Subgroups — C₃₁×#36716 #21493475

Order 7936.

2406 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#538	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2849	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2700	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2698	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2681	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2680	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2622	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2591	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2583	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2579	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2504	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2502	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2409	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2312	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#538	—	2	8	no	—
19	C₂×C₃₁×#126	C₂²	1	2	yes	—
20	C₂×C₃₁×#124	C₂²	1	2	yes	—
21	C₂×C₃₁×#123	C₂²	1	2	yes	—
22	C₂×C₃₁×#123	C₂²	1	4	yes	—
23	C₂×C₃₁×C₂²⋊Q₈	—	2	8	no	—
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
27	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
28	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#121	C₂²	1	2	yes	—
30	C₂×C₃₁×#120	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#400	C₂²	1	2	yes	complemented
38	C₃₁×#398	C₂²	1	4	yes	complemented
39	C₃₁×#398	C₂²	1	2	yes	complemented
40	C₃₁×#398	C₂²	1	4	yes	complemented
41	C₃₁×#397	C₂²	1	4	yes	—
42	C₃₁×#397	C₂²	1	4	yes	—
43	C₃₁×#397	C₂²	1	4	yes	complemented
44	C₃₁×#396	C₂²	1	4	yes	complemented
45	C₃₁×#395	C₂²	1	2	yes	complemented
46	C₃₁×#395	C₂²	1	4	yes	complemented
47	C₃₁×#394	C₂²	1	4	yes	—
48	C₃₁×#394	C₂²	1	4	yes	—
49	C₃₁×#394	C₂²	1	2	yes	complemented
50	C₃₁×#393	C₂²	1	4	yes	complemented
51	C₃₁×#391	C₂²	1	4	yes	complemented
52	C₃₁×#391	C₂²	1	2	yes	complemented
53	C₃₁×#390	C₂²	1	2	yes	complemented
54	C₃₁×#389	C₂²	1	4	yes	complemented
55	C₃₁×#388	C₂²	1	4	yes	—
56	C₃₁×#388	C₂²	1	4	yes	—
57	C₃₁×#388	C₂²	1	4	yes	complemented
58	C₃₁×#387	C₂²	1	4	yes	complemented
59	C₃₁×#386	C₂²	1	2	yes	—
60	C₃₁×#386	C₂²	1	2	yes	—
61	C₃₁×#386	C₂²	1	2	yes	—
62	C₃₁×#386	C₂²	1	4	yes	—
63	C₃₁×#386	C₂²	1	2	yes	—
64	C₃₁×#385	C₂²	1	2	yes	—
65	C₃₁×#385	C₂²	1	2	yes	—
66	C₃₁×#385	C₂²	1	2	yes	—
67	C₃₁×#385	C₂²	1	4	yes	—
68	C₃₁×#385	C₂²	1	2	yes	—
69	C₃₁×#385	C₂²	1	4	yes	complemented
70	C₃₁×#384	C₂²	1	2	yes	complemented
71	C₃₁×#382	C₂²	1	2	yes	complemented
72	C₃₁×#381	C₂²	1	4	yes	—
73	C₃₁×#380	C₂²	1	2	yes	—
74	C₃₁×#379	C₂²	1	2	yes	—
75	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
76	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
77	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₃₁	—	2	4	no	—
79	C₂²×Q₈×C₃₁	—	4	4	no	—
80	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
81	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
83	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
84	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
85	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₃₁	—	2	4	no	—
87	C₂³×C₄×C₃₁	—	4	4	no	—
88	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
90	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
91	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
92	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
93	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
94	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
95	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
96	C₃₁×#126	—	2	8	no	—
97	C₃₁×#124	—	2	8	no	—
98	C₃₁×#123	—	2	8	no	—
99	C₃₁×#123	—	4	8	no	—
100	C₃₁×C₂²⋊Q₈	—	2	4	no	—
101	C₃₁×C₂²⋊Q₈	—	2	16	no	—
102	C₃₁×C₂²⋊Q₈	—	2	16	no	—
103	C₃₁×C₂²⋊Q₈	—	4	8	no	—
104	C₃₁×#121	—	2	4	no	—
105	C₃₁×#121	—	2	8	no	—
106	C₃₁×#120	—	2	8	no	—
107	C₄×D₄×C₃₁	—	2	8	no	—
108	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
109	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	1	yes	characteristic
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
124	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
126	C₂×C₂²⋊C₄×C₃₁	D₄	1	8	yes	—
127	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
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	2	yes	—
132	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
135	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	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	4	yes	—
139	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
140	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
141	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
142	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
143	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
144	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
145	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
146	C₃₁×#95	C₂³	1	4	yes	—
147	C₃₁×#95	C₂³	1	2	yes	—
148	C₃₁×#95	C₂³	1	4	yes	—
149	C₃₁×#95	C₂³	1	4	yes	—
150	C₃₁×#95	C₂³	1	4	yes	—
151	C₃₁×#95	C₂³	1	4	yes	—
152	C₃₁×#95	C₂³	1	4	yes	—
153	C₃₁×#95	C₂³	1	4	yes	—
154	C₃₁×#95	C₂³	1	4	yes	—
155	C₃₁×#95	C₂³	1	4	yes	—
156	C₃₁×#95	C₂³	1	2	yes	—
157	C₂⁴×C₃₁	—	2	4	no	—
158	C₂⁴×C₃₁	—	4	2	no	—
159	C₂⁴×C₃₁	—	4	2	no	—
160	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
161	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
162	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
163	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
164	C₂×Q₈×C₃₁	—	2	8	no	—
165	C₂×Q₈×C₃₁	—	2	2	no	—
166	C₂×Q₈×C₃₁	—	2	8	no	—
167	C₂×Q₈×C₃₁	—	4	4	no	—
168	C₂×Q₈×C₃₁	—	4	2	no	—
169	C₂×Q₈×C₃₁	—	4	16	no	—
170	C₂×D₄×C₃₁	—	2	4	no	—
171	C₂×D₄×C₃₁	—	2	8	no	—
172	C₂×D₄×C₃₁	—	2	4	no	—
173	C₂×D₄×C₃₁	—	2	4	no	—
174	C₂×D₄×C₃₁	—	2	2	no	—
175	C₂×D₄×C₃₁	—	4	4	no	—
176	C₂×D₄×C₃₁	—	4	2	no	—
177	C₂×D₄×C₃₁	—	4	4	no	—
178	C₂²×C₄×C₃₁	—	2	2	no	—
179	C₂²×C₄×C₃₁	—	2	4	no	—
180	C₂²×C₄×C₃₁	—	2	2	no	—
181	C₂²×C₄×C₃₁	—	2	2	no	—
182	C₂²×C₄×C₃₁	—	2	2	no	—
183	C₂²×C₄×C₃₁	—	2	8	no	—
184	C₂²×C₄×C₃₁	—	2	8	no	—
185	C₂²×C₄×C₃₁	—	2	4	no	—
186	C₂²×C₄×C₃₁	—	4	4	no	—
187	C₂²×C₄×C₃₁	—	4	2	no	—
188	C₂²×C₄×C₃₁	—	4	2	no	—
189	C₂²×C₄×C₃₁	—	4	2	no	—
190	C₂²×C₄×C₃₁	—	4	2	no	—
191	C₂²×C₄×C₃₁	—	4	2	no	—
192	C₂²×C₄×C₃₁	—	4	4	no	—
193	C₂²×C₄×C₃₁	—	4	16	no	—
194	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
195	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
196	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
198	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
199	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
205	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
206	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
207	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
208	C₄⋊C₄×C₃₁	—	2	4	no	—
209	C₄⋊C₄×C₃₁	—	2	4	no	—
210	C₄⋊C₄×C₃₁	—	2	8	no	—
211	C₄⋊C₄×C₃₁	—	2	4	no	—
212	C₄⋊C₄×C₃₁	—	2	4	no	—
213	C₄⋊C₄×C₃₁	—	2	4	no	—
214	C₄⋊C₄×C₃₁	—	2	4	no	—
215	C₄⋊C₄×C₃₁	—	2	2	no	—
216	C₄⋊C₄×C₃₁	—	2	2	no	—
217	C₄⋊C₄×C₃₁	—	2	8	no	—
218	C₄⋊C₄×C₃₁	—	4	4	no	—
219	C₄⋊C₄×C₃₁	—	4	4	no	—
220	C₄⋊C₄×C₃₁	—	4	4	no	—
221	C₄⋊C₄×C₃₁	—	4	4	no	—
222	C₂²⋊C₄×C₃₁	—	2	8	no	—
223	C₂²⋊C₄×C₃₁	—	2	4	no	—
224	C₂²⋊C₄×C₃₁	—	2	4	no	—
225	C₂²⋊C₄×C₃₁	—	2	4	no	—
226	C₂²⋊C₄×C₃₁	—	2	4	no	—
227	C₂²⋊C₄×C₃₁	—	2	4	no	—
228	C₂²⋊C₄×C₃₁	—	2	4	no	—
229	C₂²⋊C₄×C₃₁	—	2	4	no	—
230	C₂²⋊C₄×C₃₁	—	2	8	no	—
231	C₂²⋊C₄×C₃₁	—	4	4	no	—
232	C₂²⋊C₄×C₃₁	—	4	4	no	—
233	C₂²⋊C₄×C₃₁	—	4	4	no	—
234	C₂²⋊C₄×C₃₁	—	4	4	no	—
235	C₂²⋊C₄×C₃₁	—	4	4	no	—
236	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
237	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
238	C₄²×C₃₁	—	2	4	no	—
239	C₄²×C₃₁	—	4	2	no	—
240	C₄²×C₃₁	—	4	2	no	—
241	#36716	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂³×C₃₁	—	2	2	no	—
243	C₂³×C₃₁	—	2	2	no	—
244	C₂³×C₃₁	—	2	8	no	—
245	C₂³×C₃₁	—	2	4	no	—
246	C₂³×C₃₁	—	2	4	no	—
247	C₂³×C₃₁	—	2	4	no	—
248	C₂³×C₃₁	—	2	2	no	—
249	C₂³×C₃₁	—	2	4	no	—
250	C₂³×C₃₁	—	2	4	no	—
251	C₂³×C₃₁	—	2	4	no	—
252	C₂³×C₃₁	—	4	4	no	—
253	C₂³×C₃₁	—	4	4	no	—
254	C₂³×C₃₁	—	4	16	no	—
255	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
256	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
257	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
258	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
259	Q₈×C₃₁	—	4	16	no	—
260	Q₈×C₃₁	—	8	2	no	—
261	D₄×C₃₁	—	4	8	no	—
262	D₄×C₃₁	—	8	2	no	—
263	D₄×C₃₁	—	8	4	no	—
264	D₄×C₃₁	—	8	4	no	—
265	C₂×C₄×C₃₁	—	2	8	no	—
266	C₂×C₄×C₃₁	—	2	2	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	2	2	no	—
269	C₂×C₄×C₃₁	—	2	2	no	—
270	C₂×C₄×C₃₁	—	2	2	no	—
271	C₂×C₄×C₃₁	—	2	2	no	—
272	C₂×C₄×C₃₁	—	2	2	no	—
273	C₂×C₄×C₃₁	—	2	2	no	—
274	C₂×C₄×C₃₁	—	2	2	no	—
275	C₂×C₄×C₃₁	—	2	2	no	—
276	C₂×C₄×C₃₁	—	2	2	no	—
277	C₂×C₄×C₃₁	—	2	4	no	—
278	C₂×C₄×C₃₁	—	2	4	no	—
279	C₂×C₄×C₃₁	—	2	4	no	—
280	C₂×C₄×C₃₁	—	2	4	no	—
281	C₂×C₄×C₃₁	—	2	4	no	—
282	C₂×C₄×C₃₁	—	2	4	no	—
283	C₂×C₄×C₃₁	—	2	4	no	—
284	C₂×C₄×C₃₁	—	2	4	no	—
285	C₂×C₄×C₃₁	—	2	4	no	—
286	C₂×C₄×C₃₁	—	2	4	no	—
287	C₂×C₄×C₃₁	—	2	4	no	—
288	C₂×C₄×C₃₁	—	2	4	no	—
289	C₂×C₄×C₃₁	—	4	4	no	—
290	C₂×C₄×C₃₁	—	4	4	no	—
291	C₂×C₄×C₃₁	—	4	16	no	—
292	C₂×C₄×C₃₁	—	4	4	no	—
293	C₂×C₄×C₃₁	—	4	4	no	—
294	C₂×C₄×C₃₁	—	4	4	no	—
295	C₂×C₄×C₃₁	—	8	4	no	—
296	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
297	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
298	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
299	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
300	C₂×#538	C₂×C₃₁	1	1	yes	characteristic
301	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#2849	C₂×C₃₁	1	2	yes	complemented
303	#2700	C₂×C₃₁	1	4	yes	complemented
304	#2698	C₂×C₃₁	1	4	yes	complemented
305	#2681	C₂×C₃₁	1	4	yes	complemented
306	#2680	C₂×C₃₁	1	4	yes	complemented
307	#2622	C₂×C₃₁	1	2	yes	complemented
308	#2591	C₂×C₃₁	1	1	yes	characteristic, complemented
309	#2583	C₂×C₃₁	1	1	yes	characteristic, complemented
310	#2579	C₂×C₃₁	1	2	yes	complemented
311	#2504	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#2502	C₂×C₃₁	1	1	yes	characteristic, complemented
313	#2409	C₂×C₃₁	1	2	yes	complemented
314	#2312	C₂×C₃₁	1	1	yes	characteristic, complemented
315	C₂²×C₃₁	—	2	2	no	—
316	C₂²×C₃₁	—	2	2	no	—
317	C₂²×C₃₁	—	2	8	no	—
318	C₂²×C₃₁	—	2	4	no	—
319	C₂²×C₃₁	—	2	4	no	—
320	C₂²×C₃₁	—	2	4	no	—
321	C₂²×C₃₁	—	4	16	no	—
322	C₂²×C₃₁	—	4	8	no	—
323	C₂²×C₃₁	—	4	2	no	—
324	C₂²×C₃₁	—	4	2	no	—
325	C₂²×C₃₁	—	4	2	no	—
326	C₂²×C₃₁	—	4	8	no	—
327	C₂²×C₃₁	#536	1	4	yes	—
328	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
329	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
330	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
331	C₂²×C₃₁	#585	1	2	yes	—
332	C₄×C₃₁	—	2	4	no	—
333	C₄×C₃₁	—	2	4	no	—
334	C₄×C₃₁	—	2	2	no	—
335	C₄×C₃₁	—	2	2	no	—
336	C₄×C₃₁	—	4	2	no	—
337	C₄×C₃₁	—	4	2	no	—
338	C₄×C₃₁	—	4	2	no	—
339	C₄×C₃₁	—	4	2	no	—
340	C₄×C₃₁	—	4	2	no	—
341	C₄×C₃₁	—	4	2	no	—
342	C₄×C₃₁	—	4	4	no	—
343	C₄×C₃₁	—	4	4	no	—
344	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
345	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
346	#538	—	2	8	no	—
347	C₂×#126	C₂²×C₃₁	1	2	yes	—
348	C₂×#124	C₂²×C₃₁	1	2	yes	—
349	C₂×#123	C₂²×C₃₁	1	2	yes	—
350	C₂×#123	C₂²×C₃₁	1	4	yes	—
351	C₂×C₂²⋊Q₈	—	2	8	no	—
352	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
353	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
354	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
355	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
356	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
357	C₂×#121	C₂²×C₃₁	1	2	yes	—
358	C₂×#120	C₂²×C₃₁	1	2	yes	—
359	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
360	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
361	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
362	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
363	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
364	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
365	#400	C₂²×C₃₁	1	2	yes	complemented
366	#398	C₂²×C₃₁	1	4	yes	complemented
367	#398	C₂²×C₃₁	1	2	yes	complemented
368	#398	C₂²×C₃₁	1	4	yes	complemented
369	#397	C₂²×C₃₁	1	4	yes	—
370	#397	C₂²×C₃₁	1	4	yes	—
371	#397	C₂²×C₃₁	1	4	yes	complemented
372	#396	C₂²×C₃₁	1	4	yes	complemented
373	#395	C₂²×C₃₁	1	2	yes	complemented
374	#395	C₂²×C₃₁	1	4	yes	complemented
375	#394	C₂²×C₃₁	1	4	yes	—
376	#394	C₂²×C₃₁	1	4	yes	—
377	#394	C₂²×C₃₁	1	2	yes	complemented
378	#393	C₂²×C₃₁	1	4	yes	complemented
379	#391	C₂²×C₃₁	1	4	yes	complemented
380	#391	C₂²×C₃₁	1	2	yes	complemented
381	#390	C₂²×C₃₁	1	2	yes	complemented
382	#389	C₂²×C₃₁	1	4	yes	complemented
383	#388	C₂²×C₃₁	1	4	yes	—
384	#388	C₂²×C₃₁	1	4	yes	—
385	#388	C₂²×C₃₁	1	4	yes	complemented
386	#387	C₂²×C₃₁	1	4	yes	complemented
387	#386	C₂²×C₃₁	1	2	yes	—
388	#386	C₂²×C₃₁	1	2	yes	—
389	#386	C₂²×C₃₁	1	2	yes	—
390	#386	C₂²×C₃₁	1	4	yes	—
391	#386	C₂²×C₃₁	1	2	yes	—
392	#385	C₂²×C₃₁	1	2	yes	—
393	#385	C₂²×C₃₁	1	2	yes	—
394	#385	C₂²×C₃₁	1	2	yes	—
395	#385	C₂²×C₃₁	1	4	yes	—
396	#385	C₂²×C₃₁	1	2	yes	—
397	#385	C₂²×C₃₁	1	4	yes	complemented
398	#384	C₂²×C₃₁	1	2	yes	complemented
399	#382	C₂²×C₃₁	1	2	yes	complemented
400	#381	C₂²×C₃₁	1	4	yes	—
401	#380	C₂²×C₃₁	1	2	yes	—
402	#379	C₂²×C₃₁	1	2	yes	—
403	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
404	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
405	C₂×C₃₁	—	2	4	no	—
406	C₂×C₃₁	—	4	4	no	—
407	C₂×C₃₁	—	8	2	no	—
408	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
409	C₂×C₃₁	#3484	1	1	yes	characteristic
410	C₂×C₃₁	#3485	1	1	yes	characteristic
411	C₂×C₃₁	#3541	1	4	yes	—
412	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
413	C₂²×Q₈	—	2	4	no	—
414	C₂²×Q₈	—	4	4	no	—
415	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
416	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
417	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
418	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
419	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
420	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
421	C₂³×C₄	—	2	4	no	—
422	C₂³×C₄	—	4	4	no	—
423	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
424	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
425	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
426	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
427	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
428	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
429	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
430	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
431	#126	—	2	8	no	—
432	#124	—	2	8	no	—
433	#123	—	2	8	no	—
434	#123	—	4	8	no	—
435	C₂²⋊Q₈	—	2	4	no	—
436	C₂²⋊Q₈	—	2	16	no	—
437	C₂²⋊Q₈	—	2	16	no	—
438	C₂²⋊Q₈	—	4	8	no	—
439	#121	—	2	4	no	—
440	#121	—	2	8	no	—
441	#120	—	2	8	no	—
442	C₄×D₄	—	2	8	no	—
443	C₂×C₄⋊C₄	—	2	4	no	—
444	C₂×C₄⋊C₄	—	2	4	no	—
445	C₂×C₄⋊C₄	—	2	4	no	—
446	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
448	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
449	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
450	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
451	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
453	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
454	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
455	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
456	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
457	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
458	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
459	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
460	C₂×C₂²⋊C₄	—	2	4	no	—
461	C₂×C₂²⋊C₄	D₄×C₃₁	1	8	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
465	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
466	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
467	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
468	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
469	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
470	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
471	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
472	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
473	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
474	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
475	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
476	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
477	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
478	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
479	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
480	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
481	#95	C₂³×C₃₁	1	2	yes	—
482	#95	C₂³×C₃₁	1	2	yes	—
483	#95	C₂³×C₃₁	1	4	yes	—
484	#95	C₂³×C₃₁	1	4	yes	—
485	#95	C₂³×C₃₁	1	4	yes	—
486	#95	C₂³×C₃₁	1	4	yes	—
487	#95	C₂³×C₃₁	1	4	yes	—
488	#95	C₂³×C₃₁	1	4	yes	—
489	#95	C₂³×C₃₁	1	4	yes	—
490	#95	C₂³×C₃₁	1	4	yes	—
491	#95	C₂³×C₃₁	1	4	yes	—
492	C₃₁	#36716	1	1	yes	characteristic, direct product factor, Sylow
493	C₂⁴	—	2	4	no	—
494	C₂⁴	—	4	2	no	—
495	C₂⁴	—	4	2	no	—
496	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
497	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
498	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
499	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
500	C₂×Q₈	—	2	8	no	—
501	C₂×Q₈	—	2	2	no	—
502	C₂×Q₈	—	2	8	no	—
503	C₂×Q₈	—	4	4	no	—
504	C₂×Q₈	—	4	2	no	—
505	C₂×Q₈	—	4	16	no	—
506	C₂×D₄	—	2	4	no	—
507	C₂×D₄	—	2	8	no	—
508	C₂×D₄	—	2	4	no	—
509	C₂×D₄	—	2	4	no	—
510	C₂×D₄	—	2	2	no	—
511	C₂×D₄	—	4	4	no	—
512	C₂×D₄	—	4	2	no	—
513	C₂×D₄	—	4	4	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	2	2	no	—
517	C₂²×C₄	—	2	2	no	—
518	C₂²×C₄	—	2	2	no	—
519	C₂²×C₄	—	2	8	no	—
520	C₂²×C₄	—	2	8	no	—
521	C₂²×C₄	—	2	4	no	—
522	C₂²×C₄	—	4	4	no	—
523	C₂²×C₄	—	4	2	no	—
524	C₂²×C₄	—	4	2	no	—
525	C₂²×C₄	—	4	2	no	—
526	C₂²×C₄	—	4	2	no	—
527	C₂²×C₄	—	4	2	no	—
528	C₂²×C₄	—	4	4	no	—
529	C₂²×C₄	—	4	16	no	—
530	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
531	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
532	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
533	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
534	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
535	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
537	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
538	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
539	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
540	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
541	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
542	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
543	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	8	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	4	no	—
550	C₄⋊C₄	—	2	4	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	8	no	—
554	C₄⋊C₄	—	4	4	no	—
555	C₄⋊C₄	—	4	4	no	—
556	C₄⋊C₄	—	4	4	no	—
557	C₄⋊C₄	—	4	4	no	—
558	C₂²⋊C₄	—	2	8	no	—
559	C₂²⋊C₄	—	2	4	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	—	2	4	no	—
562	C₂²⋊C₄	—	2	4	no	—
563	C₂²⋊C₄	—	2	4	no	—
564	C₂²⋊C₄	—	2	4	no	—
565	C₂²⋊C₄	—	2	4	no	—
566	C₂²⋊C₄	—	2	8	no	—
567	C₂²⋊C₄	—	4	4	no	—
568	C₂²⋊C₄	—	4	4	no	—
569	C₂²⋊C₄	—	4	4	no	—
570	C₂²⋊C₄	—	4	4	no	—
571	C₂²⋊C₄	—	4	4	no	—
572	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
573	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
574	C₄²	—	2	4	no	—
575	C₄²	—	4	2	no	—
576	C₄²	—	4	2	no	—
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	8	no	—
580	C₂³	—	2	4	no	—
581	C₂³	—	2	4	no	—
582	C₂³	—	2	4	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	2	4	no	—
585	C₂³	—	2	4	no	—
586	C₂³	—	2	4	no	—
587	C₂³	—	4	4	no	—
588	C₂³	—	4	4	no	—
589	C₂³	—	4	16	no	—
590	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
591	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
592	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
593	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
594	Q₈	—	4	16	no	—
595	Q₈	—	8	2	no	—
596	D₄	—	4	8	no	—
597	D₄	—	8	2	no	—
598	D₄	—	8	4	no	—
599	D₄	—	8	4	no	—
600	C₂×C₄	—	2	8	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	2	4	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	2	4	no	—
619	C₂×C₄	—	2	4	no	—
620	C₂×C₄	—	2	4	no	—
621	C₂×C₄	—	2	4	no	—
622	C₂×C₄	—	2	4	no	—
623	C₂×C₄	—	2	4	no	—
624	C₂×C₄	—	4	4	no	—
625	C₂×C₄	—	4	4	no	—
626	C₂×C₄	—	4	16	no	—
627	C₂×C₄	—	4	4	no	—
628	C₂×C₄	—	4	4	no	—
629	C₂×C₄	—	4	4	no	—
630	C₂×C₄	—	8	4	no	—
631	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
632	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
633	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
634	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	8	no	—
638	C₂²	—	2	4	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	2	4	no	—
641	C₂²	—	4	16	no	—
642	C₂²	—	4	8	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	2	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	8	no	—
647	C₂²	C₃₁×#536	1	4	yes	—
648	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
649	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
650	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
651	C₂²	C₃₁×#585	1	2	yes	—
652	C₄	—	2	4	no	—
653	C₄	—	2	4	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	2	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	2	no	—
660	C₄	—	4	2	no	—
661	C₄	—	4	2	no	—
662	C₄	—	4	4	no	—
663	C₄	—	4	4	no	—
664	C₂	—	2	4	no	—
665	C₂	—	4	4	no	—
666	C₂	—	8	2	no	—
667	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
668	C₂	C₃₁×#3484	1	1	yes	characteristic
669	C₂	C₃₁×#3485	1	1	yes	characteristic
670	C₂	C₃₁×#3541	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#538	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#2849	C₂	semidirect
4. C₃₁×#2700	C₂	semidirect
5. C₃₁×#2698	C₂	semidirect
6. C₃₁×#2681	C₂	semidirect
7. C₃₁×#2680	C₂	semidirect
8. C₃₁×#2622	C₂	semidirect
9. C₃₁×#2591	C₂	semidirect
10. C₃₁×#2583	C₂	semidirect
11. C₃₁×#2579	C₂	semidirect
12. C₃₁×#2504	C₂	semidirect
13. C₃₁×#2502	C₂	semidirect
14. C₃₁×#2409	C₂	semidirect
15. C₃₁×#2312	C₂	semidirect
27. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
37. C₃₁×#400	C₂²	semidirect
38. C₃₁×#398	C₂²	semidirect
39. C₃₁×#398	C₂²	semidirect
40. C₃₁×#398	C₂²	semidirect
43. C₃₁×#397	C₂²	semidirect
44. C₃₁×#396	C₂²	semidirect
45. C₃₁×#395	C₂²	semidirect
46. C₃₁×#395	C₂²	semidirect
49. C₃₁×#394	C₂²	semidirect
50. C₃₁×#393	C₂²	semidirect
51. C₃₁×#391	C₂²	semidirect
52. C₃₁×#391	C₂²	semidirect
53. C₃₁×#390	C₂²	semidirect
54. C₃₁×#389	C₂²	semidirect
57. C₃₁×#388	C₂²	semidirect
58. C₃₁×#387	C₂²	semidirect
69. C₃₁×#385	C₂²	semidirect
70. C₃₁×#384	C₂²	semidirect
71. C₃₁×#382	C₂²	semidirect
241. #36716	C₃₁	direct (N × H)
301. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
302. #2849	C₂×C₃₁	semidirect
303. #2700	C₂×C₃₁	semidirect
304. #2698	C₂×C₃₁	semidirect
305. #2681	C₂×C₃₁	semidirect
306. #2680	C₂×C₃₁	semidirect
307. #2622	C₂×C₃₁	semidirect
308. #2591	C₂×C₃₁	semidirect
309. #2583	C₂×C₃₁	semidirect
310. #2579	C₂×C₃₁	semidirect
311. #2504	C₂×C₃₁	semidirect
312. #2502	C₂×C₃₁	semidirect
313. #2409	C₂×C₃₁	semidirect
314. #2312	C₂×C₃₁	semidirect
355. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
365. #400	C₂²×C₃₁	semidirect
366. #398	C₂²×C₃₁	semidirect
367. #398	C₂²×C₃₁	semidirect
368. #398	C₂²×C₃₁	semidirect
371. #397	C₂²×C₃₁	semidirect
372. #396	C₂²×C₃₁	semidirect
373. #395	C₂²×C₃₁	semidirect
374. #395	C₂²×C₃₁	semidirect
377. #394	C₂²×C₃₁	semidirect
378. #393	C₂²×C₃₁	semidirect
379. #391	C₂²×C₃₁	semidirect
380. #391	C₂²×C₃₁	semidirect
381. #390	C₂²×C₃₁	semidirect
382. #389	C₂²×C₃₁	semidirect
385. #388	C₂²×C₃₁	semidirect
386. #387	C₂²×C₃₁	semidirect
397. #385	C₂²×C₃₁	semidirect
398. #384	C₂²×C₃₁	semidirect
399. #382	C₂²×C₃₁	semidirect
492. C₃₁	#36716	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×#538	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₃₁×#2849	2
C₃₁×#2700	2
C₃₁×#2698	2
C₃₁×#2681	2
C₃₁×#2680	2
C₃₁×#2622	2
C₃₁×#2591	2
C₃₁×#2583	2
C₃₁×#2579	2
C₃₁×#2504	2
C₃₁×#2502	2
C₃₁×#2409	2
C₃₁×#2312	2
#36716	31

Subgroups — C31×#36716 #21493475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36716 #21493475