₂³×C₃₁	1	2	yes	—
542	#95	C₂³×C₃₁	1	2	yes	—
543	#95	C₂³×C₃₁	1	2	yes	—
544	#95	C₂³×C₃₁	1	2	yes	—
545	#95	C₂³×C₃₁	1	2	yes	—
546	#95	C₂³×C₃₁	1	2	yes	—
547	#95	C₂³×C₃₁	1	4	yes	—
548	#95	C₂³×C₃₁	1	2	yes	—
549	#95	C₂³×C₃₁	1	4	yes	—
550	#95	C₂³×C₃₁	1	2	yes	—
551	C₃₁	#34330	1	1	yes	characteristic, direct product factor, Sylow
552	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
553	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
554	#41	—	4	32	no	—
555	C₂×Q₈	—	2	2	no	—
556	C₂×Q₈	—	2	2	no	—
557	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
558	C₂×D₄	—	2	2	no	—
559	C₂×D₄	—	2	4	no	—
560	C₂×D₄	—	2	2	no	—
561	C₂×D₄	—	2	4	no	—
562	C₂×D₄	—	2	4	no	—
563	C₂×D₄	—	2	4	no	—
564	C₂×D₄	—	4	2	no	—
565	C₂×D₄	—	4	4	no	—
566	C₂×D₄	—	4	2	no	—
567	C₂×D₄	—	4	4	no	—
568	C₂×D₄	—	4	4	no	—
569	C₂×D₄	—	4	4	no	—
570	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
571	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
572	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
573	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
574	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
575	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
576	C₂²×C₄	—	2	4	no	—
577	C₂²×C₄	—	2	4	no	—
578	C₂²×C₄	—	2	4	no	—
579	C₂²×C₄	—	2	4	no	—
580	C₂²×C₄	—	2	4	no	—
581	C₂²×C₄	—	2	8	no	—
582	C₂²×C₄	—	2	8	no	—
583	C₂²×C₄	—	4	2	no	—
584	C₂²×C₄	—	4	2	no	—
585	C₂²×C₄	—	4	2	no	—
586	C₂²×C₄	—	4	4	no	—
587	C₂²×C₄	—	4	4	no	—
588	C₂²×C₄	—	4	4	no	—
589	C₂²×C₄	—	4	4	no	—
590	C₂²×C₄	—	4	4	no	—
591	C₂²×C₄	—	4	4	no	—
592	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
593	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
594	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
595	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
596	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
597	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
598	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
599	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
600	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
601	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
602	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
603	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
604	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
605	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
606	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
607	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
608	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
609	C₄⋊C₄	—	2	2	no	—
610	C₄⋊C₄	—	2	4	no	—
611	C₄⋊C₄	—	2	2	no	—
612	C₄⋊C₄	—	2	4	no	—
613	C₄⋊C₄	—	2	4	no	—
614	C₄⋊C₄	—	2	2	no	—
615	C₄⋊C₄	—	4	2	no	—
616	C₄⋊C₄	—	4	2	no	—
617	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
618	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
619	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
620	C₄⋊C₄	#41×C₃₁	1	4	yes	complemented
621	C₂²⋊C₄	—	2	4	no	—
622	C₂²⋊C₄	—	2	8	no	—
623	C₂²⋊C₄	—	2	8	no	—
624	C₂²⋊C₄	—	2	4	no	—
625	C₂²⋊C₄	—	2	8	no	—
626	C₂²⋊C₄	—	4	4	no	—
627	C₂²⋊C₄	—	4	4	no	—
628	C₂²⋊C₄	—	4	4	no	—
629	C₂²⋊C₄	—	4	4	no	—
630	C₂²⋊C₄	—	4	4	no	—
631	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
632	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
633	C₄²	—	2	2	no	—
634	C₄²	—	2	4	no	—
635	C₄²	—	2	2	no	—
636	C₄²	—	2	4	no	—
637	C₄²	—	2	2	no	—
638	C₄²	—	2	2	no	—
639	C₄²	—	4	2	no	—
640	C₄²	—	4	2	no	—
641	C₄²	—	4	1	no	—
642	C₄²	—	4	4	no	—
643	C₄²	—	4	4	no	—
644	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
645	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
646	C₄²	#41×C₃₁	1	4	yes	complemented
647	C₂³	—	2	2	no	—
648	C₂³	—	2	4	no	—
649	C₂³	—	2	2	no	—
650	C₂³	—	2	4	no	—
651	C₂³	—	2	4	no	—
652	C₂³	—	2	2	no	—
653	C₂³	—	2	2	no	—
654	C₂³	—	2	4	no	—
655	C₂³	—	2	4	no	—
656	C₂³	—	2	2	no	—
657	C₂³	—	2	4	no	—
658	C₂³	—	2	2	no	—
659	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
660	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
661	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
662	Q₈	—	2	8	no	—
663	D₄	—	2	8	no	—
664	D₄	—	2	16	no	—
665	D₄	—	2	16	no	—
666	D₄	—	4	8	no	—
667	D₄	—	4	16	no	—
668	D₄	—	4	16	no	—
669	C₂×C₄	—	2	1	no	—
670	C₂×C₄	—	2	1	no	—
671	C₂×C₄	—	2	1	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	2	1	no	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	2	1	no	—
676	C₂×C₄	—	2	2	no	—
677	C₂×C₄	—	2	2	no	—
678	C₂×C₄	—	2	2	no	—
679	C₂×C₄	—	2	2	no	—
680	C₂×C₄	—	2	2	no	—
681	C₂×C₄	—	2	2	no	—
682	C₂×C₄	—	2	1	no	—
683	C₂×C₄	—	2	1	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	16	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	2	no	—
688	C₂×C₄	—	2	2	no	—
689	C₂×C₄	—	2	2	no	—
690	C₂×C₄	—	2	4	no	—
691	C₂×C₄	—	2	2	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	4	no	—
695	C₂×C₄	—	2	4	no	—
696	C₂×C₄	—	2	2	no	—
697	C₂×C₄	—	2	2	no	—
698	C₂×C₄	—	2	1	no	—
699	C₂×C₄	—	2	2	no	—
700	C₂×C₄	—	2	2	no	—
701	C₂×C₄	—	4	16	no	—
702	C₂×C₄	—	4	8	no	—
703	C₂×C₄	—	4	8	no	—
704	C₂×C₄	—	8	8	no	—
705	C₂×C₄	—	8	8	no	—
706	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
707	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
708	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
709	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
710	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
711	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
712	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
713	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
714	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
715	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
716	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
717	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
718	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
719	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
720	C₂²	—	2	4	no	—
721	C₂²	—	2	8	no	—
722	C₂²	—	2	4	no	—
723	C₂²	—	2	8	no	—
724	C₂²	—	2	8	no	—
725	C₂²	—	2	4	no	—
726	C₂²	—	4	2	no	—
727	C₂²	—	4	4	no	—
728	C₂²	—	4	4	no	—
729	C₂²	—	4	2	no	—
730	C₂²	—	4	2	no	—
731	C₂²	—	4	4	no	—
732	C₂²	—	4	2	no	—
733	C₂²	—	4	4	no	—
734	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
735	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
736	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
737	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
738	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
739	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
740	C₂²	C₃₁×#585	1	1	yes	characteristic
741	C₄	—	2	4	no	—
742	C₄	—	2	4	no	—
743	C₄	—	2	2	no	—
744	C₄	—	2	4	no	—
745	C₄	—	2	2	no	—
746	C₄	—	2	2	no	—
747	C₄	—	2	2	no	—
748	C₄	—	2	4	no	—
749	C₄	—	4	2	no	—
750	C₄	—	4	4	no	—
751	C₄	—	4	1	no	—
752	C₄	—	4	2	no	—
753	C₄	—	4	2	no	—
754	C₄	C₄×#41×C₃₁	1	4	yes	—
755	C₄	C₃₁×#518	1	4	yes	—
756	C₂	—	4	4	no	—
757	C₂	—	4	8	no	—
758	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
759	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
760	C₂	C₄×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
761	C₂	C₃₁×#3506	1	1	yes	characteristic
762	C₂	C₃₁×#3508	1	1	yes	characteristic
763	C₂	C₃₁×#3517	1	1	yes	characteristic
764	C₂	C₃₁×#3519	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#550	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2366	C₂	semidirect
3. C₃₁×#2366	C₂	semidirect
4. C₃₁×#2364	C₂	semidirect
5. C₃₁×#2363	C₂	semidirect
6. C₃₁×#2357	C₂	semidirect
7. C₃₁×#2355	C₂	semidirect
8. C₃₁×#2354	C₂	semidirect
9. C₃₁×#2339	C₂	semidirect
10. C₃₁×#2333	C₂	semidirect
11. C₃₁×#2328	C₂	semidirect
12. C₄×C₃₁×#127	C₂	semidirect
13. C₄×C₃₁×#121	C₂	semidirect
14. C₃₁×#2297	C₂	semidirect
15. C₃₁×#2295	C₂	semidirect
16. C₃₁×#2275	C₂	semidirect
17. C₄²×D₄×C₃₁	C₂	semidirect
19. C₃₁×#550	C₄	semidirect
86. C₂×#41×C₃₁	C₂×C₄	semidirect
100. C₃₁×#127	C₂×C₄	semidirect
101. C₃₁×#127	C₂×C₄	semidirect
102. C₃₁×#121	C₂×C₄	semidirect
103. C₃₁×#121	C₂×C₄	semidirect
104. C₄×Q₈×C₃₁	C₂×C₄	semidirect
113. C₄×D₄×C₃₁	C₂×C₄	semidirect
114. C₄×D₄×C₃₁	C₂×C₄	semidirect
237. C₄⋊C₄×C₃₁	#41	semidirect
263. C₄²×C₃₁	#41	semidirect
264. #34330	C₃₁	direct (N × H)
339. #2366	C₂×C₃₁	semidirect
340. #2366	C₂×C₃₁	semidirect
341. #2364	C₂×C₃₁	semidirect
342. #2363	C₂×C₃₁	semidirect
343. #2357	C₂×C₃₁	semidirect
344. #2355	C₂×C₃₁	semidirect
345. #2354	C₂×C₃₁	semidirect
346. #2339	C₂×C₃₁	semidirect
347. #2333	C₂×C₃₁	semidirect
348. #2328	C₂×C₃₁	semidirect
349. C₄×#127	C₂×C₃₁	semidirect
350. C₄×#121	C₂×C₃₁	semidirect
351. #2297	C₂×C₃₁	semidirect
352. #2295	C₂×C₃₁	semidirect
353. #2275	C₂×C₃₁	semidirect
354. C₄²×D₄	C₂×C₃₁	semidirect
392. #550	C₄×C₃₁	semidirect
468. C₂×#41	C₂×C₄×C₃₁	semidirect
482. #127	C₂×C₄×C₃₁	semidirect
483. #127	C₂×C₄×C₃₁	semidirect
484. #121	C₂×C₄×C₃₁	semidirect
485. #121	C₂×C₄×C₃₁	semidirect
486. C₄×Q₈	C₂×C₄×C₃₁	semidirect
495. C₄×D₄	C₂×C₄×C₃₁	semidirect
496. C₄×D₄	C₂×C₄×C₃₁	semidirect
551. C₃₁	#34330	direct (N × H)
620. C₄⋊C₄	#41×C₃₁	semidirect
646. C₄²	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#550	2
C₃₁×#2366	2
C₃₁×#2364	2
C₃₁×#2363	2
C₃₁×#2357	2
C₃₁×#2355	2
C₃₁×#2354	2
C₃₁×#2339	2
C₃₁×#2333	2
C₃₁×#2328	2
C₄×C₃₁×#127	2
C₄×C₃₁×#121	2
C₃₁×#2297	2
C₃₁×#2295	2
C₃₁×#2275	2
C₄²×D₄×C₃₁	2
#34330	31

Subgroups — C31×#34330 #21491089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34330 #21491089