Subgroups — C₃₁×#7866 #21464625

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	2	yes	maximal
2	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1276	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#117	C₄	1	2	yes	—
5	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#431	—	2	2	no	—
7	C₃₁×#431	—	2	2	no	—
8	C₃₁×#431	C₄	1	4	yes	complemented
9	C₃₁×#431	C₂²	1	4	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	complemented
11	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
12	C₂×C₄×C₈×C₃₁	—	2	2	no	—
13	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
15	C₃₁×#376	C₄	1	2	yes	—
16	C₄³×C₃₁	C₂²	1	2	yes	—
17	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	D₄	1	2	yes	—
19	C₂²×C₈×C₃₁	—	2	4	no	—
20	C₃₁×#117	—	2	4	no	—
21	C₃₁×#117	C₈	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
24	C₂×C₄²×C₃₁	—	2	4	no	—
25	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	D₄	1	2	yes	—
30	C₂×C₄²×C₃₁	D₄	1	2	yes	—
31	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
32	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×C₄⋊C₈	—	2	4	no	—
34	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
35	C₃₁×C₂²⋊C₈	—	2	4	no	—
36	C₃₁×C₂²⋊C₈	—	2	4	no	—
37	C₄×C₈×C₃₁	—	2	4	no	—
38	C₄×C₈×C₃₁	—	2	4	no	—
39	C₃₁×#95	—	2	4	no	—
40	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	—	2	4	no	—
42	C₂²×C₄×C₃₁	—	4	4	no	—
43	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
53	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₃₁	—	2	8	no	—
56	C₂×C₈×C₃₁	—	4	8	no	—
57	C₄⋊C₄×C₃₁	—	2	8	no	—
58	C₂²⋊C₄×C₃₁	—	2	8	no	—
59	C₄²×C₃₁	—	2	4	no	—
60	C₄²×C₃₁	—	4	8	no	—
61	C₄²×C₃₁	C₄²	1	1	yes	characteristic
62	C₄²×C₃₁	C₄²	1	1	yes	characteristic
63	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
66	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₄²×C₃₁	C₂×C₈	1	8	yes	complemented
69	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
70	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
72	#7866	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	—	4	2	no	—
74	C₂³×C₃₁	#95	1	2	yes	—
75	C₂³×C₃₁	C₂²⋊C₈	1	4	yes	—
76	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	—	2	4	no	—
78	C₂×C₄×C₃₁	—	2	4	no	—
79	C₂×C₄×C₃₁	—	2	4	no	—
80	C₂×C₄×C₃₁	—	2	8	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	—	4	2	no	—
84	C₂×C₄×C₃₁	—	4	8	no	—
85	C₂×C₄×C₃₁	#95	1	2	yes	—
86	C₂×C₄×C₃₁	#95	1	2	yes	—
87	C₂×C₄×C₃₁	#95	1	2	yes	—
88	C₂×C₄×C₃₁	C₄×C₈	1	4	yes	—
89	C₂×C₄×C₃₁	#97	1	4	yes	—
90	C₂×C₄×C₃₁	C₂²⋊C₈	1	4	yes	—
91	C₂×C₄×C₃₁	C₄⋊C₈	1	8	yes	—
92	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
98	C₈×C₃₁	—	4	8	no	—
99	C₂×#431	C₂×C₃₁	1	2	yes	—
100	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
101	#1276	C₂×C₃₁	1	4	yes	complemented
102	C₂²×C₃₁	—	2	2	no	—
103	C₂²×C₃₁	—	2	2	no	—
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	4	4	no	—
106	C₂²×C₃₁	#336	1	4	yes	—
107	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
108	C₂²×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
109	C₄×C₃₁	—	2	2	no	—
110	C₄×C₃₁	—	2	2	no	—
111	C₄×C₃₁	—	2	2	no	—
112	C₄×C₃₁	—	4	8	no	—
113	C₄×C₃₁	#336	1	4	yes	—
114	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
115	C₄×C₃₁	C₂×#97	1	1	yes	characteristic
116	C₄×C₃₁	#421	1	4	yes	—
117	C₄×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
118	C₂×#117	C₄×C₃₁	1	2	yes	—
119	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
120	#431	—	2	2	no	—
121	#431	—	2	2	no	—
122	#431	C₄×C₃₁	1	4	yes	complemented
123	#431	C₂²×C₃₁	1	4	yes	—
124	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	complemented
125	C₂×C₂²⋊C₈	—	2	2	no	—
126	C₂×C₄×C₈	—	2	2	no	—
127	C₄×C₄⋊C₄	—	2	2	no	—
128	C₄×C₂²⋊C₄	—	2	2	no	—
129	#376	C₄×C₃₁	1	2	yes	—
130	C₄³	C₂²×C₃₁	1	2	yes	—
131	C₂×C₃₁	—	2	2	no	—
132	C₂×C₃₁	—	2	4	no	—
133	C₂×C₃₁	C₂×#336	1	1	yes	characteristic
134	C₂×C₃₁	#1733	1	2	yes	—
135	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₂³×C₄	D₄×C₃₁	1	2	yes	—
137	C₂²×C₈	—	2	4	no	—
138	#117	—	2	4	no	—
139	#117	C₈×C₃₁	1	8	yes	complemented
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₂²⋊C₄	—	2	2	no	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
144	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×C₄²	D₄×C₃₁	1	2	yes	—
148	C₂×C₄²	D₄×C₃₁	1	2	yes	—
149	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
150	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
153	C₂²⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₄×C₈	—	2	4	no	—
156	C₄×C₈	—	2	4	no	—
157	#95	—	2	4	no	—
158	C₃₁	#7866	1	1	yes	characteristic, direct product factor, Sylow
159	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
163	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
165	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
166	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
168	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
170	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
171	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
172	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	4	8	no	—
176	C₄⋊C₄	—	2	8	no	—
177	C₂²⋊C₄	—	2	8	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	4	8	no	—
180	C₄²	C₄²×C₃₁	1	1	yes	characteristic
181	C₄²	C₄²×C₃₁	1	1	yes	characteristic
182	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
183	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
184	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
185	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
186	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
187	C₄²	C₂×C₈×C₃₁	1	8	yes	complemented
188	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
189	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
190	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
191	C₂³	—	4	2	no	—
192	C₂³	C₃₁×#95	1	2	yes	—
193	C₂³	C₃₁×C₂²⋊C₈	1	4	yes	—
194	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	C₃₁×#95	1	2	yes	—
204	C₂×C₄	C₃₁×#95	1	2	yes	—
205	C₂×C₄	C₃₁×#95	1	2	yes	—
206	C₂×C₄	C₄×C₈×C₃₁	1	4	yes	—
207	C₂×C₄	C₃₁×#97	1	4	yes	—
208	C₂×C₄	C₃₁×C₂²⋊C₈	1	4	yes	—
209	C₂×C₄	C₃₁×C₄⋊C₈	1	8	yes	—
210	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
212	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
213	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
215	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
216	C₈	—	4	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	C₃₁×#336	1	4	yes	—
222	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
223	C₂²	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	8	no	—
228	C₄	C₃₁×#336	1	4	yes	—
229	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
230	C₄	C₂×C₃₁×#97	1	1	yes	characteristic
231	C₄	C₃₁×#421	1	4	yes	—
232	C₄	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	—	2	4	no	—
235	C₂	C₂×C₃₁×#336	1	1	yes	characteristic
236	C₂	C₃₁×#1733	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#95
prime radical	C₂⁴×C₃₁	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1276	C₂	semidirect
8. C₃₁×#431	C₄	semidirect
10. C₂×C₃₁×C₄⋊C₈	C₄	semidirect
21. C₃₁×#117	C₈	semidirect
68. C₄²×C₃₁	C₂×C₈	semidirect
72. #7866	C₃₁	direct (N × H)
101. #1276	C₂×C₃₁	semidirect
122. #431	C₄×C₃₁	semidirect
124. C₂×C₄⋊C₈	C₄×C₃₁	semidirect
139. #117	C₈×C₃₁	semidirect
158. C₃₁	#7866	direct (N × H)
187. C₄²	C₂×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#431	2
C₄×C₃₁×#117	2
C₃₁×#1276	2
#7866	31

Subgroups — C31×#7866 #21464625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7866 #21464625