Subgroups — #34115268

Order 20096.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758178	C₂	1	2	yes	maximal, complemented
2	#23757650	C₂	1	1	yes	maximal, characteristic
3	#19097830	—	2	2	no	—
4	C₂×D₁₂₅₆	—	2	2	no	—
5	#19097768	C₄	1	1	yes	characteristic, complemented
6	#19097768	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₁₅₇	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#10597845	—	4	2	no	—
9	C₂×D₆₂₈	—	2	1	no	—
10	C₂×D₆₂₈	—	4	1	no	—
11	C₂×D₆₂₈	—	4	1	no	—
12	C₂×D₆₂₈	D₄	1	2	yes	—
13	D₁₂₅₆	—	4	4	no	—
14	C₂×C₈×C₁₅₇	—	2	1	no	—
15	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#10597845	—	8	2	no	—
17	C₂²×D₁₅₇	—	2	2	no	—
18	C₂²×D₁₅₇	—	4	1	no	—
19	D₆₂₈	—	4	2	no	—
20	D₆₂₈	—	4	2	no	—
21	D₆₂₈	—	8	2	no	—
22	D₆₂₈	—	8	2	no	—
23	C₂×C₄×C₁₅₇	—	2	1	no	—
24	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₁₅₇	—	2	2	no	—
26	C₂²×C₁₅₇	#99	1	1	yes	characteristic
27	C₂×D₁₅₇	—	4	2	no	—
28	C₂×D₁₅₇	—	8	2	no	—
29	C₂×D₁₅₇	—	8	2	no	—
30	C₂×D₁₅₇	—	8	1	no	—
31	#10597845	—	16	2	no	—
32	C₄×C₁₅₇	—	2	2	no	—
33	C₄×C₁₅₇	—	2	1	no	—
34	C₂×C₁₅₇	—	2	1	no	—
35	C₂×C₁₅₇	#353	1	1	yes	characteristic
36	D₁₅₇	—	8	2	no	—
37	D₁₅₇	—	16	1	no	—
38	C₁₅₇	#1408	1	1	yes	characteristic, complemented, Sylow
39	#1408	—	157	1	no	maximal, Sylow
40	#493	—	157	1	no	—
41	#353	—	157	2	no	—
42	C₂×D₈	—	314	2	no	—
43	#127	—	157	1	no	—
44	#127	—	157	1	no	—
45	#99	—	314	2	no	—
46	C₄×C₈	#10597845	1	1	yes	characteristic, complemented
47	C₂×D₄	—	314	1	no	—
48	C₂×D₄	—	628	1	no	—
49	C₂×D₄	—	628	1	no	—
50	C₂×D₄	—	157	2	no	—
51	D₈	—	628	4	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂²⋊C₄	—	628	2	no	—
54	C₄²	C₂×#10597845	1	1	yes	characteristic
55	C₂³	—	314	2	no	—
56	C₂³	—	628	1	no	—
57	D₄	—	628	2	no	—
58	D₄	—	628	2	no	—
59	D₄	—	1256	2	no	—
60	D₄	—	1256	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	1256	2	no	—
63	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₂²	—	628	2	no	—
66	C₂²	—	1256	2	no	—
67	C₂²	—	1256	2	no	—
68	C₂²	—	1256	1	no	—
69	C₂²	#19097830	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	2512	2	no	—
73	C₂	—	2	1	no	—
74	C₂	—	2512	1	no	—
75	C₂	—	1256	2	no	—
76	C₂	#23758178	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23758178
commutator	C₄²×C₁₅₇	C₂×C₄
Frattini	C₄²	C₂×#10597845
Fitting	C₄×C₈×C₁₅₇	C₄
socle	C₂×C₁₅₇	#353
prime radical	#23757650	C₂
cosocle	#19097768	C₂²
hypercenter	C₄×C₈	#10597845
hypocenter	C₁₅₇	#1408
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #23758178	C₂	semidirect
5. #19097768	C₄	semidirect
7. C₄×C₈×C₁₅₇	C₄	semidirect
38. C₁₅₇	#1408	semidirect
46. C₄×C₈	#10597845	semidirect

Maximal subgroups

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

subgroup	index
#23758178	2
#23757650	2
#1408	157