Group #34100182 — subgroups

Subgroups — #34100182

Order 20096.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	#23757311	C₂	1	2	yes	maximal, complemented
2	#23757247	C₂	1	2	yes	maximal
3	#23757030	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	#19097730	C₂²	1	4	yes	—
6	#19097705	—	2	2	no	—
7	#19097703	—	2	2	no	—
8	#19097686	—	2	4	no	—
9	C₂×#41×C₁₅₇	C₂²	1	1	yes	characteristic
10	#131×C₁₅₇	—	2	2	no	—
11	#131×C₁₅₇	—	2	2	no	—
12	C₂×M₁₆×C₁₅₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₅₇	C₄	1	2	yes	complemented
15	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₅₇	—	2	4	no	—
17	C₂²⋊Dic₁₅₇	—	2	4	no	—
18	C₄×Dic₁₅₇	—	2	4	no	—
19	#41×C₁₅₇	—	2	2	no	—
20	#41×C₁₅₇	—	2	2	no	—
21	C₂×Q₈×C₁₅₇	Q₈	1	1	yes	characteristic
22	C₂×D₄×C₁₅₇	D₄	1	2	yes	—
23	C₂×D₄×C₁₅₇	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₅₇	—	2	2	no	—
27	M₁₆×C₁₅₇	—	2	4	no	—
28	C₂×C₈×C₁₅₇	—	2	2	no	—
29	C₂×C₈×C₁₅₇	—	2	2	no	—
30	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
34	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
35	C₂×Dic₁₅₇	—	4	4	no	—
36	C₂³×C₁₅₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₅₇	—	2	2	no	—
39	D₄×C₁₅₇	—	2	4	no	—
40	D₄×C₁₅₇	—	2	2	no	—
41	C₂×C₄×C₁₅₇	—	2	2	no	—
42	C₂×C₄×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	#41	1	2	yes	—
48	C₈×C₁₅₇	—	2	2	no	—
49	C₈×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₈×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₈×C₁₅₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₁₅₇	—	2	2	no	—
53	C₂²×C₁₅₇	—	4	1	no	—
54	C₂²×C₁₅₇	—	4	2	no	—
55	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₅₇	#121	1	2	yes	—
57	C₄×C₁₅₇	—	2	2	no	—
58	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₁₅₇	#117	1	1	yes	characteristic
60	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
61	Dic₁₅₇	—	8	4	no	—
62	C₂×C₁₅₇	—	2	2	no	—
63	C₂×C₁₅₇	—	2	1	no	—
64	C₂×C₁₅₇	—	4	2	no	—
65	C₂×C₁₅₇	#380	1	1	yes	characteristic
66	C₁₅₇	#1812	1	1	yes	characteristic, complemented, Sylow
67	#1812	—	157	1	no	maximal, Sylow
68	C₂×#131	D₁₅₇	1	1	yes	characteristic
69	#427	—	157	2	no	—
70	#410	—	157	2	no	—
71	H(3,ℤ/4)	—	157	2	no	—
72	C₂×#41	C₂×D₁₅₇	1	1	yes	characteristic
73	#131	—	2	2	no	—
74	#131	—	2	2	no	—
75	C₂×M₁₆	Dic₁₅₇	1	2	yes	complemented
76	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
77	C₂²×C₈	Dic₁₅₇	1	2	yes	complemented
78	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
79	#117	—	157	4	no	—
80	#107	—	314	2	no	—
81	#106	—	314	2	no	—
82	#99	—	314	4	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	Dic₃₁₄	1	1	yes	characteristic
86	C₂×D₄	D₆₂₈	1	2	yes	—
87	C₂×D₄	Dic₃₁₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂×Dic₁₅₇	1	2	yes	—
95	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
98	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
99	C₄⋊C₄	—	314	4	no	—
100	C₂²⋊C₄	—	314	4	no	—
101	C₄²	—	314	4	no	—
102	C₂³	C₄⋊Dic₁₅₇	1	2	yes	—
103	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	628	4	no	—
110	C₂×C₄	C₄⋊Dic₁₅₇	1	2	yes	—
111	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
113	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
114	C₂×C₄	#14578954	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
117	C₈	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
118	C₈	C₂²⋊Dic₁₅₇	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
123	C₂²	#19097745	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	1256	4	no	—
126	C₄	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
127	C₄	#19097730	1	1	yes	characteristic
128	C₄	#19097750	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	#23757133	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757311

C₂

yes

maximal, complemented

#23757247

C₂

yes

maximal

#23757030

C₂

yes

maximal, complemented

C₂×#131×C₁₅₇

C₂

yes

maximal, characteristic

#19097730

C₂²

yes

—

#19097705

—

#19097703

—

#19097686

—

C₂×#41×C₁₅₇

C₂²

yes

characteristic

#131×C₁₅₇

—

#131×C₁₅₇

—

C₂×M₁₆×C₁₅₇

C₄

yes

complemented

C₂×M₁₆×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

C₄

yes

complemented

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊Dic₁₅₇

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

#41×C₁₅₇

—

#41×C₁₅₇

—

C₂×Q₈×C₁₅₇

Q₈

yes

characteristic

C₂×D₄×C₁₅₇

D₄

yes

—

C₂×D₄×C₁₅₇

Q₈

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

D₄

yes

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₄⋊C₄

yes

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₈×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₈×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₈×C₁₅₇

C₂²⋊C₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#121

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₅₇

#117

yes

characteristic

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#380

yes

characteristic

C₁₅₇

#1812

yes

characteristic, complemented, Sylow

#1812

—

157

maximal, Sylow

C₂×#131

D₁₅₇

yes

characteristic

#427

—

157

—

#410

—

157

—

H(3,ℤ/4)

—

157

—

C₂×#41

C₂×D₁₅₇

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₁₅₇

yes

complemented

C₂×M₁₆

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

Dic₁₅₇

yes

complemented

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#117

—

157

—

#107

—

314

—

#106

—

314

—

#99

—

314

—

#41

—

#41

—

C₂×Q₈

Dic₃₁₄

yes

characteristic

C₂×D₄

D₆₂₈

yes

—

C₂×D₄

Dic₃₁₄

yes

characteristic

C₂²×C₄

C₂×Dic₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₅₇

yes

—

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₅₇

yes

—

C₂×C₈

C₂²⋊D₁₅₇

yes

—

C₄⋊C₄

—

314

—

100

C₂²⋊C₄

—

314

—

101

C₄²

—

314

—

102

C₂³

C₄⋊Dic₁₅₇

yes

—

103

C₂³

C₂×D₆₂₈

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

628

—

110

C₂×C₄

C₄⋊Dic₁₅₇

yes

—

111

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

113

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

114

C₂×C₄

#14578954

yes

—

115

C₈

—

116

C₈

C₂²⋊Dic₁₅₇

yes

characteristic

117

C₈

C₂²⋊Dic₁₅₇

yes

characteristic

118

C₈

C₂²⋊Dic₁₅₇

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄⋊Dic₁₅₇

yes

characteristic

123

C₂²

#19097745

yes

—

124

C₄

—

125

C₄

—

1256

—

126

C₄

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

127

C₄

#19097730

yes

characteristic

128

C₄

#19097750

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#23757133

yes

characteristic

	subgroup	G/H
center	C₄	#19097730
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×#131×C₁₅₇	C₂
socle	C₂×C₁₅₇	#380
prime radical	C₂×#41×C₁₅₇	C₂²
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×#131	D₁₅₇
hypocenter	C₁₅₇	#1812
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#19097730

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×#131×C₁₅₇

C₂

socle

C₂×C₁₅₇

#380

prime radical

C₂×#41×C₁₅₇

C₂²

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. #23757311	C₂	semidirect
3. #23757030	C₂	semidirect
12. C₂×M₁₆×C₁₅₇	C₄	semidirect
14. C₂²×C₈×C₁₅₇	C₄	semidirect
66. C₁₅₇	#1812	semidirect
75. C₂×M₁₆	Dic₁₅₇	semidirect
77. C₂²×C₈	Dic₁₅₇	semidirect

H ≅ G/N

type

1. #23757311

C₂

semidirect

3. #23757030

C₂

semidirect

12. C₂×M₁₆×C₁₅₇

C₄

semidirect

14. C₂²×C₈×C₁₅₇

C₄

semidirect

66. C₁₅₇

#1812

semidirect

75. C₂×M₁₆

Dic₁₅₇

semidirect

77. C₂²×C₈

Dic₁₅₇

semidirect

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

subgroup	index
#23757311	2
#23757247	2
#23757030	2
C₂×#131×C₁₅₇	2
#1812	157

subgroup

index

C₂×#131×C₁₅₇

157

Subgroups — #34100182

Conjugacy classes of subgroups

Complements

Maximal subgroups