Subgroups — #34115156

Order 20096.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758196	C₂	1	1	yes	maximal, characteristic
2	#23758196	C₂	1	1	yes	maximal, characteristic
3	#23757672	C₂	1	1	yes	maximal, characteristic
4	#19097845	—	2	1	no	—
5	#19097845	—	2	1	no	—
6	#19097763	C₄	1	1	yes	characteristic
7	#19097704	—	2	1	no	—
8	#19097702	—	2	1	no	—
9	C₄×#11929434	C₂²	1	1	yes	characteristic
10	#125×C₁₅₇	C₄	1	1	yes	characteristic
11	#14578957	—	4	1	no	—
12	#14578957	—	4	1	no	—
13	C₂×#11929434	D₄	1	1	yes	characteristic
14	C₂×#11929434	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₅₇	—	2	1	no	—
16	#14578933	—	4	1	no	—
17	C₄⋊C₄×C₁₅₇	—	2	1	no	—
18	C₄⋊C₄×C₁₅₇	—	4	1	no	—
19	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₅₇	—	4	1	no	—
21	#11929434	SD₁₆	1	1	yes	characteristic
22	#11929434	SD₁₆	1	1	yes	characteristic
23	#11929434	Q₁₆	1	1	yes	characteristic
24	#11929434	Q₁₆	1	1	yes	characteristic
25	C₂×C₄×C₁₅₇	—	2	1	no	—
26	C₂×C₄×C₁₅₇	—	4	1	no	—
27	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₁₅₇	#100	1	1	yes	characteristic
29	C₄×C₁₅₇	—	4	1	no	—
30	C₄×C₁₅₇	—	8	1	no	—
31	C₄×C₁₅₇	#103	1	1	yes	characteristic
32	C₄×C₁₅₇	#103	1	1	yes	characteristic
33	Dic₁₅₇	—	8	1	no	—
34	C₂×C₁₅₇	—	2	1	no	—
35	C₂×C₁₅₇	#333	1	1	yes	characteristic
36	C₁₅₇	#1366	1	1	yes	characteristic, complemented, Sylow
37	#1366	—	157	1	no	maximal, Sylow
38	#499	—	157	1	no	—
39	#364	—	157	1	no	—
40	#364	—	157	1	no	—
41	#125	—	157	1	no	—
42	#125	#10597845	1	1	yes	characteristic
43	M₃₂	—	314	1	no	—
44	M₃₂	—	314	1	no	—
45	#107	—	314	1	no	—
46	#106	—	314	1	no	—
47	C₄×C₈	—	157	1	no	—
48	C₂×C₈	—	157	1	no	—
49	C₂×C₈	—	157	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	4	1	no	—
52	C₄⋊C₄	—	314	1	no	—
53	C₄⋊C₄	—	628	1	no	—
54	C₄²	C₂×#10597845	1	1	yes	characteristic
55	C₁₆	—	628	1	no	—
56	C₁₆	—	628	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	628	1	no	—
60	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
61	C₈	—	157	1	no	—
62	C₈	—	157	1	no	—
63	C₈	—	157	1	no	—
64	C₈	—	157	1	no	—
65	C₂²	#19097831	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	8	1	no	—
68	C₄	—	1256	1	no	—
69	C₄	#19097834	1	1	yes	characteristic
70	C₄	#19097834	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#23758133	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23758133
commutator	C₄²×C₁₅₇	C₂×C₄
Frattini	C₄²	C₂×#10597845
Fitting	#125×C₁₅₇	C₄
socle	C₂×C₁₅₇	#333
prime radical	C₂²×C₁₅₇	#100
cosocle	C₄×#11929434	C₂²
hypercenter	#125	#10597845
hypocenter	C₁₅₇	#1366
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
36. C₁₅₇	#1366	semidirect

Maximal subgroups

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

subgroup	index
#23758196	2
#23757672	2
#1366	157