Subgroups — C₃₁×#29201 #21485960

Order 7936.

770 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×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2328	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2065	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×#1864	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1836	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1768	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#107	C₂²	1	2	yes	—
15	C₂×C₃₁×#106	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
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×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#400	—	2	2	no	—
25	C₃₁×#398	—	2	2	no	—
26	C₃₁×#390	C₂²	1	2	yes	—
27	C₃₁×#389	C₂²	1	2	yes	—
28	C₃₁×#389	C₂²	1	2	yes	—
29	C₃₁×#386	—	2	2	no	—
30	C₃₁×#386	—	2	2	no	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₃₁×#382	C₂²	1	1	yes	characteristic
33	C₃₁×#382	C₂²	1	1	yes	characteristic
34	C₃₁×#376	C₂²	1	1	yes	characteristic
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	1	yes	characteristic
39	C₂²×D₄×C₃₁	—	2	2	no	—
40	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
41	C₂³×C₄×C₃₁	—	2	2	no	—
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#127	—	2	2	no	—
47	C₃₁×#127	Q₈	1	4	yes	—
48	C₄×D₄×C₃₁	—	4	2	no	—
49	C₄×D₄×C₃₁	—	4	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#107	—	4	2	no	—
63	C₃₁×#106	—	4	2	no	—
64	C₃₁×C₄⋊C₈	—	2	2	no	—
65	C₃₁×C₄⋊C₈	—	2	2	no	—
66	C₃₁×#102	—	2	4	no	—
67	C₃₁×#102	—	2	4	no	—
68	C₃₁×#102	—	4	2	no	—
69	C₃₁×#102	—	4	2	no	—
70	C₃₁×#97	—	2	2	no	—
71	C₄×C₈×C₃₁	—	2	2	no	—
72	C₃₁×#95	—	2	2	no	—
73	C₃₁×#95	—	2	2	no	—
74	C₃₁×#95	—	2	2	no	—
75	C₃₁×#95	—	2	2	no	—
76	C₃₁×#95	D₄	1	2	yes	—
77	C₃₁×#95	D₄	1	2	yes	—
78	C₂⁴×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	2	8	no	—
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	2	2	no	—
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	8	2	no	—
86	C₂×D₄×C₃₁	—	8	2	no	—
87	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	4	4	no	—
92	C₂²×C₄×C₃₁	—	8	2	no	—
93	C₂²×C₄×C₃₁	—	8	2	no	—
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	1	no	—
104	C₂×C₈×C₃₁	—	2	1	no	—
105	C₂×C₈×C₃₁	—	2	1	no	—
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	1	no	—
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	#41	1	2	yes	—
110	C₂×C₈×C₃₁	#41	1	2	yes	—
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	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	8	2	no	—
118	C₄⋊C₄×C₃₁	—	8	2	no	—
119	C₄⋊C₄×C₃₁	—	8	2	no	—
120	C₂²⋊C₄×C₃₁	—	4	4	no	—
121	C₂²⋊C₄×C₃₁	—	8	2	no	—
122	C₂²⋊C₄×C₃₁	—	8	2	no	—
123	C₄²×C₃₁	—	2	1	no	—
124	C₄²×C₃₁	—	2	1	no	—
125	C₄²×C₃₁	—	2	1	no	—
126	C₄²×C₃₁	—	2	1	no	—
127	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
128	C₄²×C₃₁	#41	1	2	yes	—
129	#29201	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	2	4	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	—	4	2	no	—
137	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
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	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	4	2	no	—
153	C₂×C₄×C₃₁	—	4	2	no	—
154	C₂×C₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	4	2	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	—	8	8	no	—
162	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
163	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
169	C₈×C₃₁	—	2	2	no	—
170	C₈×C₃₁	—	2	2	no	—
171	C₈×C₃₁	—	4	1	no	—
172	C₈×C₃₁	—	4	1	no	—
173	#2647	C₂×C₃₁	1	1	yes	characteristic, complemented
174	#2328	C₂×C₃₁	1	1	yes	characteristic
175	#2067	C₂×C₃₁	1	2	yes	complemented
176	#2065	C₂×C₃₁	1	2	yes	complemented
177	#1961	C₂×C₃₁	1	1	yes	characteristic
178	#1957	C₂×C₃₁	1	1	yes	characteristic
179	#1864	C₂×C₃₁	1	2	yes	complemented
180	#1862	C₂×C₃₁	1	2	yes	complemented
181	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1836	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#1768	C₂×C₃₁	1	1	yes	characteristic, complemented
184	C₂²×C₃₁	—	4	4	no	—
185	C₂²×C₃₁	—	4	4	no	—
186	C₂²×C₃₁	—	4	4	no	—
187	C₂²×C₃₁	—	8	2	no	—
188	C₂²×C₃₁	—	8	2	no	—
189	C₂²×C₃₁	—	8	2	no	—
190	C₂²×C₃₁	—	8	2	no	—
191	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
192	C₂²×C₃₁	#532	1	1	yes	characteristic
193	C₂²×C₃₁	#535	1	1	yes	characteristic
194	C₂²×C₃₁	#576	1	2	yes	—
195	C₂²×C₃₁	#577	1	2	yes	—
196	C₄×C₃₁	—	2	4	no	—
197	C₄×C₃₁	—	4	1	no	—
198	C₄×C₃₁	—	4	1	no	—
199	C₄×C₃₁	—	4	2	no	—
200	C₄×C₃₁	—	8	2	no	—
201	C₄×C₃₁	—	8	2	no	—
202	C₄×C₃₁	—	8	2	no	—
203	C₄×C₃₁	#553	1	1	yes	characteristic
204	C₄×C₃₁	#553	1	1	yes	characteristic
205	C₄×C₃₁	#554	1	1	yes	characteristic
206	C₄×C₃₁	#554	1	1	yes	characteristic
207	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
209	C₂×#107	C₂²×C₃₁	1	2	yes	—
210	C₂×#106	C₂²×C₃₁	1	2	yes	—
211	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×#102	C₂²×C₃₁	1	2	yes	—
214	C₂×#102	C₂²×C₃₁	1	2	yes	—
215	C₂×#102	C₂²×C₃₁	1	2	yes	—
216	C₂×#102	C₂²×C₃₁	1	2	yes	—
217	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
219	#400	—	2	2	no	—
220	#398	—	2	2	no	—
221	#390	C₂²×C₃₁	1	2	yes	—
222	#389	C₂²×C₃₁	1	2	yes	—
223	#389	C₂²×C₃₁	1	2	yes	—
224	#386	—	2	2	no	—
225	#386	—	2	2	no	—
226	#384	C₂²×C₃₁	1	1	yes	characteristic
227	#382	C₂²×C₃₁	1	1	yes	characteristic
228	#382	C₂²×C₃₁	1	1	yes	characteristic
229	#376	C₂²×C₃₁	1	1	yes	characteristic
230	#340	C₂²×C₃₁	1	2	yes	—
231	#340	C₂²×C₃₁	1	2	yes	—
232	#336	C₂²×C₃₁	1	1	yes	characteristic
233	#336	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₃₁	—	8	4	no	—
235	C₂×C₃₁	#2566	1	1	yes	characteristic
236	C₂×C₃₁	#3081	1	2	yes	—
237	C₂×C₃₁	#3174	1	1	yes	characteristic
238	C₂×C₃₁	#3176	1	1	yes	characteristic
239	C₂×C₃₁	#3224	1	2	yes	—
240	C₂²×D₄	—	2	2	no	—
241	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
242	C₂³×C₄	—	2	2	no	—
243	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
244	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
245	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
247	#127	—	2	2	no	—
248	#127	Q₈×C₃₁	1	4	yes	—
249	C₄×D₄	—	4	2	no	—
250	C₄×D₄	—	4	2	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	2	no	—
253	C₂×C₄⋊C₄	—	2	2	no	—
254	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
255	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
257	C₂×C₂²⋊C₄	—	2	2	no	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
261	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
263	#107	—	4	2	no	—
264	#106	—	4	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	2	2	no	—
267	#102	—	2	4	no	—
268	#102	—	2	4	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	—	2	2	no	—
277	#95	D₄×C₃₁	1	2	yes	—
278	#95	D₄×C₃₁	1	2	yes	—
279	C₃₁	#29201	1	1	yes	characteristic, direct product factor, Sylow
280	C₂⁴	—	2	2	no	—
281	C₂×D₄	—	2	8	no	—
282	C₂×D₄	—	2	2	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	—	8	2	no	—
288	C₂×D₄	—	8	2	no	—
289	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	4	4	no	—
294	C₂²×C₄	—	8	2	no	—
295	C₂²×C₄	—	8	2	no	—
296	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
298	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
300	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	#41×C₃₁	1	2	yes	—
312	C₂×C₈	#41×C₃₁	1	2	yes	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	8	2	no	—
320	C₄⋊C₄	—	8	2	no	—
321	C₄⋊C₄	—	8	2	no	—
322	C₂²⋊C₄	—	4	4	no	—
323	C₂²⋊C₄	—	8	2	no	—
324	C₂²⋊C₄	—	8	2	no	—
325	C₄²	—	2	1	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	2	1	no	—
329	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
330	C₄²	#41×C₃₁	1	2	yes	—
331	C₂³	—	2	4	no	—
332	C₂³	—	4	2	no	—
333	C₂³	—	4	2	no	—
334	C₂³	—	4	2	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
339	D₄	—	2	4	no	—
340	D₄	—	2	4	no	—
341	D₄	—	4	2	no	—
342	D₄	—	4	2	no	—
343	D₄	—	8	8	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	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	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	—	4	2	no	—
362	C₂×C₄	—	8	8	no	—
363	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
364	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
365	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
366	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
367	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
368	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
369	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	2	2	no	—
372	C₈	—	4	1	no	—
373	C₈	—	4	1	no	—
374	C₂²	—	4	4	no	—
375	C₂²	—	4	4	no	—
376	C₂²	—	4	4	no	—
377	C₂²	—	8	2	no	—
378	C₂²	—	8	2	no	—
379	C₂²	—	8	2	no	—
380	C₂²	—	8	2	no	—
381	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
382	C₂²	C₃₁×#532	1	1	yes	characteristic
383	C₂²	C₃₁×#535	1	1	yes	characteristic
384	C₂²	C₃₁×#576	1	2	yes	—
385	C₂²	C₃₁×#577	1	2	yes	—
386	C₄	—	2	4	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	1	no	—
389	C₄	—	4	2	no	—
390	C₄	—	8	2	no	—
391	C₄	—	8	2	no	—
392	C₄	—	8	2	no	—
393	C₄	C₃₁×#553	1	1	yes	characteristic
394	C₄	C₃₁×#553	1	1	yes	characteristic
395	C₄	C₃₁×#554	1	1	yes	characteristic
396	C₄	C₃₁×#554	1	1	yes	characteristic
397	C₂	—	8	4	no	—
398	C₂	C₃₁×#2566	1	1	yes	characteristic
399	C₂	C₃₁×#3081	1	2	yes	—
400	C₂	C₃₁×#3174	1	1	yes	characteristic
401	C₂	C₃₁×#3176	1	1	yes	characteristic
402	C₂	C₃₁×#3224	1	2	yes	—

	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×#2647	C₂	semidirect
3. C₃₁×#2067	C₂	semidirect
4. C₃₁×#2065	C₂	semidirect
7. C₃₁×#1864	C₂	semidirect
8. C₃₁×#1862	C₂	semidirect
9. C₃₁×#1846	C₂	semidirect
10. C₃₁×#1836	C₂	semidirect
11. C₃₁×#1768	C₂	semidirect
129. #29201	C₃₁	direct (N × H)
173. #2647	C₂×C₃₁	semidirect
175. #2067	C₂×C₃₁	semidirect
176. #2065	C₂×C₃₁	semidirect
179. #1864	C₂×C₃₁	semidirect
180. #1862	C₂×C₃₁	semidirect
181. #1846	C₂×C₃₁	semidirect
182. #1836	C₂×C₃₁	semidirect
183. #1768	C₂×C₃₁	semidirect
279. C₃₁	#29201	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×#2647	2
C₃₁×#2328	2
C₃₁×#2067	2
C₃₁×#2065	2
C₃₁×#1961	2
C₃₁×#1957	2
C₃₁×#1864	2
C₃₁×#1862	2
C₃₁×#1846	2
C₃₁×#1836	2
C₃₁×#1768	2
#29201	31

Subgroups — C31×#29201 #21485960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29201 #21485960