Subgroups — #34102415

Order 20096.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757675	C₂	1	2	yes	maximal
2	#23757101	C₂	1	2	yes	maximal
3	#23757099	C₂	1	2	yes	maximal
4	C₄×C₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	#19097763	—	2	2	no	—
6	#19097705	C₂²	1	4	yes	—
7	#19097703	—	2	2	no	—
8	C₂×C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₅₇	—	2	4	no	—
12	#14578933	—	4	4	no	—
13	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
15	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
16	C₁₆×C₁₅₇	Q₈	1	2	yes	—
17	C₁₆×C₁₅₇	Q₈	1	2	yes	—
18	C₂×Dic₁₅₇	—	4	4	no	—
19	C₂×C₄×C₁₅₇	D₈	1	2	yes	—
20	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
21	C₈×C₁₅₇	Q₁₆	1	2	yes	—
22	C₈×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
23	C₈×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₁₅₇	C₂×D₈	1	1	yes	characteristic
25	C₄×C₁₅₇	—	2	2	no	—
26	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₁₅₇	—	8	4	no	—
29	C₂×C₁₅₇	#500	1	1	yes	characteristic
30	C₂×C₁₅₇	#508	1	2	yes	—
31	C₁₅₇	#2253	1	1	yes	characteristic, complemented, Sylow
32	#2253	—	157	1	no	maximal, Sylow
33	#499	—	157	2	no	—
34	#367	—	157	2	no	—
35	#366	—	157	2	no	—
36	C₄×C₁₆	D₁₅₇	1	1	yes	characteristic
37	#125	—	314	2	no	—
38	C₂×C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
39	C₂×C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
40	#107	—	157	4	no	—
41	#106	—	314	2	no	—
42	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
43	C₂×C₈	D₆₂₈	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
45	C₄⋊C₄	—	314	4	no	—
46	C₄⋊C₄	—	628	4	no	—
47	C₄²	D₆₂₈	1	1	yes	characteristic
48	C₁₆	Dic₃₁₄	1	2	yes	—
49	C₁₆	Dic₃₁₄	1	2	yes	—
50	C₂×C₄	—	628	4	no	—
51	C₂×C₄	D₁₂₅₆	1	2	yes	—
52	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
53	C₈	Dic₆₂₈	1	2	yes	—
54	C₈	C₂×Dic₃₁₄	1	1	yes	characteristic
55	C₈	C₂×Dic₃₁₄	1	1	yes	characteristic
56	C₂²	C₂×D₁₂₅₆	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	1256	4	no	—
59	C₄	#19097771	1	1	yes	characteristic
60	C₄	C₂×Dic₆₂₈	1	1	yes	characteristic
61	C₂	#23757678	1	1	yes	characteristic
62	C₂	#23757705	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂₅₆
commutator, cosocle	C₂×C₈×C₁₅₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₅₇
Fitting	C₄×C₁₆×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	C₂×D₈
hypercenter	C₄×C₁₆	D₁₅₇
hypocenter	C₁₅₇	#2253
LCS	G ▷ C₂×C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
31. C₁₅₇	#2253	semidirect

Maximal subgroups

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

subgroup	index
#23757675	2
#23757101	2
#23757099	2
C₄×C₁₆×C₁₅₇	2
#2253	157