Group #34103322 — subgroups

Subgroups — #34103322

Order 20096.

408 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757172	C₂	1	4	yes	maximal
2	#23757159	C₂	1	4	yes	maximal
3	#23757159	C₂	1	4	yes	maximal
4	#23757139	C₂	1	2	yes	maximal
5	C₂×#125×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
9	C₂×C₄×Dic₁₅₇	C₂²	1	4	yes	—
10	#19097678	C₂²	1	4	yes	—
11	#19097678	C₂²	1	4	yes	—
12	#19097678	C₂²	1	4	yes	—
13	#19097678	C₂²	1	4	yes	—
14	#125×C₁₅₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₅₇	C₂³	1	4	yes	—
19	C₂²×Dic₁₅₇	C₂³	1	4	yes	—
20	C₄⋊Dic₁₅₇	—	4	4	no	—
21	C₄⋊Dic₁₅₇	—	4	2	no	—
22	C₄⋊Dic₁₅₇	—	4	2	no	—
23	C₄×Dic₁₅₇	—	4	4	no	—
24	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₁₅₇	C₂³	1	4	yes	—
27	C₄⋊C₄×C₁₅₇	C₂×C₄	1	8	yes	—
28	C₄⋊C₄×C₁₅₇	C₂×C₄	1	16	yes	—
29	C₄²×C₁₅₇	C₂×C₄	1	4	yes	—
30	C₂×Dic₁₅₇	—	2	4	no	—
31	C₂×Dic₁₅₇	—	2	4	no	—
32	C₂×Dic₁₅₇	—	2	4	no	—
33	C₂×Dic₁₅₇	—	2	4	no	—
34	C₂×Dic₁₅₇	—	2	4	no	—
35	C₂×Dic₁₅₇	—	2	4	no	—
36	C₂³×C₁₅₇	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	—	2	4	no	—
38	C₂×C₄×C₁₅₇	—	2	2	no	—
39	C₂×C₄×C₁₅₇	—	2	2	no	—
40	C₂×C₄×C₁₅₇	—	2	4	no	—
41	C₂×C₄×C₁₅₇	—	2	1	no	—
42	C₂×C₄×C₁₅₇	—	2	1	no	—
43	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₅₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₁₅₇	C₂²×C₄	1	8	yes	—
46	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
47	C₂²×C₁₅₇	H(5,𝔽₂)	1	2	yes	—
48	C₂²×C₁₅₇	2₋¹⁺⁴	1	2	yes	—
49	C₂²×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
50	C₂²×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
51	C₄×C₁₅₇	—	2	2	no	—
52	C₄×C₁₅₇	—	2	8	no	—
53	C₄×C₁₅₇	—	2	4	no	—
54	Dic₁₅₇	—	4	4	no	—
55	Dic₁₅₇	—	4	4	no	—
56	C₂×C₁₅₇	#519	1	1	yes	characteristic
57	C₂×C₁₅₇	#520	1	2	yes	—
58	C₂×C₁₅₇	#562	1	1	yes	characteristic
59	C₂×C₁₅₇	#563	1	2	yes	—
60	C₂×C₁₅₇	#564	1	1	yes	characteristic
61	C₁₅₇	#2382	1	1	yes	characteristic, complemented, Sylow
62	#2382	—	157	1	no	maximal, Sylow
63	C₂×#125	D₁₅₇	1	1	yes	characteristic
64	#389	—	157	4	no	—
65	#387	—	157	4	no	—
66	#387	—	157	4	no	—
67	#382	—	157	2	no	—
68	#125	Dic₁₅₇	1	8	yes	complemented
69	C₂×C₄⋊C₄	—	157	2	no	—
70	C₂×C₄⋊C₄	—	157	2	no	—
71	C₂×C₄⋊C₄	—	157	4	no	—
72	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
73	C₂×C₄⋊C₄	C₂×D₁₅₇	1	4	yes	—
74	C₂×C₄²	—	157	4	no	—
75	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
76	#95	—	157	4	no	—
77	#95	—	157	4	no	—
78	#95	—	157	4	no	—
79	#95	—	157	4	no	—
80	C₂²×C₄	—	157	4	no	—
81	C₂²×C₄	—	157	4	no	—
82	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₁₅₇	1	4	yes	—
84	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
85	C₄⋊C₄	—	628	4	no	—
86	C₄⋊C₄	—	628	2	no	—
87	C₄⋊C₄	—	628	2	no	—
88	C₄⋊C₄	C₂×Dic₁₅₇	1	8	yes	—
89	C₄⋊C₄	C₂×Dic₁₅₇	1	16	yes	—
90	C₄²	—	628	4	no	—
91	C₄²	C₂×Dic₁₅₇	1	4	yes	—
92	C₂³	C₂³×D₁₅₇	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	314	4	no	—
100	C₂×C₄	—	314	4	no	—
101	C₂×C₄	—	314	4	no	—
102	C₂×C₄	—	314	4	no	—
103	C₂×C₄	—	314	4	no	—
104	C₂×C₄	—	314	4	no	—
105	C₂×C₄	C₂²×Dic₁₅₇	1	2	yes	—
106	C₂×C₄	C₂²×Dic₁₅₇	1	4	yes	—
107	C₂×C₄	C₂²×Dic₁₅₇	1	8	yes	—
108	C₂²	C₂³×Dic₁₅₇	1	1	yes	characteristic
109	C₂²	#19097817	1	2	yes	—
110	C₂²	#19097818	1	2	yes	—
111	C₂²	#19097819	1	1	yes	characteristic
112	C₂²	#19097819	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	8	no	—
115	C₄	—	2	4	no	—
116	C₄	—	628	4	no	—
117	C₄	—	628	4	no	—
118	C₂	#23757751	1	1	yes	characteristic
119	C₂	#23757756	1	2	yes	—
120	C₂	#23758021	1	1	yes	characteristic
121	C₂	#23758027	1	2	yes	—
122	C₂	C₁₅₇⋊#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757172

C₂

yes

maximal

#23757159

C₂

yes

maximal

#23757159

C₂

yes

maximal

#23757139

C₂

yes

maximal

C₂×#125×C₁₅₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

#19097678

C₂²

yes

—

#19097678

C₂²

yes

—

#19097678

C₂²

yes

—

#19097678

C₂²

yes

—

#125×C₁₅₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂³

yes

—

C₂²×Dic₁₅₇

C₂³

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂⁴

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂²×C₁₅₇

C₂³×C₄

yes

characteristic

C₂²×C₁₅₇

H(5,𝔽₂)

yes

—

C₂²×C₁₅₇

2₋¹⁺⁴

yes

—

C₂²×C₁₅₇

2₋¹⁺⁴

yes

characteristic

C₂²×C₁₅₇

2₋¹⁺⁴

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#519

yes

characteristic

C₂×C₁₅₇

#520

yes

—

C₂×C₁₅₇

#562

yes

characteristic

C₂×C₁₅₇

#563

yes

—

C₂×C₁₅₇

#564

yes

characteristic

C₁₅₇

#2382

yes

characteristic, complemented, Sylow

#2382

—

157

maximal, Sylow

C₂×#125

D₁₅₇

yes

characteristic

#389

—

157

—

#387

—

157

—

#387

—

157

—

#382

—

157

—

#125

Dic₁₅₇

yes

complemented

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#95

—

157

—

#95

—

157

—

#95

—

157

—

#95

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

—

C₄⋊C₄

—

628

—

C₄⋊C₄

—

628

—

C₄⋊C₄

—

628

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₄²

—

628

—

C₄²

C₂×Dic₁₅₇

yes

—

C₂³

C₂³×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

100

C₂×C₄

—

314

—

101

C₂×C₄

—

314

—

102

C₂×C₄

—

314

—

103

C₂×C₄

—

314

—

104

C₂×C₄

—

314

—

105

C₂×C₄

C₂²×Dic₁₅₇

yes

—

106

C₂×C₄

C₂²×Dic₁₅₇

yes

—

107

C₂×C₄

C₂²×Dic₁₅₇

yes

—

108

C₂²

C₂³×Dic₁₅₇

yes

characteristic

109

C₂²

#19097817

yes

—

110

C₂²

#19097818

yes

—

111

C₂²

#19097819

yes

characteristic

112

C₂²

#19097819

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

628

—

117

C₄

—

628

—

118

C₂

#23757751

yes

characteristic

119

C₂

#23757756

yes

—

120

C₂

#23758021

yes

characteristic

121

C₂

#23758027

yes

—

122

C₂

C₁₅₇⋊#564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅₇
commutator	C₂²×C₁₅₇	C₂³×C₄
Fitting	C₂×#125×C₁₅₇	C₂
socle, prime radical, cosocle	C₂³×C₁₅₇	C₂⁴
hypercenter	C₂×#125	D₁₅₇
hypocenter	C₁₅₇	#2382
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₅₇

commutator

C₂²×C₁₅₇

C₂³×C₄

Fitting

C₂×#125×C₁₅₇

C₂

socle, prime radical, cosocle

C₂³×C₁₅₇

C₂⁴

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
14. #125×C₁₅₇	C₄	semidirect
61. C₁₅₇	#2382	semidirect
68. #125	Dic₁₅₇	semidirect

H ≅ G/N

type

14. #125×C₁₅₇

C₄

semidirect

61. C₁₅₇

#2382

semidirect

68. #125

Dic₁₅₇

semidirect

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

subgroup	index
#23757172	2
#23757159	2
#23757139	2
C₂×#125×C₁₅₇	2
#2382	157

subgroup

index

C₂×#125×C₁₅₇

157

Subgroups — #34103322

Conjugacy classes of subgroups

Complements

Maximal subgroups