Subgroups — C₃₁×#32813 #21489572

Order 7936.

704 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×#2642	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2640	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2056	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2055	C₂	1	2	yes	maximal
5	C₃₁×#2026	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2025	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2024	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2016	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2015	C₂	1	2	yes	maximal
10	C₃₁×#2013	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₃₁×#400	—	2	2	no	—
24	C₃₁×#400	—	2	2	no	—
25	C₃₁×#397	—	2	2	no	—
26	C₃₁×#393	—	2	2	no	—
27	C₃₁×#391	C₂²	1	2	yes	—
28	C₃₁×#391	C₂²	1	1	yes	characteristic
29	C₃₁×#391	C₂²	1	1	yes	characteristic
30	C₃₁×#391	C₂²	1	2	yes	—
31	C₃₁×#390	C₂²	1	1	yes	characteristic
32	C₃₁×#390	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×#336	C₂²	1	1	yes	characteristic
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₃₁	—	2	1	no	—
40	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₁₆×C₃₁	—	4	2	no	—
44	C₂×Q₁₆×C₃₁	—	4	2	no	—
45	C₂×Q₁₆×C₃₁	—	4	2	no	—
46	C₂×SD₁₆×C₃₁	—	4	2	no	—
47	C₂×SD₁₆×C₃₁	—	4	2	no	—
48	C₂×SD₁₆×C₃₁	—	4	2	no	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₃₁×C₄⋊Q₈	—	4	2	no	—
52	C₃₁×C₄⋊Q₈	—	4	2	no	—
53	C₃₁×#125	—	4	2	no	—
54	C₃₁×#124	—	4	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×C₄⋊C₈	—	4	2	no	—
69	C₃₁×#103	—	4	2	no	—
70	C₃₁×#103	—	4	2	no	—
71	C₃₁×#103	—	4	2	no	—
72	C₃₁×#102	—	4	2	no	—
73	C₃₁×#95	—	2	2	no	—
74	C₃₁×#95	—	2	2	no	—
75	C₃₁×#95	—	2	4	no	—
76	C₃₁×#95	—	2	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	D₄	1	2	yes	—
79	C₃₁×#95	D₄	1	2	yes	—
80	C₂⁴×C₃₁	—	2	1	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	—	2	1	no	—
89	C₂×Q₈×C₃₁	—	2	1	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	4	2	no	—
92	C₂×Q₈×C₃₁	—	8	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	Q₁₆×C₃₁	—	4	8	no	—
108	SD₁₆×C₃₁	—	4	8	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	4	4	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₄⋊C₄×C₃₁	—	4	4	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₂²⋊C₄×C₃₁	—	4	2	no	—
124	C₂²⋊C₄×C₃₁	—	8	2	no	—
125	C₂²⋊C₄×C₃₁	—	8	2	no	—
126	C₄²×C₃₁	—	2	1	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	1	no	—
130	#32813	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂³×C₃₁	—	4	1	no	—
132	C₂³×C₃₁	—	4	1	no	—
133	C₂³×C₃₁	—	4	1	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	—	4	2	no	—
138	Q₈×C₃₁	—	4	2	no	—
139	Q₈×C₃₁	—	4	2	no	—
140	Q₈×C₃₁	—	4	2	no	—
141	Q₈×C₃₁	—	4	2	no	—
142	Q₈×C₃₁	—	4	2	no	—
143	Q₈×C₃₁	—	8	4	no	—
144	D₄×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	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	—	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	—	4	1	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	1	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	1	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
171	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
173	C₈×C₃₁	—	4	2	no	—
174	C₈×C₃₁	—	4	2	no	—
175	#2642	C₂×C₃₁	1	1	yes	characteristic, complemented
176	#2640	C₂×C₃₁	1	1	yes	characteristic
177	#2056	C₂×C₃₁	1	2	yes	complemented
178	#2055	C₂×C₃₁	1	2	yes	—
179	#2026	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#2025	C₂×C₃₁	1	1	yes	characteristic
181	#2024	C₂×C₃₁	1	2	yes	complemented
182	#2016	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#2015	C₂×C₃₁	1	2	yes	—
184	#2013	C₂×C₃₁	1	1	yes	characteristic
185	#1832	C₂×C₃₁	1	1	yes	characteristic, complemented
186	C₂²×C₃₁	—	8	1	no	—
187	C₂²×C₃₁	—	8	1	no	—
188	C₂²×C₃₁	—	8	1	no	—
189	C₂²×C₃₁	—	8	2	no	—
190	C₂²×C₃₁	—	8	2	no	—
191	C₂²×C₃₁	#535	1	1	yes	characteristic
192	C₂²×C₃₁	#535	1	1	yes	characteristic
193	C₂²×C₃₁	#537	1	1	yes	characteristic
194	C₂²×C₃₁	#577	1	2	yes	—
195	C₂²×C₃₁	#578	1	2	yes	—
196	C₄×C₃₁	—	4	1	no	—
197	C₄×C₃₁	—	4	1	no	—
198	C₄×C₃₁	—	4	2	no	—
199	C₄×C₃₁	—	4	2	no	—
200	C₄×C₃₁	—	8	1	no	—
201	C₄×C₃₁	—	8	2	no	—
202	C₄×C₃₁	—	8	1	no	—
203	C₄×C₃₁	—	8	1	no	—
204	C₄×C₃₁	—	8	2	no	—
205	C₄×C₃₁	H(3,𝔽₄)	1	2	yes	—
206	C₄×C₃₁	#562	1	2	yes	—
207	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
208	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
209	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
214	C₂×#103	C₂²×C₃₁	1	2	yes	—
215	C₂×#103	C₂²×C₃₁	1	2	yes	—
216	C₂×#103	C₂²×C₃₁	1	2	yes	—
217	C₂×#102	C₂²×C₃₁	1	2	yes	—
218	#400	—	2	2	no	—
219	#400	—	2	2	no	—
220	#397	—	2	2	no	—
221	#393	—	2	2	no	—
222	#391	C₂²×C₃₁	1	2	yes	—
223	#391	C₂²×C₃₁	1	1	yes	characteristic
224	#391	C₂²×C₃₁	1	1	yes	characteristic
225	#391	C₂²×C₃₁	1	2	yes	—
226	#390	C₂²×C₃₁	1	1	yes	characteristic
227	#390	C₂²×C₃₁	1	1	yes	characteristic
228	#376	C₂²×C₃₁	1	1	yes	characteristic
229	#340	C₂²×C₃₁	1	2	yes	—
230	#340	C₂²×C₃₁	1	2	yes	—
231	#336	C₂²×C₃₁	1	1	yes	characteristic
232	#336	C₂²×C₃₁	1	2	yes	—
233	#336	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₃₁	—	16	1	no	—
235	C₂×C₃₁	#2819	1	1	yes	characteristic
236	C₂×C₃₁	#3222	1	1	yes	characteristic
237	C₂×C₃₁	#3223	1	1	yes	characteristic
238	C₂×C₃₁	#3225	1	2	yes	—
239	C₂×C₃₁	#3258	1	2	yes	—
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
242	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
243	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
244	C₂×Q₁₆	—	4	2	no	—
245	C₂×Q₁₆	—	4	2	no	—
246	C₂×Q₁₆	—	4	2	no	—
247	C₂×SD₁₆	—	4	2	no	—
248	C₂×SD₁₆	—	4	2	no	—
249	C₂×SD₁₆	—	4	2	no	—
250	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
252	C₄⋊Q₈	—	4	2	no	—
253	C₄⋊Q₈	—	4	2	no	—
254	#125	—	4	2	no	—
255	#124	—	4	2	no	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
260	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
261	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
262	C₂×C₂²⋊C₄	—	2	1	no	—
263	C₂×C₂²⋊C₄	—	2	1	no	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
268	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
269	C₄⋊C₈	—	4	2	no	—
270	#103	—	4	2	no	—
271	#103	—	4	2	no	—
272	#103	—	4	2	no	—
273	#102	—	4	2	no	—
274	#95	—	2	2	no	—
275	#95	—	2	2	no	—
276	#95	—	2	4	no	—
277	#95	—	2	2	no	—
278	#95	—	2	2	no	—
279	#95	D₄×C₃₁	1	2	yes	—
280	#95	D₄×C₃₁	1	2	yes	—
281	C₃₁	#32813	1	1	yes	characteristic, direct product factor, Sylow
282	C₂⁴	—	2	1	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	1	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	4	2	no	—
294	C₂×Q₈	—	8	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	1	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
305	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
308	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
309	Q₁₆	—	4	8	no	—
310	SD₁₆	—	4	8	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	4	4	no	—
322	C₄⋊C₄	—	4	2	no	—
323	C₄⋊C₄	—	4	4	no	—
324	C₂²⋊C₄	—	4	2	no	—
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	8	2	no	—
327	C₂²⋊C₄	—	8	2	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	1	no	—
332	C₂³	—	4	1	no	—
333	C₂³	—	4	1	no	—
334	C₂³	—	4	1	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
338	Q₈	—	4	2	no	—
339	Q₈	—	4	2	no	—
340	Q₈	—	4	2	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	8	4	no	—
345	D₄	—	4	2	no	—
346	D₄	—	4	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	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	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
371	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
372	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
373	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₂²	—	8	1	no	—
377	C₂²	—	8	1	no	—
378	C₂²	—	8	1	no	—
379	C₂²	—	8	2	no	—
380	C₂²	—	8	2	no	—
381	C₂²	C₃₁×#535	1	1	yes	characteristic
382	C₂²	C₃₁×#535	1	1	yes	characteristic
383	C₂²	C₃₁×#537	1	1	yes	characteristic
384	C₂²	C₃₁×#577	1	2	yes	—
385	C₂²	C₃₁×#578	1	2	yes	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	2	no	—
389	C₄	—	4	2	no	—
390	C₄	—	8	1	no	—
391	C₄	—	8	2	no	—
392	C₄	—	8	1	no	—
393	C₄	—	8	1	no	—
394	C₄	—	8	2	no	—
395	C₄	C₃₁×H(3,𝔽₄)	1	2	yes	—
396	C₄	C₃₁×#562	1	2	yes	—
397	C₂	—	16	1	no	—
398	C₂	C₃₁×#2819	1	1	yes	characteristic
399	C₂	C₃₁×#3222	1	1	yes	characteristic
400	C₂	C₃₁×#3223	1	1	yes	characteristic
401	C₂	C₃₁×#3225	1	2	yes	—
402	C₂	C₃₁×#3258	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
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×#2642	C₂	semidirect
3. C₃₁×#2056	C₂	semidirect
5. C₃₁×#2026	C₂	semidirect
7. C₃₁×#2024	C₂	semidirect
8. C₃₁×#2016	C₂	semidirect
11. C₃₁×#1832	C₂	semidirect
130. #32813	C₃₁	direct (N × H)
175. #2642	C₂×C₃₁	semidirect
177. #2056	C₂×C₃₁	semidirect
179. #2026	C₂×C₃₁	semidirect
181. #2024	C₂×C₃₁	semidirect
182. #2016	C₂×C₃₁	semidirect
185. #1832	C₂×C₃₁	semidirect
281. C₃₁	#32813	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×#2642	2
C₃₁×#2640	2
C₃₁×#2056	2
C₃₁×#2055	2
C₃₁×#2026	2
C₃₁×#2025	2
C₃₁×#2024	2
C₃₁×#2016	2
C₃₁×#2015	2
C₃₁×#2013	2
C₃₁×#1832	2
#32813	31

Subgroups — C31×#32813 #21489572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32813 #21489572