Subgroups — #34115085

Order 20096.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758169	C₂	1	2	yes	maximal
2	#23757299	C₂	1	1	yes	maximal, characteristic
3	#19097845	—	2	2	no	—
4	C₂×#14578938	C₂²	1	1	yes	characteristic
5	#19097730	C₄	1	1	yes	characteristic
6	#19097700	—	2	2	no	—
7	C₂²×C₈×C₁₅₇	C₄	1	1	yes	characteristic
8	#14578957	—	4	2	no	—
9	#14578938	—	4	1	no	—
10	C₂×#11929434	D₄	1	2	yes	—
11	C₄⋊Dic₁₅₇	—	2	1	no	—
12	C₂²⋊Dic₁₅₇	C₈	1	2	yes	—
13	C₄×Dic₁₅₇	—	2	1	no	—
14	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₅₇	—	2	1	no	—
16	C₂×C₈×C₁₅₇	—	4	1	no	—
17	C₂×Dic₁₅₇	—	2	2	no	—
18	#11929434	—	2	2	no	—
19	C₂³×C₁₅₇	C₂×C₈	1	1	yes	characteristic
20	C₂×C₄×C₁₅₇	—	2	1	no	—
21	C₂×C₄×C₁₅₇	—	2	1	no	—
22	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₅₇	M₁₆	1	1	yes	characteristic
24	C₈×C₁₅₇	—	4	1	no	—
25	C₂²×C₁₅₇	—	2	1	no	—
26	C₂²×C₁₅₇	—	4	1	no	—
27	C₂²×C₁₅₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₁₅₇	—	2	1	no	—
29	C₄×C₁₅₇	#99	1	1	yes	characteristic
30	C₄×C₁₅₇	#101	1	1	yes	characteristic
31	Dic₁₅₇	—	4	2	no	—
32	C₂×C₁₅₇	—	2	1	no	—
33	C₂×C₁₅₇	—	4	1	no	—
34	C₂×C₁₅₇	#324	1	1	yes	characteristic
35	C₁₅₇	#1328	1	1	yes	characteristic, complemented, Sylow
36	#1328	—	157	1	no	maximal, Sylow
37	#424	—	157	1	no	—
38	#349	—	157	2	no	—
39	C₂×M₁₆	—	157	1	no	—
40	C₂²×C₈	#10597845	1	1	yes	characteristic
41	#117	—	157	1	no	—
42	M₃₂	—	314	2	no	—
43	C₄⋊C₈	—	314	2	no	—
44	C₂²×C₄	C₂×#10597845	1	1	yes	characteristic
45	M₁₆	—	628	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	—	157	2	no	—
49	C₄⋊C₄	—	314	1	no	—
50	C₂²⋊C₄	—	157	2	no	—
51	C₄²	—	314	1	no	—
52	C₁₆	—	628	2	no	—
53	C₂³	D₁₅₇⋊C₈	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	314	2	no	—
57	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
58	C₂×C₄	#14578964	1	1	yes	characteristic
59	C₈	—	4	1	no	—
60	C₈	—	314	2	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#19097828	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	628	2	no	—
66	C₄	#19097829	1	1	yes	characteristic
67	C₄	#19097832	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#23758119	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19097829
commutator, prime radical	C₂³×C₁₅₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#10597845
Fitting	C₂²×C₈×C₁₅₇	C₄
socle	C₂×C₁₅₇	#324
cosocle	C₂×#14578938	C₂²
hypercenter	C₂²×C₈	#10597845
hypocenter	C₁₅₇	#1328
LCS	G ▷ C₂³×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
35. C₁₅₇	#1328	semidirect

Maximal subgroups

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

subgroup	index
#23758169	2
#23757299	2
#1328	157