Subgroups — #34102378

Order 20096.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757675	C₂	1	1	yes	maximal, characteristic, complemented
2	#23757650	C₂	1	1	yes	maximal, characteristic
3	#23757075	C₂	1	4	yes	maximal, complemented
4	#346×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₂₅₆	—	2	1	no	—
6	C₂×D₁₂₅₆	C₂²	1	2	yes	—
7	#19097768	—	2	1	no	—
8	#19097763	—	2	1	no	—
9	#19097705	C₂²	1	2	yes	—
10	#19097703	—	2	1	no	—
11	C₂×C₁₆×C₁₅₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×D₆₂₈	—	2	2	no	—
14	C₂×D₆₂₈	—	4	2	no	—
15	D₁₂₅₆	—	2	4	no	—
16	D₁₂₅₆	—	4	2	no	—
17	C₄⋊Dic₁₅₇	—	2	2	no	—
18	#14578933	—	4	2	no	—
19	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
22	C₁₆×C₁₅₇	—	2	2	no	—
23	C₂²×D₁₅₇	—	4	2	no	—
24	D₆₂₈	—	4	2	no	—
25	D₆₂₈	—	4	2	no	—
26	D₆₂₈	—	8	4	no	—
27	C₂×Dic₁₅₇	—	4	2	no	—
28	C₂×C₄×C₁₅₇	D₈	1	2	yes	—
29	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₅₇	SD₁₆	1	2	yes	—
31	C₈×C₁₅₇	#41	1	2	yes	—
32	C₂²×C₁₅₇	C₂×D₈	1	1	yes	characteristic
33	C₂×D₁₅₇	—	8	2	no	—
34	C₂×D₁₅₇	—	8	4	no	—
35	C₄×C₁₅₇	—	2	2	no	—
36	C₄×C₁₅₇	#124	1	1	yes	characteristic
37	C₄×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
38	Dic₁₅₇	—	8	2	no	—
39	C₂×C₁₅₇	#486	1	1	yes	characteristic
40	C₂×C₁₅₇	#509	1	2	yes	—
41	D₁₅₇	—	16	2	no	—
42	C₁₅₇	#2244	1	1	yes	characteristic, complemented, Sylow
43	#2244	—	157	1	no	maximal, Sylow
44	#499	—	157	1	no	—
45	#493	—	157	1	no	—
46	#357	—	157	4	no	—
47	#346	D₁₅₇	1	1	yes	characteristic, complemented
48	C₂×D₈	—	314	1	no	—
49	C₂×D₈	—	157	2	no	—
50	#127	—	314	1	no	—
51	#125	—	314	1	no	—
52	C₂×C₁₆	C₂×D₁₅₇	1	2	yes	—
53	#107	—	157	2	no	—
54	#106	—	314	1	no	—
55	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
56	C₂×D₄	—	314	2	no	—
57	C₂×D₄	—	628	2	no	—
58	D₈	—	314	4	no	—
59	D₈	—	628	2	no	—
60	C₂×C₈	D₆₂₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
62	C₄⋊C₄	—	314	2	no	—
63	C₄⋊C₄	—	628	2	no	—
64	C₄²	D₆₂₈	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	C₂³	—	628	2	no	—
67	D₄	—	628	2	no	—
68	D₄	—	628	2	no	—
69	D₄	—	1256	4	no	—
70	C₂×C₄	—	628	2	no	—
71	C₂×C₄	D₁₂₅₆	1	2	yes	—
72	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
73	C₈	#14578943	1	2	yes	—
74	C₈	#14578954	1	2	yes	—
75	C₂²	—	1256	2	no	—
76	C₂²	—	1256	4	no	—
77	C₂²	C₂×D₁₂₅₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	1256	2	no	—
80	C₄	#19097759	1	1	yes	characteristic
81	C₄	C₂×#14578943	1	1	yes	characteristic
82	C₂	—	2512	2	no	—
83	C₂	#23757620	1	1	yes	characteristic
84	C₂	#23757711	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂₅₆
commutator, cosocle	C₂×C₈×C₁₅₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₅₇
Fitting	#346×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×D₈
prime radical	#23757650	C₂
hypercenter	#346	D₁₅₇
hypocenter	C₁₅₇	#2244
LCS	G ▷ C₂×C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
1. #23757675	C₂	semidirect
3. #23757075	C₂	semidirect
4. #346×C₁₅₇	C₂	semidirect
42. C₁₅₇	#2244	semidirect
47. #346	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757675	2
#23757650	2
#23757075	2
#346×C₁₅₇	2
#2244	157