Subgroups — #34098172

Order 20096.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757650	C₂	1	1	yes	maximal, characteristic
2	#23757094	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₂₅₆	—	2	1	no	—
5	C₂×D₁₂₅₆	—	2	1	no	—
6	#19097768	C₄	1	2	yes	—
7	#19097711	—	2	1	no	—
8	M₃₂×C₁₅₇	—	2	1	no	—
9	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₂×D₆₂₈	—	2	2	no	—
11	C₂×D₆₂₈	—	4	2	no	—
12	D₁₂₅₆	—	4	1	no	—
13	D₁₂₅₆	—	4	1	no	—
14	D₁₂₅₆	—	4	1	no	—
15	D₁₂₅₆	—	4	1	no	—
16	#14578929	—	4	1	no	—
17	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
19	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₅₇	—	4	1	no	—
21	C₂²×D₁₅₇	—	4	2	no	—
22	D₆₂₈	—	4	2	no	—
23	D₆₂₈	—	4	2	no	—
24	D₆₂₈	—	8	4	no	—
25	C₂×C₄×C₁₅₇	—	2	1	no	—
26	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₈×C₁₅₇	D₈	1	1	yes	characteristic
28	C₈×C₁₅₇	D₈	1	1	yes	characteristic
29	C₈×C₁₅₇	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₅₇	SD₁₆	1	1	yes	characteristic
31	C₂²×C₁₅₇	#100	1	1	yes	characteristic
32	C₂×D₁₅₇	—	8	4	no	—
33	C₂×D₁₅₇	—	8	2	no	—
34	C₄×C₁₅₇	—	2	2	no	—
35	C₄×C₁₅₇	#102	1	1	yes	characteristic
36	C₄×C₁₅₇	#102	1	1	yes	characteristic
37	C₂×C₁₅₇	—	2	1	no	—
38	C₂×C₁₅₇	#331	1	1	yes	characteristic
39	D₁₅₇	—	16	2	no	—
40	C₁₅₇	#1357	1	1	yes	characteristic, complemented, Sylow
41	#1357	—	157	1	no	maximal, Sylow
42	#493	—	157	1	no	—
43	#364	—	157	1	no	—
44	#364	D₁₅₇	1	1	yes	characteristic, complemented
45	C₂×D₈	—	314	1	no	—
46	C₂×D₈	—	314	1	no	—
47	#127	—	157	2	no	—
48	M₃₂	—	2	1	no	—
49	M₃₂	—	314	1	no	—
50	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
51	C₂×D₄	—	314	2	no	—
52	C₂×D₄	—	628	2	no	—
53	D₈	—	628	1	no	—
54	D₈	—	628	1	no	—
55	D₈	—	628	1	no	—
56	D₈	—	628	1	no	—
57	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
58	C₂×C₈	D₆₂₈	1	1	yes	characteristic
59	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
60	C₁₆	—	4	1	no	—
61	C₁₆	—	628	1	no	—
62	C₂³	—	628	2	no	—
63	D₄	—	628	2	no	—
64	D₄	—	628	2	no	—
65	D₄	—	1256	4	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	#14578931	1	1	yes	characteristic
68	C₈	#14578939	1	1	yes	characteristic
69	C₈	D₁₂₅₆	1	1	yes	characteristic
70	C₈	#14578941	1	1	yes	characteristic
71	C₈	#14578943	1	1	yes	characteristic
72	C₂²	—	1256	4	no	—
73	C₂²	—	1256	2	no	—
74	C₂²	#19097687	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	#19097691	1	1	yes	characteristic
77	C₄	#19097693	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	2512	2	no	—
80	C₂	#23757016	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23757016
commutator	C₄²×C₁₅₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₅₇
Fitting	#364×C₁₅₇	C₂
socle	C₂×C₁₅₇	#331
prime radical	#23757650	C₂
cosocle	C₄×C₈×C₁₅₇	C₂²
hypercenter	#364	D₁₅₇
hypocenter	C₁₅₇	#1357
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #23757094	C₂	semidirect
3. #364×C₁₅₇	C₂	semidirect
40. C₁₅₇	#1357	semidirect
44. #364	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757650	2
#23757094	2
#364×C₁₅₇	2
#1357	157