Subgroups — #34100274

Order 20096.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757297	C₂	1	2	yes	maximal
2	#23757038	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578938	C₂²	1	4	yes	—
5	C₄⋊#11929434	—	2	4	no	—
6	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₅₇	C₄	1	2	yes	—
9	C₄×C₈×C₁₅₇	C₄	1	2	yes	—
10	#14578938	—	2	8	no	—
11	C₂×#11929434	—	2	4	no	—
12	C₂²×C₄×C₁₅₇	Q₈	1	2	yes	—
13	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
14	C₂×C₈×C₁₅₇	—	2	4	no	—
15	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
16	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
17	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₁₅₇	D₄	1	2	yes	—
19	#11929434	—	4	4	no	—
20	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₄×C₁₅₇	—	2	4	no	—
22	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₅₇	#41	1	4	yes	—
27	C₈×C₁₅₇	—	2	4	no	—
28	C₂²×C₁₅₇	—	2	2	no	—
29	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₁₅₇	#125	1	2	yes	—
31	C₄×C₁₅₇	—	2	2	no	—
32	C₄×C₁₅₇	#108	1	4	yes	—
33	C₄×C₁₅₇	#117	1	2	yes	—
34	C₄×C₁₅₇	#124	1	2	yes	—
35	C₂×C₁₅₇	—	2	2	no	—
36	C₂×C₁₅₇	#382	1	1	yes	characteristic
37	C₂×C₁₅₇	C₂×#108	1	2	yes	—
38	C₁₅₇	#1837	1	1	yes	characteristic, complemented, Sylow
39	#1837	—	157	1	no	maximal, Sylow
40	#423	—	157	2	no	—
41	C₂×C₄×C₈	D₁₅₇	1	1	yes	characteristic
42	#341	—	157	4	no	—
43	C₂×M₁₆	—	157	4	no	—
44	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
45	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
46	C₄⋊C₈	—	314	4	no	—
47	C₄×C₈	Dic₁₅₇	1	2	yes	—
48	C₄×C₈	Dic₁₅₇	1	2	yes	—
49	C₂²×C₄	Dic₃₁₄	1	2	yes	—
50	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
51	M₁₆	—	314	8	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	314	4	no	—
54	C₂×C₈	C₂×Dic₁₅₇	1	4	yes	—
55	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
56	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
57	C₄²	D₆₂₈	1	2	yes	—
58	C₂³	C₂×Dic₃₁₄	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	C₄⋊Dic₁₅₇	1	2	yes	—
61	C₂×C₄	C₄⋊Dic₁₅₇	1	2	yes	—
62	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
63	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
64	C₂×C₄	#14578954	1	4	yes	—
65	C₈	—	2	4	no	—
66	C₈	—	628	4	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
69	C₂²	#19097763	1	2	yes	—
70	C₄	—	2	2	no	—
71	C₄	#19097707	1	4	yes	—
72	C₄	#19097730	1	2	yes	—
73	C₄	#19097759	1	2	yes	—
74	C₂	—	2	2	no	—
75	C₂	#23757139	1	1	yes	characteristic
76	C₂	C₂×#19097707	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₂₈
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×C₄×C₈×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₅₇
hypocenter	C₁₅₇	#1837
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
38. C₁₅₇	#1837	semidirect

Maximal subgroups

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

subgroup	index
#23757297	2
#23757038	2
C₂×C₄×C₈×C₁₅₇	2
#1837	157