Subgroups — #34102446

Order 20096.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757706	C₂	1	2	yes	maximal, complemented
2	#23757110	C₂	1	2	yes	maximal, complemented
3	D₅₀₂₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097794	—	2	2	no	—
7	#19097714	—	2	2	no	—
8	D₂₅₁₂	C₂²	1	2	yes	—
9	C₁₆×D₁₅₇	C₂²	1	1	yes	characteristic
10	#19097677	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₅₇	C₂²	1	2	yes	—
12	C₃₂×C₁₅₇	C₂²	1	1	yes	characteristic
13	#14578955	—	4	2	no	—
14	#14578941	—	4	2	no	—
15	D₁₂₅₆	—	2	2	no	—
16	C₈×D₁₅₇	D₄	1	1	yes	characteristic
17	#14578929	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₅₇	—	2	2	no	—
19	C₁₆×C₁₅₇	C₂³	1	1	yes	characteristic
20	D₆₂₈	—	4	2	no	—
21	D₆₂₈	—	8	2	no	—
22	C₄×D₁₅₇	—	8	2	no	—
23	C₄×D₁₅₇	D₈	1	1	yes	characteristic
24	#11929434	D₈	1	1	yes	characteristic
25	Q₈×C₁₅₇	—	4	2	no	—
26	C₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₅₇	—	8	2	no	—
28	C₂×D₁₅₇	D₁₆	1	1	yes	characteristic
29	C₄×C₁₅₇	—	8	2	no	—
30	C₄×C₁₅₇	C₂×D₈	1	1	yes	characteristic
31	Dic₁₅₇	D₁₆	1	1	yes	characteristic
32	C₂×C₁₅₇	C₂×D₁₆	1	1	yes	characteristic
33	D₁₅₇	—	2	1	no	—
34	D₁₅₇	—	16	2	no	—
35	C₁₅₇	#2262	1	1	yes	characteristic, complemented, Sylow
36	#2262	—	157	1	no	maximal, Sylow
37	#508	—	157	2	no	—
38	Q₆₄	D₁₅₇	1	1	yes	characteristic, complemented
39	SD₆₄	—	157	2	no	—
40	D₃₂	—	157	1	no	—
41	C₂×C₃₂	—	157	1	no	—
42	#135	—	314	2	no	—
43	Q₃₂	C₂×D₁₅₇	1	2	yes	—
44	SD₃₂	—	314	2	no	—
45	D₁₆	—	157	2	no	—
46	C₂×C₁₆	—	157	1	no	—
47	C₃₂	—	157	1	no	—
48	C₃₂	C₂×D₁₅₇	1	1	yes	characteristic
49	#41	—	628	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	628	2	no	—
52	D₈	—	314	2	no	—
53	C₂×C₈	—	157	1	no	—
54	C₁₆	—	157	1	no	—
55	C₁₆	C₂²×D₁₅₇	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	628	2	no	—
58	D₄	—	1256	2	no	—
59	C₂×C₄	—	157	1	no	—
60	C₂×C₄	—	1256	2	no	—
61	C₈	—	157	1	no	—
62	C₈	D₄×D₁₅₇	1	1	yes	characteristic
63	C₂²	—	157	1	no	—
64	C₂²	—	1256	2	no	—
65	C₄	—	8	2	no	—
66	C₄	—	157	1	no	—
67	C₄	D₈×D₁₅₇	1	1	yes	characteristic
68	C₂	—	314	1	no	—
69	C₂	—	2512	2	no	—
70	C₂	D₁₆×D₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₅₇
commutator, cosocle	C₁₆×C₁₅₇	C₂³
Frattini	C₁₆	C₂²×D₁₅₇
Fitting	Q₆₄×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂×D₁₆
hypercenter	Q₆₄	D₁₅₇
hypocenter	C₁₅₇	#2262
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #23757706	C₂	semidirect
2. #23757110	C₂	semidirect
3. D₅₀₂₄	C₂	semidirect
4. C₃₂×D₁₅₇	C₂	semidirect
5. Q₆₄×C₁₅₇	C₂	semidirect
35. C₁₅₇	#2262	semidirect
38. Q₆₄	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757706	2
#23757110	2
D₅₀₂₄	2
C₃₂×D₁₅₇	2
Q₆₄×C₁₅₇	2
#2262	157