Group #34081752 — subgroups

Subgroups — #34081752

Order 19328.

370 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 141 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	#23752941	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752564	C₂	1	3	yes	maximal, complemented
3	#23752494	C₂	1	3	yes	maximal, complemented
4	#23752417	C₂	1	3	yes	maximal
5	Q₈×C₁₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14558992	C₂	1	3	yes	maximal
7	D₄×Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#14558992	C₂²	1	3	yes	—
9	#18012739	C₂²	1	3	yes	—
10	#18012730	—	2	3	no	—
11	Q₈×Dic₁₅₁	—	2	1	no	—
12	C₄×Dic₃₀₂	C₂²	1	3	yes	—
13	#18012670	C₂²	1	3	yes	—
14	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	3	yes	—
15	#18012662	C₂²	1	6	yes	—
16	#18012662	C₂²	1	3	yes	—
17	#18012660	C₂²	1	1	yes	characteristic
18	C₄×C₁₅₁⋊C₈	C₂²	1	3	yes	—
19	C₂²×Q₈×C₁₅₁	—	2	1	no	—
20	C₄⋊Q₈×C₁₅₁	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₁₅₁	—	2	3	no	—
22	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₅₁	C₂²	1	3	yes	—
24	C₂×Dic₃₀₂	C₂³	1	3	yes	—
25	#14558992	—	2	6	no	—
26	C₂×C₁₅₁⋊C₈	C₂³	1	3	yes	—
27	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₁₅₁	—	2	3	no	—
29	C₄⋊Dic₁₅₁	—	2	3	no	—
30	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
31	Dic₁₅₁⋊C₄	—	2	3	no	—
32	C₄×Dic₁₅₁	—	2	3	no	—
33	C₂×Q₈×C₁₅₁	—	2	3	no	—
34	C₂×Q₈×C₁₅₁	—	4	4	no	—
35	C₂×Q₈×C₁₅₁	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₅₁	—	2	3	no	—
38	C₄⋊C₄×C₁₅₁	—	2	3	no	—
39	C₄⋊C₄×C₁₅₁	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₅₁	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₁₅₁	—	2	3	no	—
42	C₄²×C₁₅₁	C₂³	1	3	yes	—
43	Dic₃₀₂	—	2	3	no	—
44	Dic₃₀₂	—	2	3	no	—
45	C₂×Dic₁₅₁	—	2	1	no	—
46	C₂×Dic₁₅₁	—	2	3	no	—
47	C₁₅₁⋊C₈	—	2	3	no	—
48	C₁₅₁⋊C₈	—	2	1	no	—
49	C₂³×C₁₅₁	—	2	1	no	—
50	Q₈×C₁₅₁	—	4	6	no	—
51	Q₈×C₁₅₁	SD₁₆	1	4	yes	complemented
52	D₄×C₁₅₁	—	2	1	no	—
53	D₄×C₁₅₁	#41	1	2	yes	—
54	C₂×C₄×C₁₅₁	—	2	3	no	—
55	C₂×C₄×C₁₅₁	—	4	6	no	—
56	C₂×C₄×C₁₅₁	C₂×D₄	1	3	yes	—
57	C₂×C₄×C₁₅₁	C₂×D₄	1	3	yes	—
58	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
59	C₂²×C₁₅₁	—	2	2	no	—
60	C₂²×C₁₅₁	—	4	1	no	—
61	C₂²×C₁₅₁	—	4	1	no	—
62	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₅₁	—	2	3	no	—
64	C₄×C₁₅₁	—	4	3	no	—
65	C₄×C₁₅₁	C₂×SD₁₆	1	6	yes	—
66	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
67	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₁₅₁	—	4	3	no	—
69	Dic₁₅₁	—	4	1	no	—
70	C₂×C₁₅₁	—	4	2	no	—
71	C₂×C₁₅₁	#548	1	1	yes	characteristic
72	C₂×C₁₅₁	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₁₅₁	#578	1	1	yes	characteristic
74	C₁₅₁	#3359	1	1	yes	characteristic, complemented, Sylow
75	#3359	—	151	1	no	maximal, Sylow
76	#557	—	151	1	no	—
77	D₄×Q₈	D₁₅₁	1	1	yes	characteristic
78	#487	—	151	3	no	—
79	#475	—	151	3	no	—
80	#461	—	151	3	no	—
81	C₈×Q₈	—	151	1	no	—
82	C₄×SD₁₆	—	151	3	no	—
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	—	151	3	no	—
85	C₄⋊Q₈	—	151	3	no	—
86	C₄⋊Q₈	C₂×D₁₅₁	1	3	yes	—
87	#125	—	302	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	302	1	no	—
90	C₄×Q₈	—	151	3	no	—
91	C₄×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
92	C₄×D₄	C₂×D₁₅₁	1	3	yes	—
93	#106	—	151	3	no	—
94	C₄⋊C₈	—	151	3	no	—
95	#103	—	151	3	no	—
96	#103	—	151	6	no	—
97	#102	—	151	1	no	—
98	C₄×C₈	—	151	3	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	151	3	no	—
102	C₂×Q₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
103	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	302	6	no	—
106	C₂×C₈	—	151	1	no	—
107	C₂×C₈	—	151	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	302	3	no	—
110	C₄⋊C₄	—	302	3	no	—
111	C₄⋊C₄	—	302	3	no	—
112	C₄⋊C₄	—	151	1	no	—
113	C₄⋊C₄	C₂²⋊D₁₅₁	1	3	yes	—
114	C₄⋊C₄	C₂²×D₁₅₁	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	302	3	no	—
117	C₄²	C₂²×D₁₅₁	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	4	6	no	—
120	Q₈	—	302	3	no	—
121	Q₈	—	302	3	no	—
122	Q₈	#14558992	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#14559005	1	2	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	302	1	no	—
128	C₂×C₄	—	302	3	no	—
129	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	3	yes	—
131	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
132	C₈	—	302	3	no	—
133	C₈	—	302	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	604	1	no	—
141	C₄	—	604	3	no	—
142	C₄	C₂×#14558992	1	6	yes	—
143	C₄	C₂×#14559005	1	1	yes	characteristic
144	C₄	#18012787	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#23752873	1	1	yes	characteristic
147	C₂	C₂²×#14558992	1	1	yes	characteristic
148	C₂	#23753027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752941

C₂

yes

maximal, characteristic, complemented

#23752564

C₂

yes

maximal, complemented

#23752494

C₂

yes

maximal, complemented

#23752417

C₂

yes

maximal

Q₈×C₁₅₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#14558992

C₂

yes

maximal

D₄×Q₈×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558992

C₂²

yes

—

#18012739

C₂²

yes

—

#18012730

—

Q₈×Dic₁₅₁

—

C₄×Dic₃₀₂

C₂²

yes

—

#18012670

C₂²

yes

—

C₄⋊(C₁₅₁⋊C₈)

C₂²

yes

—

#18012662

C₂²

yes

—

#18012662

C₂²

yes

—

#18012660

C₂²

yes

characteristic

C₄×C₁₅₁⋊C₈

C₂²

yes

—

C₂²×Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

C₂²

yes

—

C₂²⋊Q₈×C₁₅₁

—

C₄×Q₈×C₁₅₁

C₂²

yes

characteristic

C₄×D₄×C₁₅₁

C₂²

yes

—

C₂×Dic₃₀₂

C₂³

yes

—

#14558992

—

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₂×C₁₅₁⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂³

yes

characteristic

Dic₁₅₁⋊C₄

—

C₄×Dic₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

D₄

yes

characteristic

C₂×D₄×C₁₅₁

C₂³

yes

characteristic

C₂²×C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

—

Dic₃₀₂

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

SD₁₆

yes

complemented

D₄×C₁₅₁

—

D₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×SD₁₆

yes

—

C₄×C₁₅₁

C₂×#41

yes

characteristic

C₄×C₁₅₁

2₋¹⁺⁴

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#548

yes

characteristic

C₂×C₁₅₁

C₂²×SD₁₆

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#3359

—

151

maximal, Sylow

#557

—

151

—

D₄×Q₈

D₁₅₁

yes

characteristic

#487

—

151

—

#475

—

151

—

#461

—

151

—

C₈×Q₈

—

151

—

C₄×SD₁₆

—

151

—

C₂²×Q₈

—

C₂×SD₁₆

—

151

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

—

#125

—

302

—

C₂²⋊Q₈

—

C₄×Q₈

—

302

—

C₄×Q₈

—

151

—

C₄×Q₈

C₂×D₁₅₁

yes

characteristic

C₄×D₄

C₂×D₁₅₁

yes

—

#106

—

151

—

C₄⋊C₈

—

151

—

#103

—

151

—

#103

—

151

—

#102

—

151

—

C₄×C₈

—

151

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

151

—

102

C₂×Q₈

C₂²⋊D₁₅₁

yes

characteristic

103

C₂×D₄

C₂²×D₁₅₁

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

302

—

106

C₂×C₈

—

151

—

107

C₂×C₈

—

151

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

302

—

110

C₄⋊C₄

—

302

—

111

C₄⋊C₄

—

302

—

112

C₄⋊C₄

—

151

—

113

C₄⋊C₄

C₂²⋊D₁₅₁

yes

—

114

C₄⋊C₄

C₂²×D₁₅₁

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

302

—

117

C₄²

C₂²×D₁₅₁

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

302

—

121

Q₈

—

302

—

122

Q₈

#14558992

yes

complemented

123

D₄

—

124

D₄

#14559005

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

302

—

128

C₂×C₄

—

302

—

129

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

131

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

132

C₈

—

302

—

133

C₈

—

302

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₂²⋊D₁₅₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

604

—

141

C₄

—

604

—

142

C₄

C₂×#14558992

yes

—

143

C₄

C₂×#14559005

yes

characteristic

144

C₄

#18012787

yes

characteristic

145

C₂

—

146

C₂

#23752873

yes

characteristic

147

C₂

C₂²×#14558992

yes

characteristic

148

C₂

#23753027

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	D₄×Q₈×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂×D₄×C₁₅₁	C₂³
hypercenter	D₄×Q₈	D₁₅₁
hypocenter	C₁₅₁	#3359
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

D₄×Q₈×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂²×D₄

prime radical

C₂×D₄×C₁₅₁

C₂³

hypercenter

D₄×Q₈

D₁₅₁

hypocenter

C₁₅₁

#3359

LCS

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #23752941	C₂	semidirect
2. #23752564	C₂	semidirect
3. #23752494	C₂	semidirect
5. Q₈×C₁₅₁⋊C₈	C₂	semidirect
51. Q₈×C₁₅₁	SD₁₆	semidirect
74. C₁₅₁	#3359	semidirect
122. Q₈	#14558992	semidirect

H ≅ G/N

type

1. #23752941

C₂

semidirect

2. #23752564

C₂

semidirect

3. #23752494

C₂

semidirect

5. Q₈×C₁₅₁⋊C₈

C₂

semidirect

51. Q₈×C₁₅₁

SD₁₆

semidirect

74. C₁₅₁

#3359

semidirect

122. Q₈

#14558992

semidirect

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

subgroup	index
#23752941	2
#23752564	2
#23752494	2
#23752417	2
Q₈×C₁₅₁⋊C₈	2
C₄×#14558992	2
D₄×Q₈×C₁₅₁	2
#3359	151

subgroup

index

Q₈×C₁₅₁⋊C₈

D₄×Q₈×C₁₅₁

151

Subgroups — #34081752

Conjugacy classes of subgroups

Complements

Maximal subgroups