Subgroups — #34098190

Order 20096.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757678	C₂	1	1	yes	maximal, characteristic
2	#23757026	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097771	—	2	1	no	—
5	#19097705	—	2	1	no	—
6	#19097705	C₄	1	2	yes	—
7	C₄⋊#11929434	—	2	1	no	—
8	C₂×C₁₆×C₁₅₇	—	2	1	no	—
9	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₁₄	—	4	1	no	—
11	C₂×#11929434	—	4	1	no	—
12	C₄⋊Dic₁₅₇	—	2	2	no	—
13	C₄⋊Dic₁₅₇	—	4	1	no	—
14	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
16	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₅₇	—	4	1	no	—
18	Dic₃₁₄	—	8	1	no	—
19	Dic₃₁₄	—	8	1	no	—
20	C₂×Dic₁₅₇	—	4	2	no	—
21	#11929434	—	8	1	no	—
22	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₅₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₅₇	SD₁₆	1	1	yes	characteristic
25	C₈×C₁₅₇	—	2	1	no	—
26	C₈×C₁₅₇	SD₁₆	1	1	yes	characteristic
27	C₈×C₁₅₇	Q₁₆	1	1	yes	characteristic
28	C₂²×C₁₅₇	#102	1	1	yes	characteristic
29	C₄×C₁₅₇	—	2	1	no	—
30	C₄×C₁₅₇	#101	1	1	yes	characteristic
31	C₄×C₁₅₇	#103	1	1	yes	characteristic
32	C₄×C₁₅₇	D₁₆	1	1	yes	characteristic
33	C₄×C₁₅₇	SD₃₂	1	1	yes	characteristic
34	Dic₁₅₇	—	8	2	no	—
35	C₂×C₁₅₇	#332	1	1	yes	characteristic
36	C₂×C₁₅₇	#357	1	1	yes	characteristic
37	C₂×C₁₅₇	#362	1	1	yes	characteristic
38	C₁₅₇	#1364	1	1	yes	characteristic, complemented, Sylow
39	#1364	—	157	1	no	maximal, Sylow
40	#500	—	157	1	no	—
41	C₄⋊C₁₆	D₁₅₇	1	1	yes	characteristic
42	#335	—	157	1	no	—
43	C₄⋊Q₈	—	314	1	no	—
44	C₂×C₁₆	—	2	1	no	—
45	#107	—	314	1	no	—
46	#107	—	157	2	no	—
47	C₄⋊C₈	—	314	1	no	—
48	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
49	C₂×Q₈	—	628	1	no	—
50	C₂×C₈	—	628	1	no	—
51	C₂×C₈	C₄×D₁₅₇	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
53	C₄⋊C₄	—	314	2	no	—
54	C₄⋊C₄	—	628	1	no	—
55	C₄²	D₆₂₈	1	1	yes	characteristic
56	C₁₆	—	4	1	no	—
57	Q₈	—	1256	1	no	—
58	Q₈	—	1256	1	no	—
59	C₂×C₄	—	628	2	no	—
60	C₂×C₄	#14578931	1	1	yes	characteristic
61	C₂×C₄	D₁₂₅₆	1	1	yes	characteristic
62	C₂×C₄	#14578943	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	1256	1	no	—
65	C₈	#14578942	1	1	yes	characteristic
66	C₈	#14578944	1	1	yes	characteristic
67	C₂²	#19097693	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	1256	2	no	—
70	C₄	#19097689	1	1	yes	characteristic
71	C₄	#19097694	1	1	yes	characteristic
72	C₄	D₂₅₁₂	1	1	yes	characteristic
73	C₄	#19097716	1	1	yes	characteristic
74	C₂	#23757018	1	1	yes	characteristic
75	C₂	#23757075	1	1	yes	characteristic
76	C₂	#23757090	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19097693
commutator	C₂×C₈×C₁₅₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₅₇
Fitting	C₄⋊C₁₆×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#102
cosocle	C₄×C₈×C₁₅₇	C₂²
hypercenter	C₄⋊C₁₆	D₁₅₇
hypocenter	C₁₅₇	#1364
LCS	G ▷ C₂×C₈×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
38. C₁₅₇	#1364	semidirect

Maximal subgroups

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

subgroup	index
#23757678	2
#23757026	2
C₄⋊C₁₆×C₁₅₇	2
#1364	157