Subgroups — #34103313

Order 20096.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757153	C₂	1	12	yes	maximal, complemented
2	#23757141	C₂	1	2	yes	maximal, complemented
3	C₂×#127×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₅₇	C₂²	1	4	yes	—
5	C₂×C₂²⋊Dic₁₅₇	C₂²	1	12	yes	—
6	#19097678	C₂²	1	12	yes	—
7	C₂²×D₄×C₁₅₇	C₂²	1	6	yes	—
8	#127×C₁₅₇	C₄	1	8	yes	complemented
9	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₅₇	—	4	12	no	—
11	C₂²×Dic₁₅₇	C₂³	1	8	yes	—
12	C₂²⋊Dic₁₅₇	—	4	12	no	—
13	C₂⁴×C₁₅₇	C₂³	1	4	yes	—
14	C₂×D₄×C₁₅₇	—	4	6	no	—
15	C₂×D₄×C₁₅₇	—	4	6	no	—
16	C₂×D₄×C₁₅₇	C₂×C₄	1	24	yes	complemented
17	C₂²×C₄×C₁₅₇	C₂³	1	3	yes	—
18	C₄²×C₁₅₇	C₂×C₄	1	4	yes	complemented
19	C₂×Dic₁₅₇	—	2	24	no	—
20	C₂×Dic₁₅₇	—	4	16	no	—
21	C₂³×C₁₅₇	—	2	12	no	—
22	C₂³×C₁₅₇	—	2	12	no	—
23	C₂³×C₁₅₇	C₂²×C₄	1	8	yes	—
24	C₂³×C₁₅₇	C₂⁴	1	1	yes	characteristic
25	D₄×C₁₅₇	—	4	24	no	—
26	C₂×C₄×C₁₅₇	—	2	6	no	—
27	C₂×C₄×C₁₅₇	C₂²×C₄	1	6	yes	—
28	C₂²×C₁₅₇	—	2	24	no	—
29	C₂²×C₁₅₇	—	4	12	no	—
30	C₂²×C₁₅₇	—	4	4	no	—
31	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
32	C₂²×C₁₅₇	H(5,𝔽₂)	1	6	yes	—
33	C₄×C₁₅₇	—	2	6	no	—
34	Dic₁₅₇	—	4	8	no	—
35	C₂×C₁₅₇	—	4	8	no	—
36	C₂×C₁₅₇	#518	1	3	yes	—
37	C₂×C₁₅₇	#560	1	4	yes	—
38	C₁₅₇	#2380	1	1	yes	characteristic, complemented, Sylow
39	#2380	—	157	1	no	maximal, Sylow
40	C₂×#127	D₁₅₇	1	1	yes	characteristic
41	#386	—	157	12	no	—
42	#383	—	157	2	no	—
43	C₂²×D₄	C₂×D₁₅₇	1	6	yes	—
44	C₂³×C₄	—	157	4	no	—
45	#127	Dic₁₅₇	1	8	yes	complemented
46	C₂×C₂²⋊C₄	—	157	12	no	—
47	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
48	#95	—	157	12	no	—
49	C₂⁴	C₂²×D₁₅₇	1	4	yes	—
50	C₂×D₄	—	4	6	no	—
51	C₂×D₄	—	4	6	no	—
52	C₂×D₄	C₂×Dic₁₅₇	1	24	yes	complemented
53	C₂²×C₄	—	628	12	no	—
54	C₂²×C₄	—	157	8	no	—
55	C₂²×C₄	C₂²×D₁₅₇	1	3	yes	—
56	C₂²⋊C₄	—	628	12	no	—
57	C₄²	C₂×Dic₁₅₇	1	4	yes	complemented
58	C₂³	—	2	12	no	—
59	C₂³	—	2	12	no	—
60	C₂³	C₂²×Dic₁₅₇	1	8	yes	—
61	C₂³	C₂³×D₁₅₇	1	1	yes	characteristic
62	D₄	—	4	24	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	314	24	no	—
65	C₂×C₄	—	628	16	no	—
66	C₂×C₄	C₂²×Dic₁₅₇	1	6	yes	—
67	C₂²	—	2	24	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	4	12	no	—
70	C₂²	C₂³×Dic₁₅₇	1	1	yes	characteristic
71	C₂²	#19097816	1	6	yes	—
72	C₄	—	2	6	no	—
73	C₄	—	628	8	no	—
74	C₂	—	4	8	no	—
75	C₂	#23757748	1	3	yes	—
76	C₂	#23758010	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅₇
commutator	C₂²×C₁₅₇	C₂³×C₄
Fitting, prime radical	C₂×#127×C₁₅₇	C₂
socle, cosocle	C₂³×C₁₅₇	C₂⁴
hypercenter	C₂×#127	D₁₅₇
hypocenter	C₁₅₇	#2380
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #23757153	C₂	semidirect
2. #23757141	C₂	semidirect
8. #127×C₁₅₇	C₄	semidirect
16. C₂×D₄×C₁₅₇	C₂×C₄	semidirect
18. C₄²×C₁₅₇	C₂×C₄	semidirect
38. C₁₅₇	#2380	semidirect
45. #127	Dic₁₅₇	semidirect
52. C₂×D₄	C₂×Dic₁₅₇	semidirect
57. C₄²	C₂×Dic₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757153	2
#23757141	2
C₂×#127×C₁₅₇	2
#2380	157