Subgroups — #34099754

Order 20096.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757678	C₂	1	4	yes	maximal
2	#23757653	C₂	1	2	yes	maximal
3	C₈²×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₂₈	—	2	4	no	—
5	#19097771	C₂²	1	4	yes	—
6	#19097705	—	2	8	no	—
7	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₅₇	C₂²	1	2	yes	—
9	C₂×Dic₃₁₄	—	2	4	no	—
10	Dic₆₂₈	—	4	8	no	—
11	C₄⋊Dic₁₅₇	—	2	8	no	—
12	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
13	C₂×C₈×C₁₅₇	D₄	1	4	yes	—
14	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
15	Dic₃₁₄	—	4	8	no	—
16	C₂×Dic₁₅₇	—	4	4	no	—
17	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
19	C₈×C₁₅₇	D₈	1	4	yes	—
20	C₈×C₁₅₇	Q₁₆	1	8	yes	complemented
21	C₂²×C₁₅₇	#127	1	1	yes	characteristic
22	C₄×C₁₅₇	C₂×D₈	1	2	yes	—
23	C₄×C₁₅₇	C₂×Q₁₆	1	4	yes	—
24	Dic₁₅₇	—	8	4	no	—
25	C₂×C₁₅₇	#493	1	1	yes	characteristic
26	C₂×C₁₅₇	#494	1	2	yes	—
27	C₁₅₇	#1713	1	1	yes	characteristic, complemented, Sylow
28	#1713	—	157	1	no	maximal, Sylow
29	#500	—	157	4	no	—
30	#494	—	157	2	no	—
31	C₈²	D₁₅₇	1	1	yes	characteristic
32	C₂×Q₁₆	—	314	4	no	—
33	C₄⋊Q₈	—	157	4	no	—
34	#107	—	314	8	no	—
35	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
36	C₄×C₈	C₂×D₁₅₇	1	2	yes	—
37	C₂×Q₈	—	314	4	no	—
38	Q₁₆	—	628	8	no	—
39	C₂×C₈	D₆₂₈	1	4	yes	—
40	C₂×C₈	D₆₂₈	1	2	yes	—
41	C₄⋊C₄	—	314	8	no	—
42	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
43	Q₈	—	628	8	no	—
44	C₂×C₄	—	628	4	no	—
45	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
46	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
47	C₈	D₁₂₅₆	1	4	yes	—
48	C₈	Dic₆₂₈	1	8	yes	complemented
49	C₂²	#19097768	1	1	yes	characteristic
50	C₄	—	1256	4	no	—
51	C₄	C₂×D₁₂₅₆	1	2	yes	—
52	C₄	C₂×Dic₆₂₈	1	4	yes	—
53	C₂	#23757650	1	1	yes	characteristic
54	C₂	#23757653	1	2	yes	—

	subgroup	G/H
center	C₂²	#19097768
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	C₈²×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#127
hypercenter	C₈²	D₁₅₇
hypocenter	C₁₅₇	#1713
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₁₅₇	Q₁₆	semidirect
27. C₁₅₇	#1713	semidirect
48. C₈	Dic₆₂₈	semidirect

Maximal subgroups

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

subgroup	index
#23757678	2
#23757653	2
C₈²×C₁₅₇	2
#1713	157