Subgroups — C₃₁×#14692 #21471451

Order 7936.

2286 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#384	C₂	1	14	yes	maximal
2	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₃₁	C₂²	1	28	yes	—
4	C₂²×C₄²×C₃₁	C₂²	1	7	yes	—
5	C₃₁×#384	—	2	56	no	—
6	C₄³×C₃₁	C₄	1	8	yes	complemented
7	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
8	C₂³×C₄×C₃₁	C₂³	1	7	yes	—
9	C₂×C₄⋊C₄×C₃₁	—	2	168	no	—
10	C₂×C₄²×C₃₁	C₂×C₄	1	28	yes	—
11	C₂×C₄²×C₃₁	D₄	1	28	yes	—
12	C₂×C₄²×C₃₁	Q₈	1	28	yes	—
13	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
14	C₂²×C₄×C₃₁	—	2	56	no	—
15	C₂²×C₄×C₃₁	C₂²×C₄	1	14	yes	—
16	C₂²×C₄×C₃₁	C₂×D₄	1	42	yes	—
17	C₂²×C₄×C₃₁	C₂×Q₈	1	42	yes	—
18	C₄⋊C₄×C₃₁	—	4	112	no	—
19	C₄²×C₃₁	C₄⋊C₄	1	112	yes	complemented
20	#14692	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
22	C₂³×C₃₁	C₂²×D₄	1	7	yes	—
23	C₂³×C₃₁	C₂²×Q₈	1	7	yes	—
24	C₂×C₄×C₃₁	—	4	56	no	—
25	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	84	yes	—
26	C₂×C₄×C₃₁	#127	1	28	yes	—
27	C₂×C₄×C₃₁	C₄⋊Q₈	1	84	yes	—
28	C₂×#384	C₂×C₃₁	1	14	yes	—
29	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
30	C₂²×C₃₁	C₂²×C₄⋊C₄	1	7	yes	—
31	C₂²×C₃₁	C₂×#127	1	7	yes	—
32	C₂²×C₃₁	C₂×C₄⋊Q₈	1	21	yes	—
33	C₄×C₃₁	—	8	8	no	—
34	C₄×C₃₁	#384	1	56	yes	complemented
35	C₂²×C₄⋊C₄	C₂²×C₃₁	1	28	yes	—
36	C₂²×C₄²	C₂²×C₃₁	1	7	yes	—
37	#384	—	2	56	no	—
38	C₄³	C₄×C₃₁	1	8	yes	complemented
39	C₂×C₃₁	C₂×#384	1	7	yes	—
40	C₂×C₃₁	#2867	1	1	yes	characteristic
41	C₂×C₃₁	#2868	1	7	yes	—
42	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
43	C₂³×C₄	C₂³×C₃₁	1	7	yes	—
44	C₂×C₄⋊C₄	—	2	168	no	—
45	C₂×C₄²	C₂×C₄×C₃₁	1	28	yes	—
46	C₂×C₄²	D₄×C₃₁	1	28	yes	—
47	C₂×C₄²	Q₈×C₃₁	1	28	yes	—
48	C₃₁	#14692	1	1	yes	characteristic, direct product factor, Sylow
49	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
50	C₂²×C₄	—	2	56	no	—
51	C₂²×C₄	C₂²×C₄×C₃₁	1	14	yes	—
52	C₂²×C₄	C₂×D₄×C₃₁	1	42	yes	—
53	C₂²×C₄	C₂×Q₈×C₃₁	1	42	yes	—
54	C₄⋊C₄	—	4	112	no	—
55	C₄²	C₄⋊C₄×C₃₁	1	112	yes	complemented
56	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
57	C₂³	C₂²×D₄×C₃₁	1	7	yes	—
58	C₂³	C₂²×Q₈×C₃₁	1	7	yes	—
59	C₂×C₄	—	4	56	no	—
60	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	84	yes	—
61	C₂×C₄	C₃₁×#127	1	28	yes	—
62	C₂×C₄	C₃₁×C₄⋊Q₈	1	84	yes	—
63	C₂²	C₂²×C₄⋊C₄×C₃₁	1	7	yes	—
64	C₂²	C₂×C₃₁×#127	1	7	yes	—
65	C₂²	C₂×C₃₁×C₄⋊Q₈	1	21	yes	—
66	C₄	—	8	8	no	—
67	C₄	C₃₁×#384	1	56	yes	complemented
68	C₂	C₂×C₃₁×#384	1	7	yes	—
69	C₂	C₃₁×#2867	1	1	yes	characteristic
70	C₂	C₃₁×#2868	1	7	yes	—

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

Complements

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

N	H ≅ G/N	type
6. C₄³×C₃₁	C₄	semidirect
19. C₄²×C₃₁	C₄⋊C₄	semidirect
20. #14692	C₃₁	direct (N × H)
34. C₄×C₃₁	#384	semidirect
38. C₄³	C₄×C₃₁	semidirect
48. C₃₁	#14692	direct (N × H)
55. C₄²	C₄⋊C₄×C₃₁	semidirect
67. C₄	C₃₁×#384	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#384	2
C₂×C₄³×C₃₁	2
#14692	31

Subgroups — C31×#14692 #21471451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#14692 #21471451