Group #34079513 — subgroups

Subgroups — #34079513

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	#23752762	C₂	1	1	yes	maximal, characteristic
2	#23752582	C₂	1	2	yes	maximal, complemented
3	#23752578	C₂	1	2	yes	maximal, complemented
4	#23752569	C₂	1	4	yes	maximal, complemented
5	#23752211	C₂	1	4	yes	maximal, complemented
6	C₄×#14558988	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#125×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14559004	C₂²	1	2	yes	—
9	C₂×#14558988	C₂²	1	2	yes	—
10	#18012739	C₂²	1	1	yes	characteristic
11	#18012736	C₂²	1	1	yes	characteristic
12	#18012727	C₂²	1	2	yes	—
13	#18012713	—	2	2	no	—
14	C₄×D₆₀₄	—	2	2	no	—
15	#18012662	C₂²	1	8	yes	—
16	#18012659	C₂²	1	8	yes	—
17	#18012650	C₂²	1	2	yes	—
18	C₄×C₁₅₁⋊C₈	C₂²	1	2	yes	—
19	#125×C₁₅₁	—	2	2	no	—
20	#125×C₁₅₁	C₂²	1	2	yes	—
21	#125×C₁₅₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
25	#14559004	—	2	8	no	—
26	C₂×Dic₃₀₂	C₂³	1	2	yes	—
27	C₂×D₆₀₄	—	2	2	no	—
28	C₂×D₆₀₄	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₁₅₁	—	2	2	no	—
30	C₂×C₄×D₁₅₁	—	2	2	no	—
31	#14558988	—	2	4	no	—
32	C₂×C₁₅₁⋊C₈	C₂³	1	4	yes	—
33	C₄⋊Dic₁₅₁	—	2	2	no	—
34	(C₂×D₁₅₁)⋊C₄	—	2	4	no	—
35	C₂²×C₄×C₁₅₁	—	2	2	no	—
36	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
37	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₅₁	—	2	4	no	—
39	C₄⋊C₄×C₁₅₁	—	2	4	no	—
40	C₄⋊C₄×C₁₅₁	—	2	2	no	—
41	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
42	C₄²×C₁₅₁	D₄	1	2	yes	—
43	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
44	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
45	C₂²×D₁₅₁	—	2	2	no	—
46	Dic₃₀₂	—	2	4	no	—
47	D₆₀₄	—	2	4	no	—
48	D₆₀₄	—	4	4	no	—
49	C₂²⋊D₁₅₁	—	4	4	no	—
50	C₂×Dic₁₅₁	—	2	2	no	—
51	C₄×D₁₅₁	—	4	4	no	—
52	C₁₅₁⋊C₈	—	2	4	no	—
53	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₅₁	—	2	4	no	—
55	C₂×C₄×C₁₅₁	—	4	2	no	—
56	C₂×C₄×C₁₅₁	—	4	4	no	—
57	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₅₁	#41	1	4	yes	—
61	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₅₁	—	2	2	no	—
63	C₂²×C₁₅₁	#124	1	2	yes	—
64	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₁₅₁	—	4	2	no	—
66	C₂×D₁₅₁	—	4	4	no	—
67	C₄×C₁₅₁	—	2	2	no	—
68	C₄×C₁₅₁	—	2	2	no	—
69	C₄×C₁₅₁	—	4	4	no	—
70	C₄×C₁₅₁	#124	1	2	yes	—
71	C₄×C₁₅₁	C₂×#41	1	2	yes	—
72	Dic₁₅₁	—	4	2	no	—
73	C₂×C₁₅₁	—	2	2	no	—
74	C₂×C₁₅₁	C₂×#124	1	1	yes	characteristic
75	C₂×C₁₅₁	#575	1	2	yes	—
76	D₁₅₁	—	8	2	no	—
77	C₁₅₁	#3142	1	1	yes	characteristic, complemented, Sylow
78	#3142	—	151	1	no	maximal, Sylow
79	#532	—	151	1	no	—
80	C₂×#125	D₁₅₁	1	1	yes	characteristic, complemented
81	#491	—	151	2	no	—
82	#490	—	151	2	no	—
83	#488	—	151	4	no	—
84	#417	—	151	4	no	—
85	C₄×M₁₆	—	151	1	no	—
86	C₂×#41	—	151	2	no	—
87	C₂×M₁₆	—	151	2	no	—
88	C₄⋊Q₈	—	151	1	no	—
89	#127	—	151	1	no	—
90	#125	—	2	2	no	—
91	#125	C₂×D₁₅₁	1	2	yes	—
92	#125	C₂×D₁₅₁	1	2	yes	—
93	#124	—	151	2	no	—
94	#121	—	302	2	no	—
95	C₄×D₄	—	302	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
98	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
99	#103	—	151	8	no	—
100	#102	—	151	8	no	—
101	#97	—	151	2	no	—
102	C₄×C₈	—	151	2	no	—
103	#41	—	302	8	no	—
104	C₂×Q₈	—	151	2	no	—
105	C₂×D₄	—	302	2	no	—
106	C₂×D₄	—	302	2	no	—
107	C₂×D₄	—	151	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	302	2	no	—
110	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
112	M₁₆	—	302	4	no	—
113	C₂×C₈	—	151	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	302	2	no	—
118	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
119	C₂²⋊C₄	—	302	4	no	—
120	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
121	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
122	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
123	C₂³	—	302	2	no	—
124	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
125	Q₈	—	302	4	no	—
126	D₄	—	302	4	no	—
127	D₄	—	604	4	no	—
128	D₄	—	604	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	302	2	no	—
133	C₂×C₄	—	604	4	no	—
134	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
137	C₂×C₄	#14559005	1	4	yes	—
138	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
139	C₈	—	302	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	604	2	no	—
142	C₂²	—	604	4	no	—
143	C₂²	#18012728	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	604	2	no	—
149	C₄	#18012728	1	2	yes	—
150	C₄	C₂×#14559005	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	1208	2	no	—
153	C₂	C₂×#18012728	1	1	yes	characteristic
154	C₂	#23753010	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752762

C₂

yes

maximal, characteristic

#23752582

C₂

yes

maximal, complemented

#23752578

C₂

yes

maximal, complemented

#23752569

C₂

yes

maximal, complemented

#23752211

C₂

yes

maximal, complemented

C₄×#14558988

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14559004

C₂²

yes

—

C₂×#14558988

C₂²

yes

—

#18012739

C₂²

yes

characteristic

#18012736

C₂²

yes

characteristic

#18012727

C₂²

yes

—

#18012713

—

C₄×D₆₀₄

—

#18012662

C₂²

yes

—

#18012659

C₂²

yes

—

#18012650

C₂²

yes

—

C₄×C₁₅₁⋊C₈

C₂²

yes

—

#125×C₁₅₁

—

#125×C₁₅₁

C₂²

yes

—

#125×C₁₅₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

#14559004

—

C₂×Dic₃₀₂

C₂³

yes

—

C₂×D₆₀₄

—

C₂×D₆₀₄

C₂³

yes

—

C₂×C₂²⋊D₁₅₁

—

C₂×C₄×D₁₅₁

—

#14558988

—

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₅₁

—

(C₂×D₁₅₁)⋊C₄

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₄²×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

—

Dic₃₀₂

—

D₆₀₄

—

D₆₀₄

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₄×D₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

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

#124

yes

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#124

yes

—

C₄×C₁₅₁

C₂×#41

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×#124

yes

characteristic

C₂×C₁₅₁

#575

yes

—

D₁₅₁

—

C₁₅₁

#3142

yes

characteristic, complemented, Sylow

#3142

—

151

maximal, Sylow

#532

—

151

—

C₂×#125

D₁₅₁

yes

characteristic, complemented

#491

—

151

—

#490

—

151

—

#488

—

151

—

#417

—

151

—

C₄×M₁₆

—

151

—

C₂×#41

—

151

—

C₂×M₁₆

—

151

—

C₄⋊Q₈

—

151

—

#127

—

151

—

#125

—

#125

C₂×D₁₅₁

yes

—

#125

C₂×D₁₅₁

yes

—

#124

—

151

—

#121

—

302

—

C₄×D₄

—

302

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#103

—

151

—

100

#102

—

151

—

101

#97

—

151

—

102

C₄×C₈

—

151

—

103

#41

—

302

—

104

C₂×Q₈

—

151

—

105

C₂×D₄

—

302

—

106

C₂×D₄

—

302

—

107

C₂×D₄

—

151

—

108

C₂²×C₄

—

109

C₂²×C₄

—

302

—

110

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

111

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

112

M₁₆

—

302

—

113

C₂×C₈

—

151

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

302

—

118

C₄⋊C₄

C₂²×D₁₅₁

yes

—

119

C₂²⋊C₄

—

302

—

120

C₄²

C₂²⋊D₁₅₁

yes

—

121

C₄²

C₂²×D₁₅₁

yes

characteristic

122

C₄²

C₂²×D₁₅₁

yes

characteristic

123

C₂³

—

302

—

124

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

125

Q₈

—

302

—

126

D₄

—

302

—

127

D₄

—

604

—

128

D₄

—

604

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

302

—

133

C₂×C₄

—

604

—

134

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

136

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

137

C₂×C₄

#14559005

yes

—

138

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

139

C₈

—

302

—

140

C₂²

—

141

C₂²

—

604

—

142

C₂²

—

604

—

143

C₂²

#18012728

yes

—

144

C₂²

C₂²×C₂²⋊D₁₅₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

604

—

149

C₄

#18012728

yes

—

150

C₄

C₂×#14559005

yes

—

151

C₂

—

152

C₂

—

1208

—

153

C₂

C₂×#18012728

yes

characteristic

154

C₂

#23753010

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	C₂×#125×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	#23752762	C₂
hypercenter	C₂×#125	D₁₅₁
hypocenter	C₁₅₁	#3142
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

C₂×#125×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂²×D₄

C₂

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #23752582	C₂	semidirect
3. #23752578	C₂	semidirect
4. #23752569	C₂	semidirect
5. #23752211	C₂	semidirect
6. C₄×#14558988	C₂	semidirect
7. C₂×#125×C₁₅₁	C₂	semidirect
77. C₁₅₁	#3142	semidirect
80. C₂×#125	D₁₅₁	semidirect

H ≅ G/N

type

2. #23752582

C₂

semidirect

3. #23752578

C₂

semidirect

4. #23752569

C₂

semidirect

5. #23752211

C₂

semidirect

6. C₄×#14558988

C₂

semidirect

7. C₂×#125×C₁₅₁

C₂

semidirect

77. C₁₅₁

#3142

semidirect

80. C₂×#125

D₁₅₁

semidirect

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

subgroup	index
#23752762	2
#23752582	2
#23752578	2
#23752569	2
#23752211	2
C₄×#14558988	2
C₂×#125×C₁₅₁	2
#3142	151

subgroup

index

C₂×#125×C₁₅₁

151

Subgroups — #34079513

Conjugacy classes of subgroups

Complements

Maximal subgroups