Subgroups — #34102451

Order 20096.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757705	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₅₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757112	C₂	1	2	yes	maximal, complemented
4	D₅₀₂₄	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097793	—	2	1	no	—
7	C₂×D₁₂₅₆	—	2	1	no	—
8	Dic₁₂₅₆	C₂²	1	1	yes	characteristic, complemented
9	#19097716	—	2	1	no	—
10	D₂₅₁₂	—	2	1	no	—
11	D₂₅₁₂	C₂²	1	2	yes	—
12	D₂₅₁₂	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₅₇	C₂²	1	2	yes	complemented
15	#14578954	—	4	1	no	—
16	C₂×D₆₂₈	—	4	1	no	—
17	Dic₆₂₈	—	2	1	no	—
18	#14578943	—	4	1	no	—
19	D₁₂₅₆	—	2	2	no	—
20	D₁₂₅₆	—	2	1	no	—
21	D₁₂₅₆	—	4	1	no	—
22	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₅₇	D₄	1	1	yes	characteristic
24	C₁₆×C₁₅₇	C₂³	1	1	yes	characteristic
25	C₂²×D₁₅₇	—	8	1	no	—
26	Dic₃₁₄	—	4	1	no	—
27	D₆₂₈	—	4	2	no	—
28	D₆₂₈	—	4	1	no	—
29	D₆₂₈	—	8	1	no	—
30	C₂²⋊D₁₅₇	—	8	1	no	—
31	C₄×D₁₅₇	—	8	1	no	—
32	C₂×C₄×C₁₅₇	D₈	1	1	yes	characteristic
33	C₈×C₁₅₇	D₈	1	1	yes	characteristic
34	C₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₅₇	D₁₆	1	1	yes	characteristic
36	C₂×D₁₅₇	—	8	1	no	—
37	C₂×D₁₅₇	—	8	2	no	—
38	C₂×D₁₅₇	—	16	2	no	—
39	C₄×C₁₅₇	D₁₆	1	1	yes	characteristic
40	C₄×C₁₅₇	C₂×D₈	1	1	yes	characteristic
41	Dic₁₅₇	—	8	1	no	—
42	C₂×C₁₅₇	—	2	1	no	—
43	C₂×C₁₅₇	C₂×D₁₆	1	1	yes	characteristic
44	D₁₅₇	—	16	1	no	—
45	D₁₅₇	—	16	2	no	—
46	C₁₅₇	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	157	1	no	maximal, Sylow
48	#508	—	157	1	no	—
49	C₂×D₁₆	—	157	1	no	—
50	SD₆₄	—	157	2	no	—
51	D₃₂	—	157	2	no	—
52	M₆₄	D₁₅₇	1	1	yes	characteristic, complemented
53	#135	—	314	1	no	—
54	C₂×D₈	—	314	1	no	—
55	Q₃₂	—	157	1	no	—
56	SD₃₂	—	314	1	no	—
57	D₁₆	—	314	1	no	—
58	D₁₆	—	157	2	no	—
59	D₁₆	—	157	1	no	—
60	C₂×C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
61	C₃₂	C₂×D₁₅₇	1	2	yes	complemented
62	#41	—	628	1	no	—
63	C₂×D₄	—	628	1	no	—
64	Q₁₆	—	314	1	no	—
65	SD₁₆	—	628	1	no	—
66	D₈	—	314	2	no	—
67	D₈	—	314	1	no	—
68	D₈	—	628	1	no	—
69	C₂×C₈	D₆₂₈	1	1	yes	characteristic
70	C₁₆	D₆₂₈	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₅₇	1	1	yes	characteristic
72	C₂³	—	1256	1	no	—
73	Q₈	—	628	1	no	—
74	D₄	—	628	2	no	—
75	D₄	—	628	1	no	—
76	D₄	—	1256	1	no	—
77	D₄	—	1256	1	no	—
78	C₂×C₄	—	1256	1	no	—
79	C₂×C₄	D₁₂₅₆	1	1	yes	characteristic
80	C₈	D₁₂₅₆	1	1	yes	characteristic
81	C₈	C₂×D₆₂₈	1	1	yes	characteristic
82	C₂²	—	2512	2	no	—
83	C₂²	—	1256	1	no	—
84	C₂²	—	1256	2	no	—
85	C₂²	D₂₅₁₂	1	1	yes	characteristic
86	C₄	—	1256	1	no	—
87	C₄	D₂₅₁₂	1	1	yes	characteristic
88	C₄	C₂×D₁₂₅₆	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2512	2	no	—
91	C₂	—	2512	1	no	—
92	C₂	C₂×D₂₅₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₅₁₂
commutator, cosocle	C₁₆×C₁₅₇	C₂³
Frattini	C₁₆	C₂²×D₁₅₇
Fitting	M₆₄×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂×D₁₆
hypercenter	M₆₄	D₁₅₇
hypocenter	C₁₅₇	#2263
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #23757705	C₂	semidirect
2. C₂×D₂₅₁₂	C₂	semidirect
3. #23757112	C₂	semidirect
4. D₅₀₂₄	C₂	semidirect
5. M₆₄×C₁₅₇	C₂	semidirect
8. Dic₁₂₅₆	C₂²	semidirect
12. D₂₅₁₂	C₂²	semidirect
14. C₃₂×C₁₅₇	C₂²	semidirect
46. C₁₅₇	#2263	semidirect
52. M₆₄	D₁₅₇	semidirect
61. C₃₂	C₂×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
C₂×D₂₅₁₂	2
#23757112	2
D₅₀₂₄	2
M₆₄×C₁₅₇	2
#2263	157