Subgroups — C₃₁×#8131 #21464890

Order 7936.

574 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈×C₃₁	C₂	1	3	yes	maximal
2	C₃₁×#1273	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₃₁	C₄	1	6	yes	—
4	C₂×C₄×C₈×C₃₁	C₄	1	6	yes	—
5	C₂×C₄×C₈×C₃₁	C₂²	1	6	yes	—
6	C₄³×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂×C₄	1	12	yes	—
8	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
9	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
10	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
11	C₄×C₈×C₃₁	—	2	12	no	—
12	C₄×C₈×C₃₁	—	2	12	no	—
13	C₄×C₈×C₃₁	C₈	1	24	yes	complemented
14	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
15	C₂²×C₄×C₃₁	C₄²	1	3	yes	—
16	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
17	C₂×C₈×C₃₁	—	2	12	no	—
18	C₂×C₈×C₃₁	—	2	12	no	—
19	C₂×C₈×C₃₁	C₂×C₈	1	24	yes	—
20	C₄²×C₃₁	C₂×C₈	1	12	yes	—
21	C₄²×C₃₁	M₁₆	1	12	yes	—
22	C₄²×C₃₁	#41	1	4	yes	—
23	#8131	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
25	C₂×C₄×C₃₁	C₄×C₈	1	6	yes	—
26	C₂×C₄×C₃₁	C₄×C₈	1	6	yes	—
27	C₂×C₄×C₃₁	#97	1	12	yes	—
28	C₂×C₄×C₃₁	#117	1	6	yes	—
29	C₂×C₄×C₃₁	C₂²×C₈	1	6	yes	—
30	C₂×C₄×C₃₁	C₂×M₁₆	1	6	yes	—
31	C₈×C₃₁	—	2	24	no	—
32	C₄²×C₈	C₂×C₃₁	1	3	yes	—
33	#1273	C₂×C₃₁	1	4	yes	—
34	C₂²×C₃₁	#376	1	1	yes	characteristic
35	C₂²×C₃₁	C₂×C₄×C₈	1	3	yes	—
36	C₂²×C₃₁	C₂×#97	1	3	yes	—
37	C₄×C₃₁	C₈²	1	4	yes	—
38	C₄×C₃₁	#322	1	12	yes	—
39	C₄×C₃₁	#431	1	12	yes	—
40	C₂×C₄×C₈	C₄×C₃₁	1	6	yes	—
41	C₂×C₄×C₈	C₄×C₃₁	1	6	yes	—
42	C₂×C₄×C₈	C₂²×C₃₁	1	6	yes	—
43	C₄³	C₂²×C₃₁	1	1	yes	characteristic
44	C₂×C₃₁	C₂×C₈²	1	1	yes	characteristic
45	C₂×C₃₁	C₂×#322	1	3	yes	—
46	C₂×C₃₁	#1744	1	3	yes	—
47	C₂²×C₈	C₂×C₄×C₃₁	1	12	yes	—
48	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
49	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
50	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
51	C₄×C₈	—	2	12	no	—
52	C₄×C₈	—	2	12	no	—
53	C₄×C₈	C₈×C₃₁	1	24	yes	complemented
54	C₃₁	#8131	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
56	C₂²×C₄	C₄²×C₃₁	1	3	yes	—
57	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
58	C₂×C₈	—	2	12	no	—
59	C₂×C₈	—	2	12	no	—
60	C₂×C₈	C₂×C₈×C₃₁	1	24	yes	—
61	C₄²	C₂×C₈×C₃₁	1	12	yes	—
62	C₄²	M₁₆×C₃₁	1	12	yes	—
63	C₄²	#41×C₃₁	1	4	yes	—
64	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
65	C₂×C₄	C₄×C₈×C₃₁	1	6	yes	—
66	C₂×C₄	C₄×C₈×C₃₁	1	6	yes	—
67	C₂×C₄	C₃₁×#97	1	12	yes	—
68	C₂×C₄	C₃₁×#117	1	6	yes	—
69	C₂×C₄	C₂²×C₈×C₃₁	1	6	yes	—
70	C₂×C₄	C₂×M₁₆×C₃₁	1	6	yes	—
71	C₈	—	2	24	no	—
72	C₂²	C₃₁×#376	1	1	yes	characteristic
73	C₂²	C₂×C₄×C₈×C₃₁	1	3	yes	—
74	C₂²	C₂×C₃₁×#97	1	3	yes	—
75	C₄	C₈²×C₃₁	1	4	yes	—
76	C₄	C₃₁×#322	1	12	yes	—
77	C₄	C₃₁×#431	1	12	yes	—
78	C₂	C₂×C₈²×C₃₁	1	1	yes	characteristic
79	C₂	C₂×C₃₁×#322	1	3	yes	—
80	C₂	C₃₁×#1744	1	3	yes	—

	subgroup	G/H
center	C₄³×C₃₁	C₂²
commutator	C₂	C₂×C₈²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₈×C₃₁	C₈	semidirect
23. #8131	C₃₁	direct (N × H)
53. C₄×C₈	C₈×C₃₁	semidirect
54. C₃₁	#8131	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₈×C₃₁	2
C₃₁×#1273	2
#8131	31

Subgroups — C31×#8131 #21464890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8131 #21464890