Subgroups — #34115292

Order 20096.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758206	C₂	1	1	yes	maximal, characteristic
2	#23758202	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097842	—	2	1	no	—
5	#19097821	—	2	1	no	—
6	Q₁₆×D₁₅₇	—	2	1	no	—
7	Dic₁₂₅₆	C₄	1	1	yes	characteristic
8	#19097717	—	2	1	no	—
9	C₁₆×D₁₅₇	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₅₇	C₄	1	1	yes	characteristic
11	#14578964	—	4	1	no	—
12	Q₈×D₁₅₇	—	4	1	no	—
13	Dic₆₂₈	—	2	1	no	—
14	#14578944	—	4	1	no	—
15	C₈×D₁₅₇	D₄	1	1	yes	characteristic
16	#14578929	D₄	1	1	yes	characteristic
17	Q₁₆×C₁₅₇	—	2	1	no	—
18	C₁₆×C₁₅₇	C₂×C₄	1	1	yes	characteristic
19	#11929441	—	8	1	no	—
20	Dic₃₁₄	—	4	1	no	—
21	Dic₃₁₄	—	8	1	no	—
22	C₄×D₁₅₇	—	8	1	no	—
23	C₄×D₁₅₇	D₈	1	1	yes	characteristic
24	#11929434	SD₁₆	1	1	yes	characteristic
25	Q₈×C₁₅₇	—	4	1	no	—
26	C₈×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₁₅₇	SD₃₂	1	1	yes	characteristic
28	C₄×C₁₅₇	—	8	1	no	—
29	C₄×C₁₅₇	#102	1	1	yes	characteristic
30	Dic₁₅₇	—	8	1	no	—
31	Dic₁₅₇	D₁₆	1	1	yes	characteristic
32	C₂×C₁₅₇	#357	1	1	yes	characteristic
33	D₁₅₇	—	2	1	no	—
34	C₁₅₇	#1420	1	1	yes	characteristic, complemented, Sylow
35	#1420	—	157	1	no	maximal, Sylow
36	C₂×Q₃₂	—	157	1	no	—
37	M₆₄	—	157	1	no	—
38	#368	—	157	1	no	—
39	C₂×Q₁₆	—	314	1	no	—
40	Q₃₂	—	314	1	no	—
41	Q₃₂	—	157	1	no	—
42	Q₃₂	#10597845	1	1	yes	characteristic
43	C₂×C₁₆	—	157	1	no	—
44	#108	—	314	1	no	—
45	C₃₂	—	314	1	no	—
46	C₂×Q₈	—	628	1	no	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	314	1	no	—
49	Q₁₆	—	628	1	no	—
50	M₁₆	—	628	1	no	—
51	C₂×C₈	—	157	1	no	—
52	C₁₆	—	157	1	no	—
53	C₁₆	C₂×#10597845	1	1	yes	characteristic
54	Q₈	—	4	1	no	—
55	Q₈	—	628	1	no	—
56	Q₈	—	1256	1	no	—
57	C₂×C₄	—	157	1	no	—
58	C₂×C₄	—	1256	1	no	—
59	C₈	—	157	1	no	—
60	C₈	—	1256	1	no	—
61	C₈	C₂²⋊#10597845	1	1	yes	characteristic
62	C₂²	—	157	1	no	—
63	C₄	—	8	1	no	—
64	C₄	—	157	1	no	—
65	C₄	—	1256	1	no	—
66	C₄	#19097833	1	1	yes	characteristic
67	C₂	—	314	1	no	—
68	C₂	#23758185	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23758185
commutator	C₁₆×C₁₅₇	C₂×C₄
Frattini	C₁₆	C₂×#10597845
Fitting	Q₃₂×C₁₅₇	C₄
socle	C₂×C₁₅₇	#357
prime radical	C₂×D₁₅₇	SD₃₂
cosocle	C₁₆×D₁₅₇	C₂²
hypercenter	Q₃₂	#10597845
hypocenter	C₁₅₇	#1420
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
34. C₁₅₇	#1420	semidirect

Maximal subgroups

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

subgroup	index
#23758206	2
#23758202	2
Q₃₂×D₁₅₇	2
#1420	157