Subgroups — C₃₁×#24315 #21481074

Order 7936.

966 conjugacy classes of subgroups.

The subgroup lattice has 404 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
2	C₃₁×#3078	C₂	1	2	yes	maximal
3	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2060	C₂	1	2	yes	maximal
5	C₃₁×#2028	C₂	1	2	yes	maximal
6	C₃₁×#1994	C₂	1	2	yes	maximal
7	C₃₁×#1886	C₂	1	2	yes	maximal
8	C₃₁×#1857	C₂	1	2	yes	maximal
9	C₃₁×#1801	C₂	1	1	yes	maximal, characteristic
10	Q₈²×C₃₁	—	2	2	no	—
11	C₃₁×#557	—	2	2	no	—
12	C₃₁×#557	—	2	4	no	—
13	C₃₁×#544	—	2	2	no	—
14	C₃₁×#544	C₂²	1	2	yes	—
15	C₃₁×#533	—	2	2	no	—
16	C₃₁×#533	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃₁×#519	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#479	—	2	4	no	—
22	C₃₁×#477	—	2	2	no	—
23	C₃₁×#475	—	2	2	no	—
24	C₃₁×#428	C₂²	1	4	yes	—
25	C₃₁×#423	C₂²	1	2	yes	—
26	C₃₁×#419	C₂²	1	4	yes	—
27	C₂×C₃₁×#101	C₂²	1	2	yes	—
28	C₃₁×#391	C₂²	1	1	yes	characteristic
29	C₃₁×#391	C₂²	1	1	yes	characteristic
30	C₃₁×#389	—	2	2	no	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₄³×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#343	C₂²	1	2	yes	—
35	C₃₁×#339	C₂²	1	4	yes	—
36	C₃₁×#339	C₂²	1	4	yes	—
37	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
40	C₃₁×C₄⋊Q₈	—	2	2	no	—
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	—	2	8	no	—
43	C₃₁×C₄⋊Q₈	—	2	2	no	—
44	C₃₁×C₄⋊Q₈	—	2	4	no	—
45	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
46	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
47	C₃₁×C₄⋊Q₈	Q₈	1	4	yes	complemented
48	C₃₁×#125	—	2	4	no	—
49	C₃₁×#125	—	2	8	no	—
50	C₃₁×#125	—	2	4	no	—
51	C₃₁×#125	D₄	1	4	yes	—
52	C₃₁×C₂²⋊Q₈	—	2	2	no	—
53	C₃₁×C₂²⋊Q₈	—	2	4	no	—
54	C₃₁×C₂²⋊Q₈	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	4	no	—
56	C₄×Q₈×C₃₁	—	2	4	no	—
57	C₄×Q₈×C₃₁	—	2	4	no	—
58	C₄×Q₈×C₃₁	—	2	4	no	—
59	C₄×Q₈×C₃₁	—	2	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×Q₈×C₃₁	—	2	2	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×Q₈×C₃₁	—	2	2	no	—
65	C₃₁×#117	—	2	4	no	—
66	C₃₁×#117	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	D₄	1	2	yes	—
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#107	—	2	4	no	—
81	C₃₁×#106	—	2	4	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×#103	—	2	4	no	—
85	C₃₁×#103	—	2	4	no	—
86	C₃₁×#101	—	2	4	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₂×Q₈×C₃₁	—	2	4	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	1	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
98	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
99	C₂×Q₈×C₃₁	#41	1	2	yes	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	M₁₆×C₃₁	—	2	4	no	—
110	M₁₆×C₃₁	—	2	4	no	—
111	C₂×C₈×C₃₁	—	2	4	no	—
112	C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₄⋊C₄×C₃₁	—	2	4	no	—
114	C₄⋊C₄×C₃₁	—	2	4	no	—
115	C₄⋊C₄×C₃₁	—	2	8	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	4	no	—
121	C₄⋊C₄×C₃₁	—	2	8	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	4	4	no	—
125	C₄⋊C₄×C₃₁	—	4	2	no	—
126	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
127	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
128	C₂²⋊C₄×C₃₁	—	2	4	no	—
129	C₄²×C₃₁	—	2	4	no	—
130	C₄²×C₃₁	—	2	8	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	—	2	4	no	—
134	C₄²×C₃₁	—	2	1	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	4	2	no	—
137	C₄²×C₃₁	—	4	1	no	—
138	C₄²×C₃₁	—	4	1	no	—
139	C₄²×C₃₁	C₂×D₄	1	2	yes	—
140	C₄²×C₃₁	C₂×D₄	1	2	yes	—
141	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
142	#24315	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
144	Q₈×C₃₁	—	2	4	no	—
145	Q₈×C₃₁	—	2	4	no	—
146	Q₈×C₃₁	—	2	4	no	—
147	Q₈×C₃₁	—	4	2	no	—
148	Q₈×C₃₁	—	4	4	no	—
149	Q₈×C₃₁	—	8	2	no	—
150	Q₈×C₃₁	—	8	4	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	2	4	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	4	1	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
171	C₂×C₄×C₃₁	#120	1	2	yes	—
172	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
174	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
175	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
178	C₈×C₃₁	—	4	4	no	—
179	#3502	C₂×C₃₁	1	1	yes	characteristic
180	#3078	C₂×C₃₁	1	2	yes	—
181	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
182	#2060	C₂×C₃₁	1	2	yes	—
183	#2028	C₂×C₃₁	1	2	yes	—
184	#1994	C₂×C₃₁	1	2	yes	—
185	#1886	C₂×C₃₁	1	2	yes	—
186	#1857	C₂×C₃₁	1	2	yes	—
187	#1801	C₂×C₃₁	1	1	yes	characteristic
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	2	1	no	—
190	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
191	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
192	C₂²×C₃₁	#536	1	1	yes	characteristic
193	C₄×C₃₁	—	2	4	no	—
194	C₄×C₃₁	—	2	4	no	—
195	C₄×C₃₁	—	2	1	no	—
196	C₄×C₃₁	—	2	2	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	—	4	2	no	—
199	C₄×C₃₁	—	4	4	no	—
200	C₄×C₃₁	—	4	2	no	—
201	C₄×C₃₁	—	4	4	no	—
202	C₄×C₃₁	—	8	1	no	—
203	C₄×C₃₁	#456	1	2	yes	—
204	C₄×C₃₁	#548	1	2	yes	—
205	C₄×C₃₁	D₄×Q₈	1	2	yes	—
206	Q₈²	—	2	2	no	—
207	#557	—	2	2	no	—
208	#557	—	2	4	no	—
209	#544	—	2	2	no	—
210	#544	C₂²×C₃₁	1	2	yes	—
211	#533	—	2	2	no	—
212	#533	C₂²×C₃₁	1	2	yes	—
213	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
214	#519	C₂²×C₃₁	1	2	yes	—
215	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
217	#479	—	2	4	no	—
218	#477	—	2	2	no	—
219	#475	—	2	2	no	—
220	#428	C₂²×C₃₁	1	4	yes	—
221	#423	C₂²×C₃₁	1	2	yes	—
222	#419	C₂²×C₃₁	1	4	yes	—
223	C₂×#101	C₂²×C₃₁	1	2	yes	—
224	#391	C₂²×C₃₁	1	1	yes	characteristic
225	#391	C₂²×C₃₁	1	1	yes	characteristic
226	#389	—	2	2	no	—
227	#384	C₂²×C₃₁	1	1	yes	characteristic
228	C₄×C₄⋊C₄	—	2	2	no	—
229	C₄³	C₂²×C₃₁	1	1	yes	characteristic
230	#343	C₂²×C₃₁	1	2	yes	—
231	#339	C₂²×C₃₁	1	4	yes	—
232	#339	C₂²×C₃₁	1	4	yes	—
233	C₂×C₃₁	—	2	1	no	—
234	C₂×C₃₁	—	2	1	no	—
235	C₂×C₃₁	#2409	1	1	yes	characteristic
236	C₂×C₃₁	C₂×#456	1	1	yes	characteristic
237	C₂×C₃₁	#3018	1	1	yes	characteristic
238	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
239	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
240	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
241	C₄⋊Q₈	—	2	2	no	—
242	C₄⋊Q₈	—	2	2	no	—
243	C₄⋊Q₈	—	2	8	no	—
244	C₄⋊Q₈	—	2	2	no	—
245	C₄⋊Q₈	—	2	4	no	—
246	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
247	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
248	C₄⋊Q₈	Q₈×C₃₁	1	4	yes	complemented
249	#125	—	2	4	no	—
250	#125	—	2	8	no	—
251	#125	—	2	4	no	—
252	#125	D₄×C₃₁	1	4	yes	—
253	C₂²⋊Q₈	—	2	2	no	—
254	C₂²⋊Q₈	—	2	4	no	—
255	C₂²⋊Q₈	—	2	2	no	—
256	C₄×Q₈	—	2	4	no	—
257	C₄×Q₈	—	2	4	no	—
258	C₄×Q₈	—	2	4	no	—
259	C₄×Q₈	—	2	4	no	—
260	C₄×Q₈	—	2	2	no	—
261	C₄×Q₈	—	2	2	no	—
262	C₄×Q₈	—	2	2	no	—
263	C₄×Q₈	—	2	2	no	—
264	C₄×Q₈	—	2	2	no	—
265	C₄×Q₈	—	2	2	no	—
266	#117	—	2	4	no	—
267	#117	C₂³×C₃₁	1	4	yes	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
274	C₂×C₄²	—	2	1	no	—
275	C₂×C₄²	—	2	1	no	—
276	C₂×C₄²	—	2	1	no	—
277	C₂×C₄²	D₄×C₃₁	1	2	yes	—
278	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
279	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
281	#107	—	2	4	no	—
282	#106	—	2	4	no	—
283	C₄⋊C₈	—	2	2	no	—
284	C₄⋊C₈	—	2	2	no	—
285	#103	—	2	4	no	—
286	#103	—	2	4	no	—
287	#101	—	2	4	no	—
288	#95	—	2	2	no	—
289	#95	—	2	2	no	—
290	C₃₁	#24315	1	1	yes	characteristic, direct product factor, Sylow
291	C₂×Q₈	—	2	4	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
300	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
301	C₂×Q₈	#41×C₃₁	1	2	yes	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
308	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
311	M₁₆	—	2	4	no	—
312	M₁₆	—	2	4	no	—
313	C₂×C₈	—	2	4	no	—
314	C₄⋊C₄	—	2	4	no	—
315	C₄⋊C₄	—	2	4	no	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	2	8	no	—
318	C₄⋊C₄	—	2	4	no	—
319	C₄⋊C₄	—	2	4	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	8	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	4	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
329	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
330	C₂²⋊C₄	—	2	4	no	—
331	C₄²	—	2	4	no	—
332	C₄²	—	2	8	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	4	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	4	2	no	—
339	C₄²	—	4	1	no	—
340	C₄²	—	4	1	no	—
341	C₄²	C₂×D₄×C₃₁	1	2	yes	—
342	C₄²	C₂×D₄×C₃₁	1	2	yes	—
343	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
344	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
345	Q₈	—	2	4	no	—
346	Q₈	—	2	4	no	—
347	Q₈	—	2	4	no	—
348	Q₈	—	4	4	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	8	2	no	—
351	Q₈	—	8	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
372	C₂×C₄	C₃₁×#120	1	2	yes	—
373	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
374	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
375	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
376	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
377	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
378	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
379	C₈	—	4	4	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
383	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
384	C₂²	C₃₁×#536	1	1	yes	characteristic
385	C₄	—	2	4	no	—
386	C₄	—	2	4	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	2	no	—
389	C₄	—	4	2	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	4	no	—
392	C₄	—	4	2	no	—
393	C₄	—	4	4	no	—
394	C₄	—	8	1	no	—
395	C₄	C₃₁×#456	1	2	yes	—
396	C₄	C₃₁×#548	1	2	yes	—
397	C₄	D₄×Q₈×C₃₁	1	2	yes	—
398	C₂	—	2	1	no	—
399	C₂	—	2	1	no	—
400	C₂	C₃₁×#2409	1	1	yes	characteristic
401	C₂	C₂×C₃₁×#456	1	1	yes	characteristic
402	C₂	C₃₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
47. C₃₁×C₄⋊Q₈	Q₈	semidirect
142. #24315	C₃₁	direct (N × H)
248. C₄⋊Q₈	Q₈×C₃₁	semidirect
290. C₃₁	#24315	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×#3502	2
C₃₁×#3078	2
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#2060	2
C₃₁×#2028	2
C₃₁×#1994	2
C₃₁×#1886	2
C₃₁×#1857	2
C₃₁×#1801	2
#24315	31

Subgroups — C31×#24315 #21481074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24315 #21481074