Subgroups — C₃₁×#563 #14520744

Order 1984.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₃₁×#126	C₂	1	4	yes	maximal, complemented
3	C₃₁×#125	C₂	1	2	yes	maximal, complemented
4	C₃₁×#124	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#123	C₂	1	2	yes	maximal
6	C₃₁×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	C₂×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₃₁	C₂²	1	2	yes	—
11	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
13	C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
14	C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
15	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
17	C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
18	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	C₄²×C₃₁	C₂²	1	2	yes	—
21	C₂³×C₃₁	C₂³	1	2	yes	—
22	Q₈×C₃₁	—	4	2	no	—
23	Q₈×C₃₁	—	4	1	no	—
24	D₄×C₃₁	—	4	1	no	—
25	C₂×C₄×C₃₁	—	4	2	no	—
26	C₂×C₄×C₃₁	C₂³	1	2	yes	—
27	C₂×C₄×C₃₁	C₂³	1	4	yes	—
28	C₂×C₄×C₃₁	C₂³	1	2	yes	—
29	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₃₁	C₂³	1	4	yes	—
31	C₂²×C₃₁	—	2	2	no	—
32	C₂²×C₃₁	—	2	4	no	—
33	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
34	C₄×C₃₁	—	2	4	no	—
35	C₄×C₃₁	—	2	1	no	—
36	C₄×C₃₁	—	2	2	no	—
37	C₄×C₃₁	—	2	2	no	—
38	C₄×C₃₁	—	2	4	no	—
39	#563	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₃₁	—	4	2	no	—
41	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
42	C₂×C₃₁	2₋¹⁺⁴	1	2	yes	—
43	C₄⋊Q₈	C₂×C₃₁	1	2	yes	complemented
44	#126	C₂×C₃₁	1	4	yes	complemented
45	#125	C₂×C₃₁	1	2	yes	complemented
46	#124	C₂×C₃₁	1	1	yes	characteristic
47	#123	C₂×C₃₁	1	2	yes	—
48	C₂²⋊Q₈	C₂×C₃₁	1	4	yes	complemented
49	C₃₁	#563	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×Q₈	C₂²×C₃₁	1	2	yes	—
51	C₂×Q₈	C₂²×C₃₁	1	1	yes	characteristic
52	C₂×D₄	C₂²×C₃₁	1	1	yes	characteristic
53	C₂²×C₄	C₂²×C₃₁	1	2	yes	—
54	C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
55	C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
56	C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
57	C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
58	C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
59	C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
60	C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
61	C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
62	C₄²	C₂²×C₃₁	1	1	yes	characteristic
63	C₄²	C₂²×C₃₁	1	2	yes	—
64	C₂³	C₂³×C₃₁	1	2	yes	—
65	Q₈	—	4	2	no	—
66	Q₈	—	4	1	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	C₂³×C₃₁	1	2	yes	—
70	C₂×C₄	C₂³×C₃₁	1	4	yes	—
71	C₂×C₄	C₂³×C₃₁	1	2	yes	—
72	C₂×C₄	C₂³×C₃₁	1	1	yes	characteristic
73	C₂×C₄	C₂³×C₃₁	1	4	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂⁴×C₃₁	1	1	yes	characteristic
77	C₄	—	2	4	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	2	no	—
80	C₄	—	2	2	no	—
81	C₄	—	2	4	no	—
82	C₂	—	4	2	no	—
83	C₂	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
84	C₂	C₃₁×2₋¹⁺⁴	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×C₄⋊Q₈	C₂	semidirect
2. C₃₁×#126	C₂	semidirect
3. C₃₁×#125	C₂	semidirect
6. C₃₁×C₂²⋊Q₈	C₂	semidirect
39. #563	C₃₁	direct (N × H)
43. C₄⋊Q₈	C₂×C₃₁	semidirect
44. #126	C₂×C₃₁	semidirect
45. #125	C₂×C₃₁	semidirect
48. C₂²⋊Q₈	C₂×C₃₁	semidirect
49. C₃₁	#563	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×C₄⋊Q₈	2
C₃₁×#126	2
C₃₁×#125	2
C₃₁×#124	2
C₃₁×#123	2
C₃₁×C₂²⋊Q₈	2
#563	31

Subgroups — C31×#563 #14520744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#563 #14520744