Subgroups — #34115285

Order 20096.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₅₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#23758199	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₆×D₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097840	—	2	1	no	—
5	#19097821	—	2	1	no	—
6	D₈×D₁₅₇	—	2	1	no	—
7	D₂₅₁₂	C₄	1	1	yes	characteristic, complemented
8	#19097712	—	2	1	no	—
9	C₁₆×D₁₅₇	C₂²	1	1	yes	characteristic
10	D₁₆×C₁₅₇	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#10597845	—	4	1	no	—
12	D₄×D₁₅₇	—	4	1	no	—
13	D₁₂₅₆	—	2	1	no	—
14	#14578939	—	4	1	no	—
15	C₈×D₁₅₇	D₄	1	1	yes	characteristic
16	#14578929	D₄	1	1	yes	characteristic
17	D₈×C₁₅₇	—	2	1	no	—
18	C₁₆×C₁₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂×#10597845	—	8	1	no	—
20	C₂²×D₁₅₇	—	8	1	no	—
21	D₆₂₈	—	4	1	no	—
22	C₂²⋊D₁₅₇	—	8	1	no	—
23	C₄×D₁₅₇	D₈	1	1	yes	characteristic
24	#11929434	SD₁₆	1	1	yes	characteristic
25	D₄×C₁₅₇	—	4	1	no	—
26	C₈×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₅₇	—	8	1	no	—
28	C₂×D₁₅₇	—	8	1	no	—
29	C₂×D₁₅₇	—	16	1	no	—
30	C₂×D₁₅₇	—	16	1	no	—
31	C₂×D₁₅₇	D₁₆	1	1	yes	characteristic
32	#10597845	—	16	1	no	—
33	C₄×C₁₅₇	#102	1	1	yes	characteristic
34	Dic₁₅₇	SD₃₂	1	1	yes	characteristic
35	C₂×C₁₅₇	—	16	1	no	—
36	C₂×C₁₅₇	#357	1	1	yes	characteristic
37	D₁₅₇	—	16	1	no	—
38	D₁₅₇	D₃₂	1	1	yes	characteristic
39	D₁₅₇	SD₆₄	1	1	yes	characteristic
40	C₁₅₇	#1415	1	1	yes	characteristic, complemented, Sylow
41	#1415	—	157	1	no	maximal, Sylow
42	C₂×D₁₆	—	157	1	no	—
43	C₂×C₃₂	—	157	1	no	—
44	#366	—	157	1	no	—
45	C₂×D₈	—	314	1	no	—
46	D₁₆	—	314	1	no	—
47	D₁₆	—	157	1	no	—
48	D₁₆	#10597845	1	1	yes	characteristic, complemented
49	C₂×C₁₆	—	157	1	no	—
50	#107	—	314	1	no	—
51	C₃₂	—	314	1	no	—
52	C₂×D₄	—	628	1	no	—
53	D₈	—	2	1	no	—
54	D₈	—	314	1	no	—
55	D₈	—	628	1	no	—
56	C₂×C₈	—	157	1	no	—
57	C₄⋊C₄	—	628	1	no	—
58	C₁₆	—	157	1	no	—
59	C₁₆	C₂×#10597845	1	1	yes	characteristic
60	C₂³	—	1256	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	628	1	no	—
63	D₄	—	1256	1	no	—
64	C₂×C₄	—	157	1	no	—
65	C₂×C₄	—	1256	1	no	—
66	C₈	—	157	1	no	—
67	C₈	C₂²⋊#10597845	1	1	yes	characteristic
68	C₂²	—	8	1	no	—
69	C₂²	—	157	1	no	—
70	C₂²	—	2512	1	no	—
71	C₂²	—	2512	1	no	—
72	C₂²	—	1256	1	no	—
73	C₄	—	157	1	no	—
74	C₄	—	2512	1	no	—
75	C₄	#19097833	1	1	yes	characteristic
76	C₂	—	16	1	no	—
77	C₂	—	157	1	no	—
78	C₂	—	157	1	no	—
79	C₂	—	2512	1	no	—
80	C₂	#23758185	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23758185
commutator	C₁₆×C₁₅₇	C₂×C₄
Frattini	C₁₆	C₂×#10597845
Fitting	D₁₆×C₁₅₇	C₄
socle	C₂×C₁₅₇	#357
prime radical	D₁₆×D₁₅₇	C₂
cosocle	C₁₆×D₁₅₇	C₂²
hypercenter	D₁₆	#10597845
hypocenter	C₁₅₇	#1415
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. D₁₅₇⋊C₃₂	C₂	semidirect
2. #23758199	C₂	semidirect
7. D₂₅₁₂	C₄	semidirect
10. D₁₆×C₁₅₇	C₄	semidirect
40. C₁₅₇	#1415	semidirect
48. D₁₆	#10597845	semidirect

Maximal subgroups

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

subgroup	index
D₁₅₇⋊C₃₂	2
#23758199	2
D₁₆×D₁₅₇	2
#1415	157