Subgroups — C₃₁×#29152 #21485911

Order 7936.

798 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2660	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2328	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2025	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2023	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1836	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1804	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1802	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1768	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#398	—	2	2	no	—
24	C₃₁×#395	—	2	2	no	—
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#386	—	2	2	no	—
27	C₃₁×#386	—	2	2	no	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#384	C₂²	1	1	yes	characteristic
30	C₃₁×#382	C₂²	1	1	yes	characteristic
31	C₃₁×#382	C₂²	1	1	yes	characteristic
32	C₃₁×#382	C₂²	1	1	yes	characteristic
33	C₃₁×#376	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₂²×Q₈×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	—	2	2	no	—
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₃₁×C₄⋊Q₈	—	4	2	no	—
46	C₃₁×#127	—	2	2	no	—
47	C₃₁×#127	Q₈	1	4	yes	complemented
48	C₃₁×#125	—	4	2	no	—
49	C₄×D₄×C₃₁	—	2	4	no	—
50	C₄×D₄×C₃₁	—	2	4	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊C₈	—	2	4	no	—
68	C₃₁×#102	—	4	2	no	—
69	C₃₁×#102	—	4	2	no	—
70	C₃₁×#102	—	4	2	no	—
71	C₃₁×#102	—	4	2	no	—
72	C₃₁×#97	—	2	2	no	—
73	C₄×C₈×C₃₁	—	2	2	no	—
74	C₃₁×#95	—	2	2	no	—
75	C₃₁×#95	—	2	2	no	—
76	C₃₁×#95	—	2	2	no	—
77	C₃₁×#95	D₄	1	2	yes	—
78	C₃₁×#95	D₄	1	2	yes	—
79	C₂⁴×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	4	2	no	—
81	C₂×Q₈×C₃₁	—	8	2	no	—
82	C₂×D₄×C₃₁	—	2	8	no	—
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	8	2	no	—
89	C₂×D₄×C₃₁	—	8	2	no	—
90	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	4	4	no	—
97	C₂²×C₄×C₃₁	—	8	2	no	—
98	C₂²×C₄×C₃₁	—	8	2	no	—
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	4	2	no	—
117	C₄⋊C₄×C₃₁	—	4	2	no	—
118	C₄⋊C₄×C₃₁	—	4	2	no	—
119	C₄⋊C₄×C₃₁	—	8	2	no	—
120	C₄⋊C₄×C₃₁	—	8	2	no	—
121	C₂²⋊C₄×C₃₁	—	4	4	no	—
122	C₂²⋊C₄×C₃₁	—	8	2	no	—
123	C₂²⋊C₄×C₃₁	—	8	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
126	C₄²×C₃₁	#41	1	2	yes	—
127	C₄²×C₃₁	#41	1	4	yes	—
128	#29152	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₃₁	—	2	4	no	—
130	C₂³×C₃₁	—	4	2	no	—
131	C₂³×C₃₁	—	4	2	no	—
132	C₂³×C₃₁	—	4	2	no	—
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	—	4	2	no	—
135	C₂³×C₃₁	—	4	2	no	—
136	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	Q₈×C₃₁	—	8	4	no	—
138	D₄×C₃₁	—	2	4	no	—
139	D₄×C₃₁	—	2	4	no	—
140	D₄×C₃₁	—	4	2	no	—
141	D₄×C₃₁	—	4	2	no	—
142	D₄×C₃₁	—	8	8	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	—	4	1	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	4	1	no	—
155	C₂×C₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	8	8	no	—
163	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
164	C₂×C₄×C₃₁	#124	1	4	yes	—
165	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
167	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
169	C₈×C₃₁	—	4	2	no	—
170	C₈×C₃₁	—	4	2	no	—
171	#2660	C₂×C₃₁	1	1	yes	characteristic, complemented
172	#2328	C₂×C₃₁	1	1	yes	characteristic
173	#2025	C₂×C₃₁	1	2	yes	complemented
174	#2023	C₂×C₃₁	1	2	yes	complemented
175	#1961	C₂×C₃₁	1	1	yes	characteristic
176	#1957	C₂×C₃₁	1	1	yes	characteristic
177	#1836	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#1835	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#1804	C₂×C₃₁	1	2	yes	complemented
180	#1802	C₂×C₃₁	1	2	yes	complemented
181	#1768	C₂×C₃₁	1	1	yes	characteristic, complemented
182	C₂²×C₃₁	—	4	4	no	—
183	C₂²×C₃₁	—	4	4	no	—
184	C₂²×C₃₁	—	4	4	no	—
185	C₂²×C₃₁	—	8	2	no	—
186	C₂²×C₃₁	—	8	2	no	—
187	C₂²×C₃₁	—	8	2	no	—
188	C₂²×C₃₁	—	8	2	no	—
189	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
190	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
191	C₂²×C₃₁	#535	1	1	yes	characteristic
192	C₂²×C₃₁	#576	1	2	yes	—
193	C₂²×C₃₁	#577	1	2	yes	—
194	C₄×C₃₁	—	2	4	no	—
195	C₄×C₃₁	—	2	4	no	—
196	C₄×C₃₁	—	4	1	no	—
197	C₄×C₃₁	—	4	1	no	—
198	C₄×C₃₁	—	8	2	no	—
199	C₄×C₃₁	—	8	1	no	—
200	C₄×C₃₁	—	8	1	no	—
201	C₄×C₃₁	—	8	1	no	—
202	C₄×C₃₁	—	8	1	no	—
203	C₄×C₃₁	#554	1	2	yes	—
204	C₄×C₃₁	#555	1	2	yes	—
205	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
206	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
207	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
209	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
210	C₂×#102	C₂²×C₃₁	1	2	yes	—
211	C₂×#102	C₂²×C₃₁	1	2	yes	—
212	C₂×#102	C₂²×C₃₁	1	2	yes	—
213	C₂×#102	C₂²×C₃₁	1	2	yes	—
214	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
216	#398	—	2	2	no	—
217	#395	—	2	2	no	—
218	#390	C₂²×C₃₁	1	2	yes	—
219	#386	—	2	2	no	—
220	#386	—	2	2	no	—
221	#384	C₂²×C₃₁	1	1	yes	characteristic
222	#384	C₂²×C₃₁	1	1	yes	characteristic
223	#382	C₂²×C₃₁	1	1	yes	characteristic
224	#382	C₂²×C₃₁	1	1	yes	characteristic
225	#382	C₂²×C₃₁	1	1	yes	characteristic
226	#376	C₂²×C₃₁	1	1	yes	characteristic
227	#340	C₂²×C₃₁	1	2	yes	—
228	#340	C₂²×C₃₁	1	2	yes	—
229	#340	C₂²×C₃₁	1	2	yes	—
230	#340	C₂²×C₃₁	1	2	yes	—
231	#336	C₂²×C₃₁	1	2	yes	—
232	C₂×C₃₁	—	8	4	no	—
233	C₂×C₃₁	#2563	1	1	yes	characteristic
234	C₂×C₃₁	#3081	1	2	yes	—
235	C₂×C₃₁	#3136	1	2	yes	—
236	C₂×C₃₁	#3220	1	1	yes	characteristic
237	C₂×C₃₁	#3221	1	1	yes	characteristic
238	C₂²×Q₈	—	2	1	no	—
239	C₂²×D₄	—	2	2	no	—
240	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
241	C₂³×C₄	—	2	2	no	—
242	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
243	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
244	C₄⋊Q₈	—	4	2	no	—
245	#127	—	2	2	no	—
246	#127	Q₈×C₃₁	1	4	yes	complemented
247	#125	—	4	2	no	—
248	C₄×D₄	—	2	4	no	—
249	C₄×D₄	—	2	4	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	2	no	—
254	C₂×C₄⋊C₄	—	2	2	no	—
255	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
259	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
260	C₂×C₂²⋊C₄	—	2	2	no	—
261	C₂×C₂²⋊C₄	—	2	2	no	—
262	C₂×C₂²⋊C₄	—	2	2	no	—
263	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
264	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
266	C₄⋊C₈	—	2	4	no	—
267	#102	—	4	2	no	—
268	#102	—	4	2	no	—
269	#102	—	4	2	no	—
270	#102	—	4	2	no	—
271	#97	—	2	2	no	—
272	C₄×C₈	—	2	2	no	—
273	#95	—	2	2	no	—
274	#95	—	2	2	no	—
275	#95	—	2	2	no	—
276	#95	D₄×C₃₁	1	2	yes	—
277	#95	D₄×C₃₁	1	2	yes	—
278	C₃₁	#29152	1	1	yes	characteristic, direct product factor, Sylow
279	C₂⁴	—	2	2	no	—
280	C₂×Q₈	—	4	2	no	—
281	C₂×Q₈	—	8	2	no	—
282	C₂×D₄	—	2	8	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	8	2	no	—
289	C₂×D₄	—	8	2	no	—
290	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	4	4	no	—
297	C₂²×C₄	—	8	2	no	—
298	C₂²×C₄	—	8	2	no	—
299	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
303	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	—	4	2	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄⋊C₄	—	8	2	no	—
320	C₄⋊C₄	—	8	2	no	—
321	C₂²⋊C₄	—	4	4	no	—
322	C₂²⋊C₄	—	8	2	no	—
323	C₂²⋊C₄	—	8	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
326	C₄²	#41×C₃₁	1	2	yes	—
327	C₄²	#41×C₃₁	1	4	yes	—
328	C₂³	—	2	4	no	—
329	C₂³	—	4	2	no	—
330	C₂³	—	4	2	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	2	no	—
333	C₂³	—	4	2	no	—
334	C₂³	—	4	2	no	—
335	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
336	Q₈	—	8	4	no	—
337	D₄	—	2	4	no	—
338	D₄	—	2	4	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	8	8	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	8	8	no	—
362	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
363	C₂×C₄	C₃₁×#124	1	4	yes	—
364	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
365	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
366	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
367	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₂²	—	4	4	no	—
371	C₂²	—	4	4	no	—
372	C₂²	—	4	4	no	—
373	C₂²	—	8	2	no	—
374	C₂²	—	8	2	no	—
375	C₂²	—	8	2	no	—
376	C₂²	—	8	2	no	—
377	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
378	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
379	C₂²	C₃₁×#535	1	1	yes	characteristic
380	C₂²	C₃₁×#576	1	2	yes	—
381	C₂²	C₃₁×#577	1	2	yes	—
382	C₄	—	2	4	no	—
383	C₄	—	2	4	no	—
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	8	1	no	—
388	C₄	—	8	1	no	—
389	C₄	—	8	1	no	—
390	C₄	—	8	1	no	—
391	C₄	C₃₁×#554	1	2	yes	—
392	C₄	C₃₁×#555	1	2	yes	—
393	C₂	—	8	4	no	—
394	C₂	C₃₁×#2563	1	1	yes	characteristic
395	C₂	C₃₁×#3081	1	2	yes	—
396	C₂	C₃₁×#3136	1	2	yes	—
397	C₂	C₃₁×#3220	1	1	yes	characteristic
398	C₂	C₃₁×#3221	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2660	C₂	semidirect
3. C₃₁×#2025	C₂	semidirect
4. C₃₁×#2023	C₂	semidirect
7. C₃₁×#1836	C₂	semidirect
8. C₃₁×#1835	C₂	semidirect
9. C₃₁×#1804	C₂	semidirect
10. C₃₁×#1802	C₂	semidirect
11. C₃₁×#1768	C₂	semidirect
47. C₃₁×#127	Q₈	semidirect
128. #29152	C₃₁	direct (N × H)
171. #2660	C₂×C₃₁	semidirect
173. #2025	C₂×C₃₁	semidirect
174. #2023	C₂×C₃₁	semidirect
177. #1836	C₂×C₃₁	semidirect
178. #1835	C₂×C₃₁	semidirect
179. #1804	C₂×C₃₁	semidirect
180. #1802	C₂×C₃₁	semidirect
181. #1768	C₂×C₃₁	semidirect
246. #127	Q₈×C₃₁	semidirect
278. C₃₁	#29152	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2660	2
C₃₁×#2328	2
C₃₁×#2025	2
C₃₁×#2023	2
C₃₁×#1961	2
C₃₁×#1957	2
C₃₁×#1836	2
C₃₁×#1835	2
C₃₁×#1804	2
C₃₁×#1802	2
C₃₁×#1768	2
#29152	31

Subgroups — C31×#29152 #21485911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29152 #21485911