Group #34111935 — subgroups

Subgroups — #34111935

Order 20096.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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	#23758074	C₂	1	4	yes	maximal, complemented
2	#23757540	C₂	1	4	yes	maximal, complemented
3	#23757440	C₂	1	4	yes	maximal, complemented
4	#23757391	C₂	1	2	yes	maximal, complemented
5	#493×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097816	C₂²	1	4	yes	complemented
7	#19097797	C₂²	1	8	yes	complemented
8	#19097795	—	2	8	no	—
9	#19097793	C₂²	1	2	yes	—
10	#19097790	—	2	4	no	—
11	D₈×D₁₅₇	—	2	4	no	—
12	#19097777	C₂²	1	4	yes	complemented
13	#19097707	C₂²	1	2	yes	complemented
14	#19097699	C₂²	1	4	yes	complemented
15	#19097688	C₂²	1	4	yes	complemented
16	C₂×D₈×C₁₅₇	C₂²	1	4	yes	complemented
17	#127×C₁₅₇	C₂²	1	2	yes	complemented
18	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
19	#14578954	C₂³	1	4	yes	—
20	#14578953	—	2	8	no	—
21	C₂×C₂²⋊D₁₅₇	—	4	4	no	—
22	D₄×D₁₅₇	—	2	8	no	—
23	Dic₆₂₈	D₄	1	2	yes	complemented
24	#14578943	D₄	1	4	yes	complemented
25	#14578942	—	2	8	no	—
26	D₁₂₅₆	D₄	1	2	yes	complemented
27	#14578939	—	2	8	no	—
28	#14578938	C₂³	1	4	yes	—
29	#14578937	—	2	4	no	—
30	C₈×D₁₅₇	—	2	4	no	—
31	C₂×D₄×C₁₅₇	—	4	2	no	—
32	C₂×D₄×C₁₅₇	C₂³	1	4	yes	—
33	D₈×C₁₅₇	—	2	4	no	—
34	D₈×C₁₅₇	—	2	4	no	—
35	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
36	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
37	C₂²×D₁₅₇	—	4	4	no	—
38	Dic₃₁₄	C₂×D₄	1	4	yes	—
39	D₆₂₈	C₂×D₄	1	4	yes	—
40	C₂²⋊D₁₅₇	—	2	4	no	—
41	C₂²⋊D₁₅₇	—	4	8	no	—
42	C₂×Dic₁₅₇	—	4	4	no	—
43	C₄×D₁₅₇	—	2	4	no	—
44	#11929434	—	2	4	no	—
45	C₂³×C₁₅₇	—	2	4	no	—
46	D₄×C₁₅₇	—	2	8	no	—
47	D₄×C₁₅₇	—	8	4	no	—
48	C₂×C₄×C₁₅₇	—	2	1	no	—
49	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
50	C₈×C₁₅₇	C₂×D₄	1	2	yes	—
51	C₈×C₁₅₇	C₂×D₄	1	2	yes	—
52	C₂²×C₁₅₇	—	4	4	no	—
53	C₂²×C₁₅₇	—	8	4	no	—
54	C₂²×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₁₅₇	—	2	4	no	—
56	C₂×D₁₅₇	—	16	4	no	—
57	C₄×C₁₅₇	—	2	2	no	—
58	C₄×C₁₅₇	C₂²×D₄	1	2	yes	—
59	Dic₁₅₇	—	2	4	no	—
60	C₂×C₁₅₇	—	2	1	no	—
61	C₂×C₁₅₇	—	8	4	no	—
62	C₂×C₁₅₇	D₄²	1	1	yes	characteristic
63	D₁₅₇	—	4	4	no	—
64	C₁₅₇	#3292	1	1	yes	characteristic, complemented, Sylow
65	#3292	—	157	1	no	maximal, Sylow
66	#576	—	157	4	no	—
67	#493	D₁₅₇	1	1	yes	characteristic, complemented
68	#472	—	157	4	no	—
69	#453	—	157	4	no	—
70	#443	—	157	2	no	—
71	H(5,𝔽₂)	—	157	4	no	—
72	AGL(1,ℤ/8)	—	314	8	no	—
73	AGL(1,ℤ/8)	—	157	8	no	—
74	#135	—	314	4	no	—
75	#135	—	157	2	no	—
76	C₂×D₈	—	314	4	no	—
77	C₂×D₈	C₂×D₁₅₇	1	4	yes	complemented
78	#131	—	157	4	no	—
79	#127	C₂×D₁₅₇	1	2	yes	complemented
80	#108	—	157	2	no	—
81	#104	—	157	4	no	—
82	#100	—	157	4	no	—
83	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic, complemented
84	#41	—	314	8	no	—
85	#41	—	157	4	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	314	8	no	—
88	C₂×D₄	—	628	4	no	—
89	C₂×D₄	C₂²×D₁₅₇	1	4	yes	—
90	Q₁₆	—	157	2	no	—
91	SD₁₆	—	314	8	no	—
92	SD₁₆	—	157	4	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	314	8	no	—
96	D₈	—	157	2	no	—
97	M₁₆	—	314	4	no	—
98	M₁₆	—	157	4	no	—
99	C₂×C₈	—	314	4	no	—
100	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
101	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	628	4	no	—
104	Q₈	—	157	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	314	4	no	—
108	D₄	—	628	8	no	—
109	D₄	—	157	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	314	4	no	—
112	C₂×C₄	—	628	4	no	—
113	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
114	C₈	—	314	4	no	—
115	C₈	D₄×D₁₅₇	1	2	yes	—
116	C₈	D₄×D₁₅₇	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	314	4	no	—
120	C₂²	—	2512	4	no	—
121	C₂²	#19097816	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	314	4	no	—
124	C₄	C₂×D₄×D₁₅₇	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	628	4	no	—
128	C₂	#23757908	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23758074

C₂

yes

maximal, complemented

#23757540

C₂

yes

maximal, complemented

#23757440

C₂

yes

maximal, complemented

#23757391

C₂

yes

maximal, complemented

#493×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#19097816

C₂²

yes

complemented

#19097797

C₂²

yes

complemented

#19097795

—

#19097793

C₂²

yes

—

#19097790

—

D₈×D₁₅₇

—

#19097777

C₂²

yes

complemented

#19097707

C₂²

yes

complemented

#19097699

C₂²

yes

complemented

#19097688

C₂²

yes

complemented

C₂×D₈×C₁₅₇

C₂²

yes

complemented

#127×C₁₅₇

C₂²

yes

complemented

C₄×C₈×C₁₅₇

C₂²

yes

characteristic, complemented

#14578954

C₂³

yes

—

#14578953

—

C₂×C₂²⋊D₁₅₇

—

D₄×D₁₅₇

—

Dic₆₂₈

D₄

yes

complemented

#14578943

D₄

yes

complemented

#14578942

—

D₁₂₅₆

D₄

yes

complemented

#14578939

—

#14578938

C₂³

yes

—

#14578937

—

C₈×D₁₅₇

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

C₂³

yes

—

D₈×C₁₅₇

—

D₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₄²×C₁₅₇

C₂³

yes

characteristic

C₂²×D₁₅₇

—

Dic₃₁₄

C₂×D₄

yes

—

D₆₂₈

C₂×D₄

yes

—

C₂²⋊D₁₅₇

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

C₂³×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

C₂×D₄

yes

—

C₈×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

H(5,𝔽₂)

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂²×D₄

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

D₄²

yes

characteristic

D₁₅₇

—

C₁₅₇

#3292

yes

characteristic, complemented, Sylow

#3292

—

157

maximal, Sylow

#576

—

157

—

#493

D₁₅₇

yes

characteristic, complemented

#472

—

157

—

#453

—

157

—

#443

—

157

—

H(5,𝔽₂)

—

157

—

AGL(1,ℤ/8)

—

314

—

AGL(1,ℤ/8)

—

157

—

#135

—

314

—

#135

—

157

—

C₂×D₈

—

314

—

C₂×D₈

C₂×D₁₅₇

yes

complemented

#131

—

157

—

#127

C₂×D₁₅₇

yes

complemented

#108

—

157

—

#104

—

157

—

#100

—

157

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic, complemented

#41

—

314

—

#41

—

157

—

C₂×D₄

—

C₂×D₄

—

314

—

C₂×D₄

—

628

—

C₂×D₄

C₂²×D₁₅₇

yes

—

Q₁₆

—

157

—

SD₁₆

—

314

—

SD₁₆

—

157

—

D₈

—

D₈

—

D₈

—

314

—

D₈

—

157

—

M₁₆

—

314

—

M₁₆

—

157

—

C₂×C₈

—

314

—

100

C₂×C₈

C₂²×D₁₅₇

yes

—

101

C₄²

C₂²×D₁₅₇

yes

characteristic

102

C₂³

—

103

C₂³

—

628

—

104

Q₈

—

157

—

105

D₄

—

106

D₄

—

107

D₄

—

314

—

108

D₄

—

628

—

109

D₄

—

157

—

110

C₂×C₄

—

111

C₂×C₄

—

314

—

112

C₂×C₄

—

628

—

113

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

114

C₈

—

314

—

115

C₈

D₄×D₁₅₇

yes

—

116

C₈

D₄×D₁₅₇

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

314

—

120

C₂²

—

2512

—

121

C₂²

#19097816

yes

characteristic

122

C₄

—

123

C₄

—

314

—

124

C₄

C₂×D₄×D₁₅₇

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

628

—

128

C₂

#23757908

yes

characteristic

	subgroup	G/H
center	C₂	#23757908
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#493×C₁₅₇	C₂
socle	C₂×C₁₅₇	D₄²
hypercenter	#493	D₁₅₇
hypocenter	C₁₅₇	#3292
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#23757908

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

#493×C₁₅₇

C₂

socle

C₂×C₁₅₇

D₄²

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #23758074	C₂	semidirect
2. #23757540	C₂	semidirect
3. #23757440	C₂	semidirect
4. #23757391	C₂	semidirect
5. #493×C₁₅₇	C₂	semidirect
6. #19097816	C₂²	semidirect
7. #19097797	C₂²	semidirect
12. #19097777	C₂²	semidirect
13. #19097707	C₂²	semidirect
14. #19097699	C₂²	semidirect
15. #19097688	C₂²	semidirect
16. C₂×D₈×C₁₅₇	C₂²	semidirect
17. #127×C₁₅₇	C₂²	semidirect
18. C₄×C₈×C₁₅₇	C₂²	semidirect
23. Dic₆₂₈	D₄	semidirect
24. #14578943	D₄	semidirect
26. D₁₂₅₆	D₄	semidirect
64. C₁₅₇	#3292	semidirect
67. #493	D₁₅₇	semidirect
77. C₂×D₈	C₂×D₁₅₇	semidirect
79. #127	C₂×D₁₅₇	semidirect
83. C₄×C₈	C₂×D₁₅₇	semidirect

H ≅ G/N

type

1. #23758074

C₂

semidirect

2. #23757540

C₂

semidirect

3. #23757440

C₂

semidirect

4. #23757391

C₂

semidirect

5. #493×C₁₅₇

C₂

semidirect

6. #19097816

C₂²

semidirect

7. #19097797

C₂²

semidirect

12. #19097777

C₂²

semidirect

13. #19097707

C₂²

semidirect

14. #19097699

C₂²

semidirect

15. #19097688

C₂²

semidirect

16. C₂×D₈×C₁₅₇

C₂²

semidirect

17. #127×C₁₅₇

C₂²

semidirect

18. C₄×C₈×C₁₅₇

C₂²

semidirect

23. Dic₆₂₈

D₄

semidirect

24. #14578943

D₄

semidirect

26. D₁₂₅₆

D₄

semidirect

64. C₁₅₇

#3292

semidirect

67. #493

D₁₅₇

semidirect

77. C₂×D₈

C₂×D₁₅₇

semidirect

79. #127

C₂×D₁₅₇

semidirect

83. C₄×C₈

C₂×D₁₅₇

semidirect

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

subgroup	index
#23758074	2
#23757540	2
#23757440	2
#23757391	2
#493×C₁₅₇	2
#3292	157

subgroup

index

#493×C₁₅₇

157

Subgroups — #34111935

Conjugacy classes of subgroups

Complements

Maximal subgroups