Group #34079455 — subgroups

Subgroups — #34079455

Order 19328.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#23752769	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752580	C₂	1	2	yes	maximal, complemented
3	#23752576	C₂	1	2	yes	maximal, complemented
4	#23752571	C₂	1	4	yes	maximal, complemented
5	#23752208	C₂	1	4	yes	maximal, complemented
6	C₂×#18012650	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
9	#18012739	C₂²	1	2	yes	—
10	#18012731	C₂²	1	2	yes	—
11	#18012718	—	2	2	no	—
12	C₄×Dic₃₀₂	—	2	2	no	—
13	#18012698	C₂²	1	2	yes	—
14	#18012663	C₂²	1	8	yes	—
15	#18012660	C₂²	1	8	yes	—
16	#18012650	C₂²	1	2	yes	—
17	#18012650	C₂²	1	2	yes	—
18	C₂×#41×C₁₅₁	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
20	#127×C₁₅₁	C₂²	1	1	yes	characteristic
21	#124×C₁₅₁	C₂²	1	2	yes	—
22	#121×C₁₅₁	—	2	2	no	—
23	C₄×D₄×C₁₅₁	—	2	2	no	—
24	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₃₀₂	—	2	2	no	—
26	C₂×C₁₅₁⋊C₈	—	2	4	no	—
27	C₂×C₁₅₁⋊C₈	C₂³	1	4	yes	—
28	C₄⋊Dic₁₅₁	—	2	2	no	—
29	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
30	Dic₁₅₁⋊C₄	—	2	4	no	—
31	C₂²⋊Dic₁₅₁	—	2	2	no	—
32	C₄×Dic₁₅₁	—	2	2	no	—
33	#41×C₁₅₁	—	2	8	no	—
34	C₂×Q₈×C₁₅₁	C₂³	1	2	yes	—
35	C₂×D₄×C₁₅₁	—	2	2	no	—
36	C₂×D₄×C₁₅₁	—	2	2	no	—
37	C₂×D₄×C₁₅₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₅₁	—	2	2	no	—
39	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
40	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₅₁	—	2	2	no	—
42	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
43	C₄²×C₁₅₁	D₄	1	2	yes	—
44	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
45	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
46	Dic₃₀₂	—	4	4	no	—
47	C₂×Dic₁₅₁	—	2	4	no	—
48	C₁₅₁⋊C₈	—	2	4	no	—
49	C₂³×C₁₅₁	—	2	2	no	—
50	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₅₁	—	2	4	no	—
52	D₄×C₁₅₁	—	2	4	no	—
53	D₄×C₁₅₁	—	4	4	no	—
54	D₄×C₁₅₁	—	4	4	no	—
55	C₂×C₄×C₁₅₁	—	2	2	no	—
56	C₂×C₄×C₁₅₁	—	4	2	no	—
57	C₂×C₄×C₁₅₁	—	4	4	no	—
58	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₅₁	#41	1	4	yes	—
62	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
63	C₂²×C₁₅₁	—	2	2	no	—
64	C₂²×C₁₅₁	—	4	2	no	—
65	C₂²×C₁₅₁	—	4	4	no	—
66	C₂²×C₁₅₁	#124	1	2	yes	—
67	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₁₅₁	—	2	2	no	—
69	C₄×C₁₅₁	—	2	2	no	—
70	C₄×C₁₅₁	—	4	2	no	—
71	C₄×C₁₅₁	#124	1	2	yes	—
72	C₄×C₁₅₁	C₂×#41	1	2	yes	—
73	Dic₁₅₁	—	4	4	no	—
74	C₂×C₁₅₁	—	2	2	no	—
75	C₂×C₁₅₁	—	8	2	no	—
76	C₂×C₁₅₁	C₂×#124	1	1	yes	characteristic
77	C₂×C₁₅₁	#575	1	2	yes	—
78	C₁₅₁	#3135	1	1	yes	characteristic, complemented, Sylow
79	#3135	—	151	1	no	maximal, Sylow
80	#533	—	151	1	no	—
81	#532	D₁₅₁	1	1	yes	characteristic
82	#491	—	151	2	no	—
83	#490	—	151	2	no	—
84	#489	—	151	4	no	—
85	#416	—	151	4	no	—
86	C₂×#97	—	151	1	no	—
87	C₂×#41	C₂×D₁₅₁	1	2	yes	—
88	C₂²×C₈	—	151	2	no	—
89	C₄⋊Q₈	—	151	2	no	—
90	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
91	#127	C₂×D₁₅₁	1	1	yes	characteristic
92	#125	—	151	2	no	—
93	#124	C₂×D₁₅₁	1	2	yes	—
94	C₂²⋊Q₈	—	302	2	no	—
95	#121	—	2	2	no	—
96	C₄×Q₈	—	302	2	no	—
97	C₄×D₄	—	2	2	no	—
98	#117	—	151	2	no	—
99	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
100	#103	—	151	8	no	—
101	#102	—	151	8	no	—
102	#97	—	151	2	no	—
103	#97	—	151	2	no	—
104	#41	—	2	8	no	—
105	C₂×Q₈	—	302	2	no	—
106	C₂×Q₈	C₂²×D₁₅₁	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂²×D₁₅₁	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
113	C₂×C₈	—	302	4	no	—
114	C₂×C₈	—	151	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	302	4	no	—
117	C₄⋊C₄	—	302	2	no	—
118	C₄⋊C₄	—	151	4	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	302	2	no	—
121	C₄²	—	302	2	no	—
122	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
123	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
124	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	604	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	302	4	no	—
136	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
139	C₂×C₄	#14559003	1	4	yes	—
140	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
141	C₈	—	302	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	4	no	—
145	C₂²	#18012726	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	604	4	no	—
151	C₄	#18012726	1	2	yes	—
152	C₄	C₂×#14559003	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	8	2	no	—
155	C₂	C₂×#18012726	1	1	yes	characteristic
156	C₂	#23753010	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752769

C₂

yes

maximal, characteristic, complemented

#23752580

C₂

yes

maximal, complemented

#23752576

C₂

yes

maximal, complemented

#23752571

C₂

yes

maximal, complemented

#23752208

C₂

yes

maximal, complemented

C₂×#18012650

C₂

yes

maximal, characteristic, complemented

#532×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

#18012739

C₂²

yes

—

#18012731

C₂²

yes

—

#18012718

—

C₄×Dic₃₀₂

—

#18012698

C₂²

yes

—

#18012663

C₂²

yes

—

#18012660

C₂²

yes

—

#18012650

C₂²

yes

—

#18012650

C₂²

yes

—

C₂×#41×C₁₅₁

C₂²

yes

—

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

#127×C₁₅₁

C₂²

yes

characteristic

#124×C₁₅₁

C₂²

yes

—

#121×C₁₅₁

—

C₄×D₄×C₁₅₁

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂³

yes

—

Dic₁₅₁⋊C₄

—

C₂²⋊Dic₁₅₁

—

C₄×Dic₁₅₁

—

#41×C₁₅₁

—

C₂×Q₈×C₁₅₁

C₂³

yes

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

C₂³

yes

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₄²×C₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

Q₈×C₁₅₁

—

D₄×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

#41

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#124

yes

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#124

yes

—

C₄×C₁₅₁

C₂×#41

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×#124

yes

characteristic

C₂×C₁₅₁

#575

yes

—

C₁₅₁

#3135

yes

characteristic, complemented, Sylow

#3135

—

151

maximal, Sylow

#533

—

151

—

#532

D₁₅₁

yes

characteristic

#491

—

151

—

#490

—

151

—

#489

—

151

—

#416

—

151

—

C₂×#97

—

151

—

C₂×#41

C₂×D₁₅₁

yes

—

C₂²×C₈

—

151

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

#127

C₂×D₁₅₁

yes

characteristic

#125

—

151

—

#124

C₂×D₁₅₁

yes

—

C₂²⋊Q₈

—

302

—

#121

—

C₄×Q₈

—

302

—

C₄×D₄

—

#117

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

100

#103

—

151

—

101

#102

—

151

—

102

#97

—

151

—

103

#97

—

151

—

104

#41

—

105

C₂×Q₈

—

302

—

106

C₂×Q₈

C₂²×D₁₅₁

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₁₅₁

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

112

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

113

C₂×C₈

—

302

—

114

C₂×C₈

—

151

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

302

—

117

C₄⋊C₄

—

302

—

118

C₄⋊C₄

—

151

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

302

—

121

C₄²

—

302

—

122

C₄²

C₂²⋊D₁₅₁

yes

—

123

C₄²

C₂²×D₁₅₁

yes

characteristic

124

C₄²

C₂²×D₁₅₁

yes

characteristic

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

127

Q₈

—

128

Q₈

—

604

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

302

—

136

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

139

C₂×C₄

#14559003

yes

—

140

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

141

C₈

—

302

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#18012726

yes

—

146

C₂²

C₂²×C₂²⋊D₁₅₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

604

—

151

C₄

#18012726

yes

—

152

C₄

C₂×#14559003

yes

—

153

C₂

—

154

C₂

—

155

C₂

C₂×#18012726

yes

characteristic

156

C₂

#23753010

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting, prime radical	#532×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
hypercenter	#532	D₁₅₁
hypocenter	C₁₅₁	#3135
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting, prime radical

#532×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂²×D₄

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #23752769	C₂	semidirect
2. #23752580	C₂	semidirect
3. #23752576	C₂	semidirect
4. #23752571	C₂	semidirect
5. #23752208	C₂	semidirect
6. C₂×#18012650	C₂	semidirect
78. C₁₅₁	#3135	semidirect

H ≅ G/N

type

1. #23752769

C₂

semidirect

2. #23752580

C₂

semidirect

3. #23752576

C₂

semidirect

4. #23752571

C₂

semidirect

5. #23752208

C₂

semidirect

6. C₂×#18012650

C₂

semidirect

78. C₁₅₁

#3135

semidirect

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

subgroup	index
#23752769	2
#23752580	2
#23752576	2
#23752571	2
#23752208	2
C₂×#18012650	2
#532×C₁₅₁	2
#3135	151

subgroup

index

#532×C₁₅₁

151

Subgroups — #34079455

Conjugacy classes of subgroups

Complements

Maximal subgroups