Subgroups — #34115144

Order 20096.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758196	C₂	1	2	yes	maximal, complemented
2	#23757649	C₂	1	1	yes	maximal, characteristic
3	#19097845	—	2	2	no	—
4	C₂×#14578939	—	2	2	no	—
5	#19097768	C₄	1	1	yes	characteristic
6	C₄×#11929434	C₂²	1	1	yes	characteristic
7	#127×C₁₅₇	C₄	1	1	yes	characteristic
8	#14578957	—	4	2	no	—
9	C₂×D₆₂₈	—	2	1	no	—
10	C₂×D₆₂₈	—	4	1	no	—
11	#14578939	—	4	2	no	—
12	#14578939	—	4	2	no	—
13	C₂×#11929434	D₄	1	2	yes	—
14	C₂×D₄×C₁₅₇	—	2	1	no	—
15	C₂×D₄×C₁₅₇	—	4	1	no	—
16	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×D₁₅₇	—	4	1	no	—
18	D₆₂₈	—	4	2	no	—
19	D₆₂₈	—	8	2	no	—
20	#11929434	D₈	1	2	yes	—
21	#11929434	SD₁₆	1	2	yes	—
22	C₂³×C₁₅₇	—	4	1	no	—
23	D₄×C₁₅₇	—	4	2	no	—
24	D₄×C₁₅₇	—	8	2	no	—
25	C₂×C₄×C₁₅₇	—	2	1	no	—
26	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₅₇	—	8	2	no	—
28	C₂²×C₁₅₇	—	8	1	no	—
29	C₂²×C₁₅₇	#100	1	1	yes	characteristic
30	C₂×D₁₅₇	—	8	2	no	—
31	C₂×D₁₅₇	—	8	1	no	—
32	C₄×C₁₅₇	—	2	2	no	—
33	C₄×C₁₅₇	#102	1	2	yes	—
34	C₂×C₁₅₇	—	2	1	no	—
35	C₂×C₁₅₇	—	16	1	no	—
36	C₂×C₁₅₇	#331	1	1	yes	characteristic
37	D₁₅₇	—	16	1	no	—
38	C₁₅₇	#1357	1	1	yes	characteristic, complemented, Sylow
39	#1357	—	157	1	no	maximal, Sylow
40	#493	—	157	1	no	—
41	#364	—	157	2	no	—
42	C₂×D₈	—	314	2	no	—
43	#127	—	157	1	no	—
44	#127	#10597845	1	1	yes	characteristic
45	M₃₂	—	314	2	no	—
46	C₄×C₈	—	157	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	314	1	no	—
50	C₂×D₄	—	628	1	no	—
51	D₈	—	628	2	no	—
52	D₈	—	628	2	no	—
53	C₂×C₈	—	157	2	no	—
54	C₄²	C₂×#10597845	1	1	yes	characteristic
55	C₁₆	—	628	2	no	—
56	C₂³	—	4	1	no	—
57	C₂³	—	628	1	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	2	no	—
60	D₄	—	628	2	no	—
61	D₄	—	1256	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
64	C₈	—	157	2	no	—
65	C₈	—	157	2	no	—
66	C₂²	—	8	2	no	—
67	C₂²	—	8	1	no	—
68	C₂²	—	1256	2	no	—
69	C₂²	—	1256	1	no	—
70	C₂²	#19097831	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	#19097833	1	2	yes	—
73	C₂	—	2	1	no	—
74	C₂	—	16	1	no	—
75	C₂	—	2512	1	no	—
76	C₂	#23758131	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23758131
commutator	C₄²×C₁₅₇	C₂×C₄
Frattini	C₄²	C₂×#10597845
Fitting	#127×C₁₅₇	C₄
socle	C₂×C₁₅₇	#331
prime radical	#23757649	C₂
cosocle	C₄×#11929434	C₂²
hypercenter	#127	#10597845
hypocenter	C₁₅₇	#1357
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #23758196	C₂	semidirect
38. C₁₅₇	#1357	semidirect

Maximal subgroups

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

subgroup	index
#23758196	2
#23757649	2
#1357	157