Subgroups — #14521792

Order 1984.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758530	C₂	1	2	yes	maximal, complemented
3	#11758514	C₂	1	3	yes	maximal, complemented
4	#11758513	C₂	1	3	yes	maximal, complemented
5	#11758462	C₂	1	6	yes	maximal, complemented
6	C₂⁴×C₃₁	C₂²	1	1	yes	characteristic, complemented
7	C₂³×D₃₁	C₂²	1	1	yes	characteristic, complemented
8	C₂×D₄×C₃₁	C₂²	1	3	yes	—
9	C₂×C₂²⋊D₃₁	C₂²	1	6	yes	—
10	C₂×Dic₆₂	C₂²	1	3	yes	complemented
11	C₂²⋊C₄×C₃₁	C₂²	1	3	yes	complemented
12	C₂²⋊Dic₃₁	C₂²	1	6	yes	complemented
13	C₂²⋊Dic₃₁	C₂²	1	3	yes	complemented
14	(C₂×D₃₁)⋊C₄	C₂²	1	6	yes	complemented
15	C₄×Dic₃₁	C₂²	1	3	yes	complemented
16	C₂³×C₃₁	—	4	3	no	—
17	C₂³×C₃₁	C₂³	1	1	yes	characteristic
18	C₂³×C₃₁	C₂³	1	3	yes	—
19	C₂²×D₃₁	—	4	3	no	—
20	C₂²×D₃₁	C₂³	1	2	yes	—
21	D₄×C₃₁	—	4	3	no	—
22	C₂×C₄×C₃₁	C₂³	1	3	yes	—
23	C₂²⋊D₃₁	—	4	6	no	—
24	C₂×Dic₃₁	C₂³	1	6	yes	—
25	Dic₆₂	—	4	3	no	—
26	C₂²×C₃₁	—	2	6	no	—
27	C₂²×C₃₁	—	2	3	no	—
28	C₂²×C₃₁	—	2	3	no	—
29	C₂²×C₃₁	—	4	4	no	—
30	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
31	C₂×D₃₁	—	2	6	no	—
32	C₂×D₃₁	—	4	4	no	—
33	C₄×C₃₁	—	2	3	no	—
34	Dic₃₁	—	2	6	no	—
35	H(3,𝔽₄)	—	31	1	no	maximal, Sylow
36	C₂×C₃₁	—	4	1	no	—
37	C₂×C₃₁	—	4	3	no	—
38	C₂×C₃₁	H(5,𝔽₂)	1	3	yes	—
39	D₃₁	—	4	2	no	—
40	#124	—	31	3	no	—
41	#124	—	31	6	no	—
42	#120	—	31	2	no	—
43	#120	—	31	3	no	—
44	#120	D₃₁	1	1	yes	characteristic, complemented
45	C₃₁	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
46	C₂⁴	—	31	1	no	—
47	C₂⁴	C₂×D₃₁	1	1	yes	characteristic, complemented
48	C₂×Q₈	—	31	3	no	—
49	C₂×D₄	—	31	6	no	—
50	C₂×D₄	C₂×D₃₁	1	3	yes	—
51	C₂²⋊C₄	—	31	3	no	—
52	C₂²⋊C₄	—	31	6	no	—
53	C₂²⋊C₄	—	31	6	no	—
54	C₂²⋊C₄	C₂×D₃₁	1	3	yes	complemented
55	C₄²	—	31	3	no	—
56	C₂³	—	4	3	no	—
57	C₂³	—	31	2	no	—
58	C₂³	—	124	3	no	—
59	C₂³	C₂²×D₃₁	1	1	yes	characteristic
60	C₂³	C₂²×D₃₁	1	3	yes	—
61	Q₈	—	124	3	no	—
62	D₄	—	4	3	no	—
63	D₄	—	124	6	no	—
64	C₂×C₄	—	31	6	no	—
65	C₂×C₄	C₂²×D₃₁	1	3	yes	—
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	62	6	no	—
71	C₂²	—	124	4	no	—
72	C₂²	C₂³×D₃₁	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	62	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	124	2	no	—
78	C₂	#11758561	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#120	C₂
hypercenter	#120	D₃₁
hypocenter	C₃₁	H(3,𝔽₄)
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. C₃₁×#120	C₂	semidirect
2. #11758530	C₂	semidirect
3. #11758514	C₂	semidirect
4. #11758513	C₂	semidirect
5. #11758462	C₂	semidirect
6. C₂⁴×C₃₁	C₂²	semidirect
7. C₂³×D₃₁	C₂²	semidirect
10. C₂×Dic₆₂	C₂²	semidirect
11. C₂²⋊C₄×C₃₁	C₂²	semidirect
12. C₂²⋊Dic₃₁	C₂²	semidirect
13. C₂²⋊Dic₃₁	C₂²	semidirect
14. (C₂×D₃₁)⋊C₄	C₂²	semidirect
15. C₄×Dic₃₁	C₂²	semidirect
44. #120	D₃₁	semidirect
45. C₃₁	H(3,𝔽₄)	semidirect
47. C₂⁴	C₂×D₃₁	semidirect
54. C₂²⋊C₄	C₂×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#120	2
#11758530	2
#11758514	2
#11758513	2
#11758462	2
H(3,𝔽₄)	31