Subgroups — C₃₁×#33302 #21490061

Order 7936.

1422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#494	C₂	1	12	yes	maximal
2	C₃₁×#2868	C₂	1	2	yes	maximal
3	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₁₆×C₃₁	C₂²	1	16	yes	—
5	C₂×C₃₁×C₄⋊Q₈	C₂²	1	12	yes	—
6	C₃₁×#494	—	2	48	no	—
7	C₂×C₄×C₈×C₃₁	C₂²	1	6	yes	—
8	C₃₁×#384	—	2	8	no	—
9	C₄³×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
11	C₂×Q₁₆×C₃₁	—	2	96	no	—
12	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
13	C₃₁×C₄⋊Q₈	—	2	48	no	—
14	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
15	C₂×C₄²×C₃₁	D₄	1	4	yes	—
16	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
17	C₄×C₈×C₃₁	D₄	1	24	yes	—
18	C₂×Q₈×C₃₁	—	2	48	no	—
19	C₂²×C₄×C₃₁	—	2	8	no	—
20	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
21	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
22	Q₁₆×C₃₁	—	4	64	no	—
23	C₂×C₈×C₃₁	C₂×D₄	1	24	yes	—
24	C₄⋊C₄×C₃₁	—	4	48	no	—
25	C₄²×C₃₁	Q₁₆	1	16	yes	complemented
26	C₄²×C₃₁	C₂×D₄	1	12	yes	—
27	#33302	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
29	Q₈×C₃₁	—	4	32	no	—
30	C₂×C₄×C₃₁	—	4	24	no	—
31	C₂×C₄×C₃₁	#127	1	12	yes	—
32	C₂×C₄×C₃₁	C₂×Q₁₆	1	24	yes	—
33	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
34	C₈×C₃₁	#127	1	16	yes	—
35	C₂×#494	C₂×C₃₁	1	12	yes	—
36	#2868	C₂×C₃₁	1	2	yes	—
37	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
38	C₂²×C₃₁	C₂×#127	1	3	yes	—
39	C₂²×C₃₁	C₂²×Q₁₆	1	4	yes	—
40	C₄×C₃₁	—	8	8	no	—
41	C₄×C₃₁	#494	1	24	yes	complemented
42	C₄×C₃₁	C₂×#127	1	4	yes	—
43	C₂²×Q₁₆	C₂²×C₃₁	1	16	yes	—
44	C₂×C₄⋊Q₈	C₂²×C₃₁	1	12	yes	—
45	#494	—	2	48	no	—
46	C₂×C₄×C₈	C₂²×C₃₁	1	6	yes	—
47	#384	—	2	8	no	—
48	C₄³	C₂²×C₃₁	1	1	yes	characteristic
49	C₂×C₃₁	#2867	1	1	yes	characteristic
50	C₂×C₃₁	C₂×#494	1	6	yes	—
51	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
52	C₂×Q₁₆	—	2	96	no	—
53	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
54	C₄⋊Q₈	—	2	48	no	—
55	C₂×C₄⋊C₄	—	2	24	no	—
56	C₂×C₄²	D₄×C₃₁	1	4	yes	—
57	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
58	C₄×C₈	D₄×C₃₁	1	24	yes	—
59	C₃₁	#33302	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×Q₈	—	2	48	no	—
61	C₂²×C₄	—	2	8	no	—
62	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
63	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
64	Q₁₆	—	4	64	no	—
65	C₂×C₈	C₂×D₄×C₃₁	1	24	yes	—
66	C₄⋊C₄	—	4	48	no	—
67	C₄²	Q₁₆×C₃₁	1	16	yes	complemented
68	C₄²	C₂×D₄×C₃₁	1	12	yes	—
69	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
70	Q₈	—	4	32	no	—
71	C₂×C₄	—	4	24	no	—
72	C₂×C₄	C₃₁×#127	1	12	yes	—
73	C₂×C₄	C₂×Q₁₆×C₃₁	1	24	yes	—
74	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
75	C₈	C₃₁×#127	1	16	yes	—
76	C₂²	C₂×C₃₁×#127	1	3	yes	—
77	C₂²	C₂²×Q₁₆×C₃₁	1	4	yes	—
78	C₄	—	8	8	no	—
79	C₄	C₃₁×#494	1	24	yes	complemented
80	C₄	C₂×C₃₁×#127	1	4	yes	—
81	C₂	C₃₁×#2867	1	1	yes	characteristic
82	C₂	C₂×C₃₁×#494	1	6	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₄²×C₃₁	Q₁₆	semidirect
27. #33302	C₃₁	direct (N × H)
41. C₄×C₃₁	#494	semidirect
59. C₃₁	#33302	direct (N × H)
67. C₄²	Q₁₆×C₃₁	semidirect
79. C₄	C₃₁×#494	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#494	2
C₃₁×#2868	2
C₄²×C₈×C₃₁	2
#33302	31

Subgroups — C31×#33302 #21490061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33302 #21490061