Group #34077762 — subgroups

Subgroups — #34077762

Order 19328.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#14559004	C₂	1	1	yes	maximal, characteristic
2	#23752271	C₂	1	4	yes	maximal, complemented
3	#23752267	C₂	1	4	yes	maximal, complemented
4	#23752240	C₂	1	2	yes	maximal, complemented
5	#23752180	C₂	1	2	yes	maximal
6	C₂×#18012650	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14559004	C₂²	1	1	yes	characteristic
9	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₃₀₂	C₄	1	2	yes	—
11	C₄×D₆₀₄	C₄	1	2	yes	—
12	C₄×C₂²⋊D₁₅₁	C₄	1	4	yes	—
13	#18012698	C₂²	1	1	yes	characteristic
14	#18012696	C₂²	1	2	yes	—
15	C₄²×D₁₅₁	C₂²	1	2	yes	—
16	Dic₁₅₁⋊C₈	C₂²	1	8	yes	—
17	(C₂×D₁₅₁)⋊C₈	C₂²	1	8	yes	—
18	#18012650	C₂²	1	2	yes	—
19	#18012650	C₂²	1	2	yes	—
20	C₂×M₁₆×C₁₅₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
22	#97×C₁₅₁	C₂²	1	2	yes	—
23	C₄×C₈×C₁₅₁	C₂²	1	2	yes	—
24	#14559004	—	2	4	no	—
25	C₂×Dic₃₀₂	C₂×C₄	1	1	yes	characteristic
26	C₂×D₆₀₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₅₁	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₅₁	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
30	C₂×C₁₅₁⋊C₈	—	2	4	no	—
31	C₂×C₁₅₁⋊C₈	C₂³	1	4	yes	—
32	C₄⋊Dic₁₅₁	C₂×C₄	1	2	yes	—
33	Dic₁₅₁⋊C₄	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₁₅₁	C₂×C₄	1	2	yes	—
35	(C₂×D₁₅₁)⋊C₄	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₅₁	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₅₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₅₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₅₁	—	2	4	no	—
41	C₂×C₈×C₁₅₁	C₂³	1	4	yes	—
42	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
43	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
44	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
45	C₂²×D₁₅₁	C₂²×C₄	1	2	yes	—
46	Dic₃₀₂	—	2	2	no	—
47	D₆₀₄	—	2	2	no	—
48	C₂²⋊D₁₅₁	—	2	4	no	—
49	C₂×Dic₁₅₁	C₂²×C₄	1	2	yes	—
50	C₂×Dic₁₅₁	C₂²×C₄	1	4	yes	—
51	C₄×D₁₅₁	—	2	4	no	—
52	C₄×D₁₅₁	—	4	4	no	—
53	C₁₅₁⋊C₈	—	2	4	no	—
54	C₂³×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₅₁	—	4	2	no	—
56	C₂×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₅₁	#41	1	4	yes	—
60	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₅₁	—	2	4	no	—
62	C₂²×C₁₅₁	—	2	2	no	—
63	C₂²×C₁₅₁	#117	1	2	yes	—
64	C₂²×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₅₁	—	2	2	no	—
66	C₂×D₁₅₁	—	2	4	no	—
67	C₄×C₁₅₁	—	2	2	no	—
68	C₄×C₁₅₁	—	2	2	no	—
69	C₄×C₁₅₁	#117	1	2	yes	—
70	C₄×C₁₅₁	C₂×#41	1	2	yes	—
71	Dic₁₅₁	—	2	2	no	—
72	Dic₁₅₁	—	2	4	no	—
73	C₂×C₁₅₁	—	2	2	no	—
74	C₂×C₁₅₁	C₂×#117	1	1	yes	characteristic
75	C₂×C₁₅₁	#568	1	2	yes	—
76	D₁₅₁	—	4	2	no	—
77	C₁₅₁	#2923	1	1	yes	characteristic, complemented, Sylow
78	#2923	—	151	1	no	maximal, Sylow
79	C₄×#41	—	151	1	no	—
80	#433	—	151	4	no	—
81	#432	—	151	4	no	—
82	#424	—	151	2	no	—
83	#408	—	151	2	no	—
84	C₄×M₁₆	D₁₅₁	1	1	yes	characteristic, complemented
85	C₂×#97	—	151	1	no	—
86	C₂×#41	—	151	1	no	—
87	C₂×M₁₆	C₂×D₁₅₁	1	2	yes	—
88	C₂²×C₈	—	151	2	no	—
89	C₄×Q₈	—	151	2	no	—
90	C₄×D₄	—	151	4	no	—
91	C₄×D₄	—	151	2	no	—
92	#117	—	151	1	no	—
93	#117	—	151	2	no	—
94	C₂×C₄²	—	151	2	no	—
95	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
96	C₄⋊C₈	—	151	8	no	—
97	C₂²⋊C₈	—	151	8	no	—
98	#97	—	151	2	no	—
99	#97	—	151	2	no	—
100	#97	C₂×D₁₅₁	1	2	yes	—
101	C₄×C₈	C₂×D₁₅₁	1	2	yes	—
102	#41	—	302	4	no	—
103	C₂×Q₈	—	151	1	no	—
104	C₂×D₄	—	151	2	no	—
105	C₂×D₄	—	151	1	no	—
106	C₂²×C₄	—	151	2	no	—
107	C₂²×C₄	—	151	4	no	—
108	C₂²×C₄	C₄×D₁₅₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	302	4	no	—
112	C₂×C₈	—	151	4	no	—
113	C₂×C₈	C₂²×D₁₅₁	1	4	yes	—
114	C₄⋊C₄	—	151	4	no	—
115	C₄⋊C₄	—	151	2	no	—
116	C₂²⋊C₄	—	151	2	no	—
117	C₂²⋊C₄	—	151	4	no	—
118	C₄²	—	151	4	no	—
119	C₄²	—	151	2	no	—
120	C₄²	C₄×D₁₅₁	1	2	yes	—
121	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
122	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
123	C₂³	—	151	2	no	—
124	C₂³	C₂×C₄×D₁₅₁	1	1	yes	characteristic
125	Q₈	—	302	2	no	—
126	D₄	—	302	4	no	—
127	D₄	—	302	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	302	4	no	—
130	C₂×C₄	—	604	4	no	—
131	C₂×C₄	—	151	4	no	—
132	C₂×C₄	—	151	2	no	—
133	C₂×C₄	C₂×C₄×D₁₅₁	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₁₅₁	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
136	C₂×C₄	#14559004	1	4	yes	—
137	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	302	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	302	4	no	—
142	C₂²	—	302	2	no	—
143	C₂²	#18012696	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	302	4	no	—
148	C₄	—	302	2	no	—
149	C₄	#18012696	1	2	yes	—
150	C₄	C₂×#14559004	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	604	2	no	—
153	C₂	C₂×#18012696	1	1	yes	characteristic
154	C₂	#23752978	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14559004

C₂

yes

maximal, characteristic

#23752271

C₂

yes

maximal, complemented

#23752267

C₂

yes

maximal, complemented

#23752240

C₂

yes

maximal, complemented

#23752180

C₂

yes

maximal

C₂×#18012650

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14559004

C₂²

yes

characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₄×Dic₃₀₂

C₄

yes

—

C₄×D₆₀₄

C₄

yes

—

C₄×C₂²⋊D₁₅₁

C₄

yes

—

#18012698

C₂²

yes

characteristic

#18012696

C₂²

yes

—

C₄²×D₁₅₁

C₂²

yes

—

Dic₁₅₁⋊C₈

C₂²

yes

—

(C₂×D₁₅₁)⋊C₈

C₂²

yes

—

#18012650

C₂²

yes

—

#18012650

C₂²

yes

—

C₂×M₁₆×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

#97×C₁₅₁

C₂²

yes

—

C₄×C₈×C₁₅₁

C₂²

yes

—

#14559004

—

C₂×Dic₃₀₂

C₂×C₄

yes

characteristic

C₂×D₆₀₄

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₁₅₁

C₂×C₄

yes

—

C₂×C₄×D₁₅₁

C₂×C₄

yes

—

C₂×C₄×D₁₅₁

C₂³

yes

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₅₁

C₂×C₄

yes

—

Dic₁₅₁⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₁₅₁

C₂×C₄

yes

—

(C₂×D₁₅₁)⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₅₁

C₂×C₄

yes

—

C₄×Dic₁₅₁

C₂³

yes

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂³

yes

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

C₂²×C₄

yes

—

Dic₃₀₂

—

D₆₀₄

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

C₂²×C₄

yes

—

C₂×Dic₁₅₁

C₂²×C₄

yes

—

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#117

yes

—

C₂²×C₁₅₁

C₂³×C₄

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#117

yes

—

C₄×C₁₅₁

C₂×#41

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×#117

yes

characteristic

C₂×C₁₅₁

#568

yes

—

D₁₅₁

—

C₁₅₁

#2923

yes

characteristic, complemented, Sylow

#2923

—

151

maximal, Sylow

C₄×#41

—

151

—

#433

—

151

—

#432

—

151

—

#424

—

151

—

#408

—

151

—

C₄×M₁₆

D₁₅₁

yes

characteristic, complemented

C₂×#97

—

151

—

C₂×#41

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

—

C₂²×C₈

—

151

—

C₄×Q₈

—

151

—

C₄×D₄

—

151

—

C₄×D₄

—

151

—

#117

—

151

—

#117

—

151

—

C₂×C₄²

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

151

—

C₂²⋊C₈

—

151

—

#97

—

151

—

#97

—

151

—

100

#97

C₂×D₁₅₁

yes

—

101

C₄×C₈

C₂×D₁₅₁

yes

—

102

#41

—

302

—

103

C₂×Q₈

—

151

—

104

C₂×D₄

—

151

—

105

C₂×D₄

—

151

—

106

C₂²×C₄

—

151

—

107

C₂²×C₄

—

151

—

108

C₂²×C₄

C₄×D₁₅₁

yes

—

109

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

302

—

112

C₂×C₈

—

151

—

113

C₂×C₈

C₂²×D₁₅₁

yes

—

114

C₄⋊C₄

—

151

—

115

C₄⋊C₄

—

151

—

116

C₂²⋊C₄

—

151

—

117

C₂²⋊C₄

—

151

—

118

C₄²

—

151

—

119

C₄²

—

151

—

120

C₄²

C₄×D₁₅₁

yes

—

121

C₄²

C₂²×D₁₅₁

yes

characteristic

122

C₄²

C₂²×D₁₅₁

yes

characteristic

123

C₂³

—

151

—

124

C₂³

C₂×C₄×D₁₅₁

yes

characteristic

125

Q₈

—

302

—

126

D₄

—

302

—

127

D₄

—

302

—

128

C₂×C₄

—

129

C₂×C₄

—

302

—

130

C₂×C₄

—

604

—

131

C₂×C₄

—

151

—

132

C₂×C₄

—

151

—

133

C₂×C₄

C₂×C₄×D₁₅₁

yes

—

134

C₂×C₄

C₂×C₄×D₁₅₁

yes

—

135

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

136

C₂×C₄

#14559004

yes

—

137

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

138

C₈

—

139

C₈

—

302

—

140

C₂²

—

141

C₂²

—

302

—

142

C₂²

—

302

—

143

C₂²

#18012696

yes

—

144

C₂²

C₂²×C₄×D₁₅₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

302

—

148

C₄

—

302

—

149

C₄

#18012696

yes

—

150

C₄

C₂×#14559004

yes

—

151

C₂

—

152

C₂

—

604

—

153

C₂

C₂×#18012696

yes

characteristic

154

C₂

#23752978

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₁
commutator, socle	C₂²×C₁₅₁	C₂³×C₄
Fitting	C₄×M₁₆×C₁₅₁	C₂
prime radical	C₂×#14559004	C₂²
cosocle	C₂×C₄×C₁₅₁	C₂⁴
hypercenter	C₄×M₁₆	D₁₅₁
hypocenter	C₁₅₁	#2923
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₅₁

commutator, socle

C₂²×C₁₅₁

C₂³×C₄

Fitting

C₄×M₁₆×C₁₅₁

C₂

prime radical

C₂×#14559004

C₂²

cosocle

C₂×C₄×C₁₅₁

C₂⁴

hypercenter

C₄×M₁₆

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #23752271	C₂	semidirect
3. #23752267	C₂	semidirect
4. #23752240	C₂	semidirect
6. C₂×#18012650	C₂	semidirect
7. C₄×M₁₆×C₁₅₁	C₂	semidirect
77. C₁₅₁	#2923	semidirect
84. C₄×M₁₆	D₁₅₁	semidirect

H ≅ G/N

type

2. #23752271

C₂

semidirect

3. #23752267

C₂

semidirect

4. #23752240

C₂

semidirect

6. C₂×#18012650

C₂

semidirect

7. C₄×M₁₆×C₁₅₁

C₂

semidirect

77. C₁₅₁

#2923

semidirect

84. C₄×M₁₆

D₁₅₁

semidirect

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

subgroup	index
C₄×#14559004	2
#23752271	2
#23752267	2
#23752240	2
#23752180	2
C₂×#18012650	2
C₄×M₁₆×C₁₅₁	2
#2923	151

subgroup

index

C₄×M₁₆×C₁₅₁

151

Subgroups — #34077762

Conjugacy classes of subgroups

Complements

Maximal subgroups