Subgroups — #34098321

Order 20096.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757705	C₂	1	1	yes	maximal, characteristic
2	#23757103	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#19097793	—	2	1	no	—
5	Dic₁₂₅₆	C₄	1	1	yes	characteristic
6	#19097716	—	2	1	no	—
7	D₂₅₁₂	C₄	1	1	yes	characteristic
8	#19097707	—	2	1	no	—
9	C₂×C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₅₇	—	2	1	no	—
11	#14578954	—	4	1	no	—
12	Dic₆₂₈	—	2	1	no	—
13	#14578943	—	4	1	no	—
14	D₁₂₅₆	—	2	1	no	—
15	#14578938	—	4	1	no	—
16	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₅₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₅₇	D₄	1	1	yes	characteristic
19	Dic₃₁₄	—	4	1	no	—
20	D₆₂₈	—	4	1	no	—
21	C₂²⋊D₁₅₇	—	8	1	no	—
22	C₄×D₁₅₇	—	8	1	no	—
23	#11929434	—	8	1	no	—
24	C₂×C₄×C₁₅₇	D₈	1	1	yes	characteristic
25	C₈×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₅₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₅₇	SD₃₂	1	1	yes	characteristic
28	C₂×D₁₅₇	—	8	1	no	—
29	C₄×C₁₅₇	#102	1	1	yes	characteristic
30	C₄×C₁₅₇	D₁₆	1	1	yes	characteristic
31	Dic₁₅₇	—	8	1	no	—
32	C₂×C₁₅₇	—	2	1	no	—
33	C₂×C₁₅₇	#357	1	1	yes	characteristic
34	D₁₅₇	—	16	1	no	—
35	C₁₅₇	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	157	1	no	maximal, Sylow
37	#508	—	157	1	no	—
38	C₂×C₃₂	D₁₅₇	1	1	yes	characteristic, complemented
39	#368	—	157	1	no	—
40	#135	—	314	1	no	—
41	Q₃₂	—	157	1	no	—
42	SD₃₂	—	314	1	no	—
43	D₁₆	—	157	1	no	—
44	C₂×C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
45	#108	—	314	1	no	—
46	C₃₂	—	2	1	no	—
47	#41	—	628	1	no	—
48	Q₁₆	—	314	1	no	—
49	SD₁₆	—	628	1	no	—
50	D₈	—	314	1	no	—
51	M₁₆	—	628	1	no	—
52	C₂×C₈	D₆₂₈	1	1	yes	characteristic
53	C₁₆	C₄×D₁₅₇	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₁₅₇	1	1	yes	characteristic
55	Q₈	—	628	1	no	—
56	D₄	—	628	1	no	—
57	D₄	—	1256	1	no	—
58	C₂×C₄	—	1256	1	no	—
59	C₂×C₄	D₁₂₅₆	1	1	yes	characteristic
60	C₈	—	1256	1	no	—
61	C₈	#14578931	1	1	yes	characteristic
62	C₈	#14578943	1	1	yes	characteristic
63	C₂²	—	1256	1	no	—
64	C₂²	#19097716	1	1	yes	characteristic
65	C₄	—	1256	1	no	—
66	C₄	#19097693	1	1	yes	characteristic
67	C₄	D₂₅₁₂	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	2512	1	no	—
70	C₂	#23757075	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₅₁₂
commutator	C₁₆×C₁₅₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₅₇
Fitting	C₂×C₃₂×C₁₅₇	C₂
socle	C₂×C₁₅₇	#357
prime radical	#23757705	C₂
cosocle	C₂×C₁₆×C₁₅₇	C₂²
hypercenter	C₂×C₃₂	D₁₅₇
hypocenter	C₁₅₇	#1417
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #23757103	C₂	semidirect
3. C₂×C₃₂×C₁₅₇	C₂	semidirect
35. C₁₅₇	#1417	semidirect
38. C₂×C₃₂	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757705	2
#23757103	2
C₂×C₃₂×C₁₅₇	2
#1417	157