Subgroups — #34102382

Order 20096.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757675	C₂	1	1	yes	maximal, characteristic
2	#23757653	C₂	1	1	yes	maximal, characteristic
3	#23757078	C₂	1	4	yes	maximal
4	#346×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₂₈	—	2	1	no	—
6	C₂×Dic₆₂₈	C₂²	1	2	yes	—
7	#19097771	—	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×Dic₃₁₄	—	2	2	no	—
14	Dic₆₂₈	—	2	4	no	—
15	Dic₆₂₈	—	4	2	no	—
16	C₄⋊Dic₁₅₇	—	2	2	no	—
17	C₄⋊Dic₁₅₇	—	4	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	Dic₃₁₄	—	4	2	no	—
24	Dic₃₁₄	—	4	2	no	—
25	C₂×Dic₁₅₇	—	4	2	no	—
26	C₂×Dic₁₅₇	—	4	2	no	—
27	C₂×C₄×C₁₅₇	D₈	1	2	yes	—
28	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₅₇	SD₁₆	1	2	yes	—
30	C₈×C₁₅₇	#41	1	2	yes	—
31	C₂²×C₁₅₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₅₇	—	2	2	no	—
33	C₄×C₁₅₇	#124	1	1	yes	characteristic
34	C₄×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
35	Dic₁₅₇	—	8	2	no	—
36	Dic₁₅₇	—	8	2	no	—
37	C₂×C₁₅₇	#486	1	1	yes	characteristic
38	C₂×C₁₅₇	#510	1	2	yes	—
39	C₁₅₇	#2245	1	1	yes	characteristic, complemented, Sylow
40	#2245	—	157	1	no	maximal, Sylow
41	#499	—	157	1	no	—
42	#494	—	157	1	no	—
43	#358	—	157	4	no	—
44	#346	D₁₅₇	1	1	yes	characteristic
45	C₂×Q₁₆	—	314	1	no	—
46	C₂×Q₁₆	—	157	2	no	—
47	C₄⋊Q₈	—	314	1	no	—
48	#125	—	314	1	no	—
49	C₂×C₁₆	C₂×D₁₅₇	1	2	yes	—
50	#107	—	157	2	no	—
51	#106	—	314	1	no	—
52	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
53	C₂×Q₈	—	314	2	no	—
54	Q₁₆	—	314	4	no	—
55	Q₁₆	—	628	2	no	—
56	C₂×C₈	D₆₂₈	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
58	C₄⋊C₄	—	314	2	no	—
59	C₄⋊C₄	—	628	2	no	—
60	C₄⋊C₄	—	628	2	no	—
61	C₄²	D₆₂₈	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	Q₈	—	628	2	no	—
64	Q₈	—	628	2	no	—
65	C₂×C₄	—	628	2	no	—
66	C₂×C₄	—	628	2	no	—
67	C₂×C₄	D₁₂₅₆	1	2	yes	—
68	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
69	C₈	#14578943	1	2	yes	—
70	C₈	#14578954	1	2	yes	—
71	C₂²	C₂×D₁₂₅₆	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	1256	2	no	—
74	C₄	—	1256	2	no	—
75	C₄	#19097759	1	1	yes	characteristic
76	C₄	C₂×#14578943	1	1	yes	characteristic
77	C₂	#23757620	1	1	yes	characteristic
78	C₂	#23757716	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, prime radical	C₂²×C₁₅₇	C₂×D₈
hypercenter	#346	D₁₅₇
hypocenter	C₁₅₇	#2245
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
39. C₁₅₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#23757675	2
#23757653	2
#23757078	2
#346×C₁₅₇	2
#2245	157