Subgroups — #14521786

Order 1984.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758516	C₂	1	6	yes	maximal, complemented
3	#11758514	C₂	1	3	yes	maximal, complemented
4	#11758512	C₂	1	3	yes	maximal, complemented
5	#11758453	C₂	1	2	yes	maximal, complemented
6	C₂³×D₃₁	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₃₁	C₂²	1	3	yes	—
8	C₂×D₄×C₃₁	C₂²	1	3	yes	complemented
9	C₂×C₂²⋊D₃₁	C₂²	1	6	yes	—
10	C₂²×Dic₃₁	C₂²	1	3	yes	complemented
11	C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
12	C₂²⋊Dic₃₁	C₂²	1	3	yes	—
13	C₂²⋊Dic₃₁	C₂²	1	3	yes	complemented
14	(C₂×D₃₁)⋊C₄	C₂²	1	6	yes	—
15	Dic₃₁⋊C₄	C₂²	1	6	yes	complemented
16	C₂³×C₃₁	C₂³	1	1	yes	characteristic
17	C₂³×C₃₁	C₂³	1	3	yes	—
18	C₂²×D₃₁	—	4	3	no	—
19	C₂²×D₃₁	C₂³	1	2	yes	—
20	D₄×C₃₁	—	4	3	no	—
21	D₄×C₃₁	—	4	3	no	—
22	C₂×C₄×C₃₁	C₂³	1	3	yes	—
23	C₂²⋊D₃₁	—	4	6	no	—
24	C₂×Dic₃₁	—	4	3	no	—
25	C₂×Dic₃₁	C₂³	1	6	yes	—
26	C₂²×C₃₁	—	2	3	no	—
27	C₂²×C₃₁	—	2	6	no	—
28	C₂²×C₃₁	—	2	3	no	—
29	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
30	C₂×D₃₁	—	2	6	no	—
31	C₂×D₃₁	—	4	4	no	—
32	C₄×C₃₁	—	2	3	no	—
33	Dic₃₁	—	2	6	no	—
34	#560	—	31	1	no	maximal, Sylow
35	C₂×C₃₁	—	4	1	no	—
36	C₂×C₃₁	—	4	3	no	—
37	C₂×C₃₁	H(5,𝔽₂)	1	3	yes	—
38	D₃₁	—	4	2	no	—
39	#127	D₃₁	1	1	yes	characteristic, complemented
40	#126	—	31	2	no	—
41	#123	—	31	3	no	—
42	#121	—	31	6	no	—
43	#120	—	31	3	no	—
44	C₃₁	#560	1	1	yes	characteristic, complemented, Sylow
45	C₂⁴	—	31	1	no	—
46	C₂×D₄	—	31	6	no	—
47	C₂×D₄	C₂×D₃₁	1	3	yes	—
48	C₂×D₄	C₂×D₃₁	1	3	yes	complemented
49	C₂²×C₄	—	31	3	no	—
50	C₄⋊C₄	—	31	6	no	—
51	C₂²⋊C₄	—	31	3	no	—
52	C₂²⋊C₄	—	31	3	no	—
53	C₂²⋊C₄	—	31	6	no	—
54	C₄²	C₂×D₃₁	1	1	yes	characteristic, complemented
55	C₂³	—	31	2	no	—
56	C₂³	—	124	3	no	—
57	C₂³	C₂²×D₃₁	1	1	yes	characteristic
58	C₂³	C₂²×D₃₁	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	124	6	no	—
62	C₂×C₄	—	31	6	no	—
63	C₂×C₄	—	124	3	no	—
64	C₂×C₄	C₂²×D₃₁	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	62	6	no	—
69	C₂²	—	124	4	no	—
70	C₂²	C₂³×D₃₁	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	62	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	124	2	no	—
76	C₂	#11758561	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#127	C₂
hypercenter	#127	D₃₁
hypocenter	C₃₁	#560
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₃₁×#127	C₂	semidirect
2. #11758516	C₂	semidirect
3. #11758514	C₂	semidirect
4. #11758512	C₂	semidirect
5. #11758453	C₂	semidirect
8. C₂×D₄×C₃₁	C₂²	semidirect
10. C₂²×Dic₃₁	C₂²	semidirect
11. C₄²×C₃₁	C₂²	semidirect
13. C₂²⋊Dic₃₁	C₂²	semidirect
15. Dic₃₁⋊C₄	C₂²	semidirect
39. #127	D₃₁	semidirect
44. C₃₁	#560	semidirect
48. C₂×D₄	C₂×D₃₁	semidirect
54. C₄²	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×#127	2
#11758516	2
#11758514	2
#11758512	2
#11758453	2
#560	31