Group #34070417 — subgroups

Subgroups — #34070417

Order 19328.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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	#23753012	C₂	1	1	yes	maximal, characteristic
2	#23752240	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752206	C₂	1	2	yes	maximal, complemented
4	#23752185	C₂	1	2	yes	maximal, complemented
5	H(3,ℤ/4)×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14559004	C₂²	1	2	yes	—
7	#18012772	—	2	2	no	—
8	#18012767	—	2	2	no	—
9	#18012761	—	2	2	no	—
10	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
11	#18012698	C₂²	1	1	yes	characteristic
12	Dic₁₅₁⋊C₈	—	2	2	no	—
13	#18012662	—	2	2	no	—
14	#18012659	—	2	2	no	—
15	#18012653	—	2	4	no	—
16	C₂×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
17	#117×C₁₅₁	C₂²	1	2	yes	—
18	#14559004	—	2	2	no	—
19	#14559004	—	2	2	no	—
20	#14559004	—	2	2	no	—
21	#14559004	—	2	2	no	—
22	C₂×Dic₃₀₂	D₄	1	2	yes	—
23	C₂×D₆₀₄	D₄	1	2	yes	—
24	C₂×C₂²⋊D₁₅₁	—	2	2	no	—
25	C₂×C₄×D₁₅₁	—	2	2	no	—
26	Dic₆₀₄	—	2	2	no	—
27	#14558993	—	2	4	no	—
28	D₁₂₀₈	—	2	2	no	—
29	C₂×C₁₅₁⋊C₈	—	2	2	no	—
30	C₂×C₁₅₁⋊C₈	—	2	1	no	—
31	C₄⋊Dic₁₅₁	—	2	1	no	—
32	C₂²⋊Dic₁₅₁	D₄	1	2	yes	—
33	C₄×Dic₁₅₁	—	2	1	no	—
34	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₅₁	—	2	2	no	—
36	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
37	C₄⋊C₄×C₁₅₁	—	2	2	no	—
38	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
39	C₄²×C₁₅₁	—	2	2	no	—
40	C₂²×D₁₅₁	—	2	2	no	—
41	Dic₃₀₂	—	2	2	no	—
42	Dic₃₀₂	—	2	2	no	—
43	D₆₀₄	—	2	2	no	—
44	D₆₀₄	—	2	2	no	—
45	C₂²⋊D₁₅₁	—	4	2	no	—
46	C₂²⋊D₁₅₁	—	4	2	no	—
47	C₂×Dic₁₅₁	—	2	2	no	—
48	C₂×Dic₁₅₁	—	2	2	no	—
49	C₄×D₁₅₁	—	4	2	no	—
50	C₄×D₁₅₁	—	4	2	no	—
51	C₁₅₁⋊C₈	—	4	1	no	—
52	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₁	—	4	2	no	—
54	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₅₁	#41	1	2	yes	—
57	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
58	C₈×C₁₅₁	—	2	2	no	—
59	C₂²×C₁₅₁	—	4	1	no	—
60	C₂²×C₁₅₁	#121	1	2	yes	—
61	C₂²×C₁₅₁	#121	1	1	yes	characteristic
62	C₂×D₁₅₁	—	4	2	no	—
63	C₂×D₁₅₁	—	8	2	no	—
64	C₄×C₁₅₁	—	8	2	no	—
65	C₄×C₁₅₁	#120	1	1	yes	characteristic
66	C₄×C₁₅₁	#123	1	2	yes	—
67	C₄×C₁₅₁	#124	1	1	yes	characteristic
68	Dic₁₅₁	—	4	2	no	—
69	Dic₁₅₁	—	4	2	no	—
70	C₂×C₁₅₁	—	2	1	no	—
71	C₂×C₁₅₁	—	2	2	no	—
72	C₂×C₁₅₁	#394	1	1	yes	characteristic
73	D₁₅₁	—	8	2	no	—
74	C₁₅₁	#2053	1	1	yes	characteristic, complemented, Sylow
75	#2053	—	151	1	no	maximal, Sylow
76	#575	—	151	1	no	—
77	#424	—	151	1	no	—
78	#416	—	151	2	no	—
79	#410	—	151	2	no	—
80	H(3,ℤ/4)	D₁₅₁	1	1	yes	characteristic, complemented
81	C₂×#41	—	151	2	no	—
82	#137	—	302	2	no	—
83	AGL(1,ℤ/8)	—	302	2	no	—
84	#135	—	302	2	no	—
85	C₂×M₁₆	C₂×D₁₅₁	1	1	yes	characteristic
86	C₂²×C₈	—	151	1	no	—
87	#117	—	151	1	no	—
88	#117	C₂×D₁₅₁	1	2	yes	—
89	C₄⋊C₈	—	302	2	no	—
90	#103	—	302	2	no	—
91	#102	—	302	2	no	—
92	#99	—	302	4	no	—
93	#41	—	302	2	no	—
94	#41	—	302	2	no	—
95	#41	—	302	2	no	—
96	#41	—	302	2	no	—
97	C₂×Q₈	—	151	2	no	—
98	C₂×D₄	—	302	2	no	—
99	C₂×D₄	—	151	2	no	—
100	C₂²×C₄	—	302	2	no	—
101	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
102	Q₁₆	—	302	2	no	—
103	SD₁₆	—	302	4	no	—
104	D₈	—	302	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	302	2	no	—
107	C₂×C₈	—	302	1	no	—
108	C₂×C₈	D₆₀₄	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	302	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	151	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	302	1	no	—
115	C₂³	—	302	2	no	—
116	C₂³	D₄×D₁₅₁	1	1	yes	characteristic
117	Q₈	—	302	2	no	—
118	Q₈	—	302	2	no	—
119	D₄	—	302	2	no	—
120	D₄	—	302	2	no	—
121	D₄	—	604	2	no	—
122	D₄	—	604	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	302	2	no	—
125	C₂×C₄	—	302	2	no	—
126	C₂×C₄	—	604	2	no	—
127	C₂×C₄	—	604	2	no	—
128	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
129	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
130	C₂×C₄	#14559004	1	2	yes	—
131	C₂×C₄	#14559005	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	604	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	604	2	no	—
136	C₂²	—	1208	2	no	—
137	C₂²	#18012712	1	2	yes	—
138	C₂²	#18012714	1	1	yes	characteristic
139	C₄	—	8	2	no	—
140	C₄	—	604	2	no	—
141	C₄	—	604	2	no	—
142	C₄	#18012709	1	1	yes	characteristic
143	C₄	#18012724	1	2	yes	—
144	C₄	#18012727	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	1208	2	no	—
148	C₂	#23752130	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23753012

C₂

yes

maximal, characteristic

#23752240

C₂

yes

maximal, characteristic, complemented

#23752206

C₂

yes

maximal, complemented

#23752185

C₂

yes

maximal, complemented

H(3,ℤ/4)×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14559004

C₂²

yes

—

#18012772

—

#18012767

—

#18012761

—

C₂²×C₁₅₁⋊C₈

C₂²

yes

characteristic

#18012698

C₂²

yes

characteristic

Dic₁₅₁⋊C₈

—

#18012662

—

#18012659

—

#18012653

—

C₂×M₁₆×C₁₅₁

C₂²

yes

characteristic

#117×C₁₅₁

C₂²

yes

—

—

—

—

—

C₂×Dic₃₀₂

D₄

yes

—

C₂×D₆₀₄

D₄

yes

—

C₂×C₂²⋊D₁₅₁

—

C₂×C₄×D₁₅₁

—

Dic₆₀₄

—

#14558993

—

D₁₂₀₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₂²⋊Dic₁₅₁

D₄

yes

—

C₄×Dic₁₅₁

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₂²×D₁₅₁

—

Dic₃₀₂

—

Dic₃₀₂

—

D₆₀₄

—

D₆₀₄

—

C₂²⋊D₁₅₁

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

characteristic

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#121

yes

—

C₂²×C₁₅₁

#121

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#120

yes

characteristic

C₄×C₁₅₁

#123

yes

—

C₄×C₁₅₁

#124

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#394

yes

characteristic

D₁₅₁

—

C₁₅₁

#2053

yes

characteristic, complemented, Sylow

#2053

—

151

maximal, Sylow

#575

—

151

—

#424

—

151

—

#416

—

151

—

#410

—

151

—

H(3,ℤ/4)

D₁₅₁

yes

characteristic, complemented

C₂×#41

—

151

—

#137

—

302

—

AGL(1,ℤ/8)

—

302

—

#135

—

302

—

C₂×M₁₆

C₂×D₁₅₁

yes

characteristic

C₂²×C₈

—

151

—

#117

—

151

—

#117

C₂×D₁₅₁

yes

—

C₄⋊C₈

—

302

—

#103

—

302

—

#102

—

302

—

#99

—

302

—

#41

—

302

—

#41

—

302

—

#41

—

302

—

#41

—

302

—

C₂×Q₈

—

151

—

C₂×D₄

—

302

—

C₂×D₄

—

151

—

100

C₂²×C₄

—

302

—

101

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

102

Q₁₆

—

302

—

103

SD₁₆

—

302

—

104

D₈

—

302

—

105

M₁₆

—

106

C₂×C₈

—

302

—

107

C₂×C₈

—

302

—

108

C₂×C₈

D₆₀₄

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

302

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

151

—

113

C₄²

—

114

C₄²

—

302

—

115

C₂³

—

302

—

116

C₂³

D₄×D₁₅₁

yes

characteristic

117

Q₈

—

302

—

118

Q₈

—

302

—

119

D₄

—

302

—

120

D₄

—

302

—

121

D₄

—

604

—

122

D₄

—

604

—

123

C₂×C₄

—

124

C₂×C₄

—

302

—

125

C₂×C₄

—

302

—

126

C₂×C₄

—

604

—

127

C₂×C₄

—

604

—

128

C₂×C₄

D₄×D₁₅₁

yes

—

129

C₂×C₄

C₂×D₆₀₄

yes

characteristic

130

C₂×C₄

#14559004

yes

—

131

C₂×C₄

#14559005

yes

characteristic

132

C₈

—

133

C₈

—

604

—

134

C₂²

—

135

C₂²

—

604

—

136

C₂²

—

1208

—

137

C₂²

#18012712

yes

—

138

C₂²

#18012714

yes

characteristic

139

C₄

—

140

C₄

—

604

—

141

C₄

—

604

—

142

C₄

#18012709

yes

characteristic

143

C₄

#18012724

yes

—

144

C₄

#18012727

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

1208

—

148

C₂

#23752130

yes

characteristic

	subgroup	G/H
center	C₄	#18012727
commutator, cosocle	C₂²×C₄×C₁₅₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	H(3,ℤ/4)×C₁₅₁	C₂
socle	C₂×C₁₅₁	#394
prime radical	#23753012	C₂
hypercenter	H(3,ℤ/4)	D₁₅₁
hypocenter	C₁₅₁	#2053
LCS	G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#18012727

commutator, cosocle

C₂²×C₄×C₁₅₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

H(3,ℤ/4)×C₁₅₁

C₂

C₂×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #23752240	C₂	semidirect
3. #23752206	C₂	semidirect
4. #23752185	C₂	semidirect
5. H(3,ℤ/4)×C₁₅₁	C₂	semidirect
74. C₁₅₁	#2053	semidirect
80. H(3,ℤ/4)	D₁₅₁	semidirect

H ≅ G/N

type

2. #23752240

C₂

semidirect

3. #23752206

C₂

semidirect

4. #23752185

C₂

semidirect

5. H(3,ℤ/4)×C₁₅₁

C₂

semidirect

74. C₁₅₁

#2053

semidirect

80. H(3,ℤ/4)

D₁₅₁

semidirect

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

subgroup	index
#23753012	2
#23752240	2
#23752206	2
#23752185	2
H(3,ℤ/4)×C₁₅₁	2
#2053	151

subgroup

index

H(3,ℤ/4)×C₁₅₁

151

Subgroups — #34070417

Conjugacy classes of subgroups

Complements

Maximal subgroups