Group #34070231 — subgroups

Subgroups — #34070231

Order 19328.

342 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂×#18012769	C₂	1	2	yes	maximal
2	C₂²×Dic₆₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#18012657	C₂	1	2	yes	maximal, complemented
4	#23751977	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₀₂	C₂²	1	2	yes	—
7	#18012769	—	2	4	no	—
8	#18012769	—	2	4	no	—
9	C₂×Dic₆₀₄	—	2	2	no	—
10	C₂×Dic₆₀₄	—	2	4	no	—
11	C₂×#14558994	—	2	2	no	—
12	C₂×#14558992	—	2	2	no	—
13	C₂×#14558988	C₂²	1	2	yes	—
14	#18012657	—	2	4	no	—
15	C₂×#41×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₁₅₁	—	2	2	no	—
19	C₂×Dic₃₀₂	—	2	2	no	—
20	C₂×Dic₃₀₂	—	2	2	no	—
21	C₂×Dic₃₀₂	—	2	2	no	—
22	C₂×Dic₃₀₂	—	2	2	no	—
23	C₂×Dic₃₀₂	—	2	2	no	—
24	C₂×Dic₃₀₂	D₄	1	4	yes	—
25	C₂²×Dic₁₅₁	—	2	2	no	—
26	Dic₆₀₄	—	2	8	no	—
27	#14558994	—	2	8	no	—
28	#14558992	—	2	8	no	—
29	#14558988	—	2	2	no	—
30	#14558988	D₄	1	4	yes	—
31	C₂×C₁₅₁⋊C₈	—	2	2	no	—
32	#41×C₁₅₁	—	2	2	no	—
33	#41×C₁₅₁	—	2	2	no	—
34	C₂×Q₈×C₁₅₁	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₅₁	—	2	1	no	—
36	C₂×D₄×C₁₅₁	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₅₁	—	2	1	no	—
38	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
39	M₁₆×C₁₅₁	—	2	2	no	—
40	C₂×C₈×C₁₅₁	—	2	2	no	—
41	C₂×C₈×C₁₅₁	—	2	1	no	—
42	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
43	Dic₃₀₂	—	2	4	no	—
44	Dic₃₀₂	—	2	4	no	—
45	Dic₃₀₂	—	4	2	no	—
46	Dic₃₀₂	—	4	2	no	—
47	C₂×Dic₁₅₁	—	2	4	no	—
48	C₂×Dic₁₅₁	—	4	2	no	—
49	C₂×Dic₁₅₁	—	4	2	no	—
50	C₁₅₁⋊C₈	—	2	4	no	—
51	C₂³×C₁₅₁	—	2	1	no	—
52	C₂³×C₁₅₁	#41	1	1	yes	characteristic
53	Q₈×C₁₅₁	—	2	2	no	—
54	D₄×C₁₅₁	—	2	2	no	—
55	D₄×C₁₅₁	—	4	2	no	—
56	C₂×C₄×C₁₅₁	—	2	1	no	—
57	C₂×C₄×C₁₅₁	—	4	1	no	—
58	C₂×C₄×C₁₅₁	—	4	1	no	—
59	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
63	C₈×C₁₅₁	—	2	2	no	—
64	C₈×C₁₅₁	—	4	1	no	—
65	C₂²×C₁₅₁	—	2	2	no	—
66	C₂²×C₁₅₁	—	4	1	no	—
67	C₂²×C₁₅₁	—	4	2	no	—
68	C₂²×C₁₅₁	#121	1	2	yes	—
69	C₂²×C₁₅₁	#121	1	1	yes	characteristic
70	C₄×C₁₅₁	—	4	1	no	—
71	C₄×C₁₅₁	#120	1	1	yes	characteristic
72	C₄×C₁₅₁	#120	1	2	yes	—
73	C₄×C₁₅₁	#127	1	1	yes	characteristic
74	Dic₁₅₁	—	4	4	no	—
75	C₂×C₁₅₁	—	2	2	no	—
76	C₂×C₁₅₁	—	8	1	no	—
77	C₂×C₁₅₁	#392	1	1	yes	characteristic
78	C₂×C₁₅₁	#473	1	2	yes	—
79	C₁₅₁	#2020	1	1	yes	characteristic, complemented, Sylow
80	#2020	—	151	1	no	maximal, Sylow
81	C₂×#137	—	151	2	no	—
82	C₂²×Q₁₆	—	151	1	no	—
83	C₂×#101	—	151	2	no	—
84	#408	D₁₅₁	1	1	yes	characteristic
85	#341	—	151	1	no	—
86	C₂×#41	C₂×D₁₅₁	1	1	yes	characteristic
87	C₂²×Q₈	—	151	2	no	—
88	#137	—	302	4	no	—
89	#137	—	302	4	no	—
90	C₂×Q₁₆	—	302	4	no	—
91	C₂×Q₁₆	—	302	2	no	—
92	C₂×Q₁₆	—	302	2	no	—
93	C₂×SD₁₆	—	302	2	no	—
94	C₂×M₁₆	—	151	2	no	—
95	C₂×M₁₆	C₂×D₁₅₁	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
97	#101	—	302	4	no	—
98	C₂²⋊C₈	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	—	302	2	no	—
102	C₂×Q₈	—	302	2	no	—
103	C₂×Q₈	—	302	2	no	—
104	C₂×Q₈	—	302	2	no	—
105	C₂×Q₈	—	302	2	no	—
106	C₂×Q₈	—	151	4	no	—
107	C₂×Q₈	D₆₀₄	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	D₆₀₄	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	302	2	no	—
112	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
113	Q₁₆	—	302	8	no	—
114	Q₁₆	—	302	8	no	—
115	SD₁₆	—	302	8	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	302	2	no	—
118	M₁₆	—	151	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	302	2	no	—
122	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
123	C₂³	—	2	1	no	—
124	C₂³	#14559004	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	302	4	no	—
127	Q₈	—	302	4	no	—
128	Q₈	—	604	2	no	—
129	Q₈	—	604	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	302	4	no	—
136	C₂×C₄	—	604	2	no	—
137	C₂×C₄	—	604	2	no	—
138	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
139	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
141	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	4	1	no	—
144	C₈	—	302	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	#18012712	1	2	yes	—
149	C₂²	#18012713	1	1	yes	characteristic
150	C₄	—	4	1	no	—
151	C₄	—	604	4	no	—
152	C₄	#18012707	1	1	yes	characteristic
153	C₄	#18012709	1	2	yes	—
154	C₄	#18012735	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	8	1	no	—
157	C₂	#23752121	1	1	yes	characteristic
158	C₂	#23752487	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012769

C₂

yes

maximal

C₂²×Dic₆₀₄

C₂

yes

maximal, characteristic, complemented

C₂×#18012657

C₂

yes

maximal, complemented

#23751977

C₂

yes

maximal, characteristic, complemented

#408×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

C₂²

yes

—

#18012769

—

#18012769

—

C₂×Dic₆₀₄

—

C₂×Dic₆₀₄

—

C₂×#14558994

—

C₂×#14558992

—

C₂×#14558988

C₂²

yes

—

#18012657

—

C₂×#41×C₁₅₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₅₁

C₂²

yes

characteristic

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

C₂²⋊C₈×C₁₅₁

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

D₄

yes

—

C₂²×Dic₁₅₁

—

Dic₆₀₄

—

—

—

—

D₄

yes

—

C₂×C₁₅₁⋊C₈

—

#41×C₁₅₁

—

#41×C₁₅₁

—

C₂×Q₈×C₁₅₁

D₄

yes

characteristic

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

Dic₃₀₂

—

Dic₃₀₂

—

Dic₃₀₂

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

#41

yes

characteristic

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#121

yes

—

C₂²×C₁₅₁

#121

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

#120

yes

characteristic

C₄×C₁₅₁

#120

yes

—

C₄×C₁₅₁

#127

yes

characteristic

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#392

yes

characteristic

C₂×C₁₅₁

#473

yes

—

C₁₅₁

#2020

yes

characteristic, complemented, Sylow

#2020

—

151

maximal, Sylow

C₂×#137

—

151

—

C₂²×Q₁₆

—

151

—

C₂×#101

—

151

—

#408

D₁₅₁

yes

characteristic

#341

—

151

—

C₂×#41

C₂×D₁₅₁

yes

characteristic

C₂²×Q₈

—

151

—

#137

—

302

—

#137

—

302

—

C₂×Q₁₆

—

302

—

C₂×Q₁₆

—

302

—

C₂×Q₁₆

—

302

—

C₂×SD₁₆

—

302

—

C₂×M₁₆

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

characteristic

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

#101

—

302

—

C₂²⋊C₈

—

#41

—

100

#41

—

101

C₂×Q₈

—

302

—

102

C₂×Q₈

—

302

—

103

C₂×Q₈

—

302

—

104

C₂×Q₈

—

302

—

105

C₂×Q₈

—

302

—

106

C₂×Q₈

—

151

—

107

C₂×Q₈

D₆₀₄

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

D₆₀₄

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

302

—

112

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

113

Q₁₆

—

302

—

114

Q₁₆

—

302

—

115

SD₁₆

—

302

—

116

M₁₆

—

117

M₁₆

—

302

—

118

M₁₆

—

151

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

302

—

122

C₂×C₈

C₂²⋊D₁₅₁

yes

—

123

C₂³

—

124

C₂³

#14559004

yes

characteristic

125

Q₈

—

126

Q₈

—

302

—

127

Q₈

—

302

—

128

Q₈

—

604

—

129

Q₈

—

604

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

302

—

136

C₂×C₄

—

604

—

137

C₂×C₄

—

604

—

138

C₂×C₄

D₄×D₁₅₁

yes

—

139

C₂×C₄

D₄×D₁₅₁

yes

—

140

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

141

C₂×C₄

C₂×D₆₀₄

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

302

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#18012712

yes

—

149

C₂²

#18012713

yes

characteristic

150

C₄

—

151

C₄

—

604

—

152

C₄

#18012707

yes

characteristic

153

C₄

#18012709

yes

—

154

C₄

#18012735

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#23752121

yes

characteristic

158

C₂

#23752487

yes

—

	subgroup	G/H
center	C₂²	#18012713
commutator, cosocle	C₂²×C₄×C₁₅₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#408×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#121
prime radical	C₂×#41×C₁₅₁	C₂²
hypercenter	#408	D₁₅₁
hypocenter	C₁₅₁	#2020
LCS	G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#18012713

commutator, cosocle

C₂²×C₄×C₁₅₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#408×C₁₅₁

C₂

socle

C₂²×C₁₅₁

#121

prime radical

C₂×#41×C₁₅₁

C₂²

D₁₅₁

C₁₅₁

G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. C₂²×Dic₆₀₄	C₂	semidirect
3. C₂×#18012657	C₂	semidirect
4. #23751977	C₂	semidirect
79. C₁₅₁	#2020	semidirect

H ≅ G/N

type

2. C₂²×Dic₆₀₄

C₂

semidirect

3. C₂×#18012657

C₂

semidirect

4. #23751977

C₂

semidirect

79. C₁₅₁

#2020

semidirect

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

subgroup	index
C₂×#18012769	2
C₂²×Dic₆₀₄	2
C₂×#18012657	2
#23751977	2
#408×C₁₅₁	2
#2020	151

subgroup

index

C₂²×Dic₆₀₄

#408×C₁₅₁

151

Subgroups — #34070231

Conjugacy classes of subgroups

Complements

Maximal subgroups