Subgroups — C₃₁×#7902 #21464661

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1279	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#117	C₄	1	2	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#432	—	2	2	no	—
8	C₃₁×#432	C₄	1	4	yes	complemented
9	C₃₁×#431	—	2	2	no	—
10	C₃₁×#431	C₄	1	4	yes	complemented
11	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₃₁×#97	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
21	C₃₁×#376	—	2	1	no	—
22	C₄³×C₃₁	—	2	1	no	—
23	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₃₁×#117	—	2	8	no	—
29	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄²×C₃₁	—	2	2	no	—
32	C₂×C₄²×C₃₁	—	2	2	no	—
33	C₂×C₄²×C₃₁	—	2	2	no	—
34	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	D₄	1	2	yes	—
40	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×C₄⋊C₈	—	2	4	no	—
43	C₃₁×C₄⋊C₈	—	2	2	no	—
44	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₃₁×C₂²⋊C₈	—	2	4	no	—
46	C₃₁×C₂²⋊C₈	—	2	4	no	—
47	C₃₁×#97	—	2	4	no	—
48	C₄×C₈×C₃₁	—	2	2	no	—
49	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
50	C₃₁×#95	—	2	2	no	—
51	C₃₁×#95	—	2	2	no	—
52	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	4	4	no	—
72	C₂×C₈×C₃₁	—	4	4	no	—
73	C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₄⋊C₄×C₃₁	M₁₆	1	8	yes	—
75	C₂²⋊C₄×C₃₁	—	2	8	no	—
76	C₄²×C₃₁	—	2	8	no	—
77	C₄²×C₃₁	—	4	4	no	—
78	C₄²×C₃₁	—	4	4	no	—
79	C₄²×C₃₁	C₄²	1	2	yes	—
80	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
83	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
84	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
85	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
89	#7902	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	—	2	2	no	—
92	C₂³×C₃₁	—	2	2	no	—
93	C₂³×C₃₁	#95	1	2	yes	—
94	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	8	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	4	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	#95	1	2	yes	—
109	C₂×C₄×C₃₁	#95	1	2	yes	—
110	C₂×C₄×C₃₁	#95	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
118	C₂×C₄×C₃₁	#135	1	4	yes	—
119	C₈×C₃₁	—	4	4	no	—
120	C₈×C₃₁	—	4	4	no	—
121	C₂×#432	C₂×C₃₁	1	1	yes	characteristic
122	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
123	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
124	#1279	C₂×C₃₁	1	4	yes	complemented
125	C₂²×C₃₁	—	2	2	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	4	4	no	—
129	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
130	C₂²×C₃₁	#407	1	2	yes	—
131	C₂²×C₃₁	#416	1	2	yes	—
132	C₂²×C₃₁	#421	1	2	yes	—
133	C₄×C₃₁	—	2	2	no	—
134	C₄×C₃₁	—	2	1	no	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	—	2	1	no	—
137	C₄×C₃₁	—	2	2	no	—
138	C₄×C₃₁	—	4	8	no	—
139	C₄×C₃₁	C₄×M₁₆	1	2	yes	—
140	C₄×C₃₁	#416	1	2	yes	—
141	C₄×C₃₁	#423	1	2	yes	—
142	C₄×C₃₁	#427	1	2	yes	—
143	C₂×#117	C₄×C₃₁	1	2	yes	—
144	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
145	#432	—	2	2	no	—
146	#432	C₄×C₃₁	1	4	yes	complemented
147	#431	—	2	2	no	—
148	#431	C₄×C₃₁	1	4	yes	complemented
149	C₂×C₄⋊C₈	—	2	1	no	—
150	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
151	C₂×C₂²⋊C₈	—	2	1	no	—
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×#97	—	2	1	no	—
154	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
155	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
156	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
157	C₄×C₂²⋊C₄	—	2	1	no	—
158	C₄×C₂²⋊C₄	—	2	1	no	—
159	#376	—	2	1	no	—
160	C₄³	—	2	1	no	—
161	C₂×C₃₁	—	4	2	no	—
162	C₂×C₃₁	#1488	1	4	yes	—
163	C₂×C₃₁	#1728	1	1	yes	characteristic
164	C₂×C₃₁	#1733	1	1	yes	characteristic
165	C₂×C₃₁	#1735	1	1	yes	characteristic
166	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
168	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	#117	—	2	8	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
173	C₂×C₂²⋊C₄	—	2	2	no	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
180	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₃₁	1	2	yes	—
183	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
184	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
188	C₂²⋊C₈	—	2	4	no	—
189	C₂²⋊C₈	—	2	4	no	—
190	#97	—	2	4	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	C₃₁	#7902	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
202	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
206	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
207	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
208	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
209	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	4	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	M₁₆×C₃₁	1	8	yes	—
219	C₂²⋊C₄	—	2	8	no	—
220	C₄²	—	2	8	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	C₄²×C₃₁	1	2	yes	—
224	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
225	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
226	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
227	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
228	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
229	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
230	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
231	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₃₁×#95	1	2	yes	—
237	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₃₁×#95	1	2	yes	—
252	C₂×C₄	C₃₁×#95	1	2	yes	—
253	C₂×C₄	C₃₁×#95	1	2	yes	—
254	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
261	C₂×C₄	C₃₁×#135	1	4	yes	—
262	C₈	—	4	4	no	—
263	C₈	—	4	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	4	no	—
268	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
269	C₂²	C₃₁×#407	1	2	yes	—
270	C₂²	C₃₁×#416	1	2	yes	—
271	C₂²	C₃₁×#421	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	8	no	—
278	C₄	C₄×M₁₆×C₃₁	1	2	yes	—
279	C₄	C₃₁×#416	1	2	yes	—
280	C₄	C₃₁×#423	1	2	yes	—
281	C₄	C₃₁×#427	1	2	yes	—
282	C₂	—	4	2	no	—
283	C₂	C₃₁×#1488	1	4	yes	—
284	C₂	C₃₁×#1728	1	1	yes	characteristic
285	C₂	C₃₁×#1733	1	1	yes	characteristic
286	C₂	C₃₁×#1735	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1279	C₂	semidirect
8. C₃₁×#432	C₄	semidirect
10. C₃₁×#431	C₄	semidirect
89. #7902	C₃₁	direct (N × H)
124. #1279	C₂×C₃₁	semidirect
146. #432	C₄×C₃₁	semidirect
148. #431	C₄×C₃₁	semidirect
195. C₃₁	#7902	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×#432	2
C₂×C₃₁×#431	2
C₄×C₃₁×#117	2
C₃₁×#1279	2
#7902	31

Subgroups — C31×#7902 #21464661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7902 #21464661