Group #34068616 — subgroups

Subgroups — #34068616

Order 19328.

146 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 57 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	#23752610	C₂	1	1	yes	maximal, characteristic
2	#23752507	C₂	1	1	yes	maximal, characteristic
3	#23752422	C₂	1	1	yes	maximal, characteristic
4	#23751965	C₂	1	1	yes	maximal, characteristic
5	#23751960	C₂	1	1	yes	maximal, characteristic
6	#23751957	C₂	1	1	yes	maximal, characteristic
7	#446×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#14558994	—	2	1	no	—
9	#18012739	C₂²	1	1	yes	characteristic
10	#18012739	C₂²	1	1	yes	characteristic
11	#18012672	—	2	1	no	—
12	#18012671	—	2	1	no	—
13	#18012671	—	2	1	no	—
14	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
16	#18012663	—	2	1	no	—
17	#18012662	—	2	1	no	—
18	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₅₁	—	2	1	no	—
20	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	#97×C₁₅₁	—	2	1	no	—
22	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₀₂	—	2	1	no	—
24	C₂×Dic₃₀₂	—	2	1	no	—
25	#14558994	—	4	2	no	—
26	C₂×C₁₅₁⋊C₈	—	2	1	no	—
27	C₂×C₁₅₁⋊C₈	—	2	1	no	—
28	C₄⋊Dic₁₅₁	—	2	1	no	—
29	C₄⋊Dic₁₅₁	—	2	1	no	—
30	C₄⋊Dic₁₅₁	—	2	1	no	—
31	C₄⋊Dic₁₅₁	—	2	1	no	—
32	C₂×Q₈×C₁₅₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₅₁	—	2	1	no	—
34	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₅₁	—	2	1	no	—
37	C₄⋊C₄×C₁₅₁	D₄	1	1	yes	characteristic
38	C₄²×C₁₅₁	—	2	1	no	—
39	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
40	Dic₃₀₂	—	4	1	no	—
41	Dic₃₀₂	—	4	1	no	—
42	Dic₃₀₂	—	8	1	no	—
43	C₂×Dic₁₅₁	—	4	1	no	—
44	C₂×Dic₁₅₁	—	4	1	no	—
45	C₁₅₁⋊C₈	—	4	1	no	—
46	C₁₅₁⋊C₈	—	4	1	no	—
47	Q₈×C₁₅₁	—	2	1	no	—
48	Q₈×C₁₅₁	—	2	1	no	—
49	C₂×C₄×C₁₅₁	—	2	1	no	—
50	C₂×C₄×C₁₅₁	—	2	1	no	—
51	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
54	C₈×C₁₅₁	—	2	1	no	—
55	C₈×C₁₅₁	—	4	1	no	—
56	C₈×C₁₅₁	Q₁₆	1	2	yes	complemented
57	C₂²×C₁₅₁	#121	1	1	yes	characteristic
58	C₄×C₁₅₁	—	2	1	no	—
59	C₄×C₁₅₁	—	4	1	no	—
60	C₄×C₁₅₁	—	4	1	no	—
61	C₄×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₄×C₁₅₁	#137	1	1	yes	characteristic
64	C₄×C₁₅₁	#137	1	1	yes	characteristic
65	Dic₁₅₁	—	8	1	no	—
66	Dic₁₅₁	—	8	1	no	—
67	C₂×C₁₅₁	#462	1	1	yes	characteristic
68	C₂×C₁₅₁	#468	1	1	yes	characteristic
69	C₂×C₁₅₁	#471	1	1	yes	characteristic
70	C₁₅₁	#1692	1	1	yes	characteristic, complemented, Sylow
71	#1692	—	151	1	no	maximal, Sylow
72	#498	—	151	1	no	—
73	#477	—	151	1	no	—
74	#462	—	151	1	no	—
75	#446	D₁₅₁	1	1	yes	characteristic
76	#335	—	151	1	no	—
77	#333	—	151	1	no	—
78	#332	—	151	1	no	—
79	C₂×Q₁₆	—	302	1	no	—
80	C₄⋊Q₈	—	151	1	no	—
81	C₄⋊Q₈	—	151	1	no	—
82	C₄×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
83	#107	—	302	1	no	—
84	#106	—	302	1	no	—
85	#106	—	302	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	151	1	no	—
88	C₄⋊C₈	—	151	1	no	—
89	C₄⋊C₈	C₂×D₁₅₁	1	1	yes	characteristic
90	#103	—	302	1	no	—
91	#103	—	302	1	no	—
92	#97	—	2	1	no	—
93	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
94	C₂×Q₈	—	302	1	no	—
95	C₂×Q₈	—	302	1	no	—
96	C₂×Q₈	D₆₀₄	1	1	yes	characteristic
97	Q₁₆	—	604	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	302	1	no	—
100	C₂×C₈	—	302	1	no	—
101	C₂×C₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	302	1	no	—
105	C₄⋊C₄	—	302	1	no	—
106	C₄⋊C₄	—	302	1	no	—
107	C₄⋊C₄	—	302	1	no	—
108	C₄⋊C₄	D₆₀₄	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	604	1	no	—
114	Q₈	—	604	1	no	—
115	Q₈	—	1208	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	604	1	no	—
119	C₂×C₄	—	604	1	no	—
120	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
121	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
122	C₂×C₄	#14559004	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	604	1	no	—
126	C₈	—	604	1	no	—
127	C₈	#14558994	1	2	yes	complemented
128	C₂²	#18012713	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	1208	1	no	—
133	C₄	—	1208	1	no	—
134	C₄	C₂×#14558994	1	1	yes	characteristic
135	C₄	#18012767	1	1	yes	characteristic
136	C₄	#18012769	1	1	yes	characteristic
137	C₄	#18012772	1	1	yes	characteristic
138	C₂	#23752422	1	1	yes	characteristic
139	C₂	#23752458	1	1	yes	characteristic
140	C₂	#23752475	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752610

C₂

yes

maximal, characteristic

#23752507

C₂

yes

maximal, characteristic

#23752422

C₂

yes

maximal, characteristic

#23751965

C₂

yes

maximal, characteristic

#23751960

C₂

yes

maximal, characteristic

#23751957

C₂

yes

maximal, characteristic

#446×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558994

—

#18012739

C₂²

yes

characteristic

#18012739

C₂²

yes

—

—

—

C₄⋊(C₁₅₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₅₁⋊C₈)

C₂²

yes

characteristic

#18012663

—

#18012662

—

C₄×Q₈×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₁

—

C₄⋊C₈×C₁₅₁

C₂²

yes

characteristic

#97×C₁₅₁

—

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

#14558994

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂×Q₈×C₁₅₁

D₄

yes

characteristic

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

characteristic

C₂×C₈×C₁₅₁

D₄

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

D₄

yes

characteristic

C₄²×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

Dic₃₀₂

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

characteristic

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

Q₁₆

yes

complemented

C₂²×C₁₅₁

#121

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×Q₁₆

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#1692

—

151

maximal, Sylow

#498

—

151

—

#477

—

151

—

#462

—

151

—

#446

D₁₅₁

yes

characteristic

#335

—

151

—

#333

—

151

—

#332

—

151

—

C₂×Q₁₆

—

302

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

—

151

—

C₄×Q₈

C₂×D₁₅₁

yes

characteristic

#107

—

302

—

#106

—

302

—

#106

—

302

—

C₄⋊C₈

—

C₄⋊C₈

—

151

—

C₄⋊C₈

—

151

—

C₄⋊C₈

C₂×D₁₅₁

yes

characteristic

#103

—

302

—

#103

—

302

—

#97

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

D₆₀₄

yes

characteristic

Q₁₆

—

604

—

C₂×C₈

—

C₂×C₈

—

302

—

100

C₂×C₈

—

302

—

101

C₂×C₈

C₂²⋊D₁₅₁

yes

characteristic

102

C₂×C₈

C₂²⋊D₁₅₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

302

—

105

C₄⋊C₄

—

302

—

106

C₄⋊C₄

—

302

—

107

C₄⋊C₄

—

302

—

108

C₄⋊C₄

D₆₀₄

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₁₅₁

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

604

—

114

Q₈

—

604

—

115

Q₈

—

1208

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

604

—

119

C₂×C₄

—

604

—

120

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

121

C₂×C₄

C₂×D₆₀₄

yes

characteristic

122

C₂×C₄

#14559004

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

604

—

126

C₈

—

604

—

127

C₈

#14558994

yes

complemented

128

C₂²

#18012713

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

1208

—

133

C₄

—

1208

—

134

C₄

C₂×#14558994

yes

characteristic

135

C₄

#18012767

yes

characteristic

136

C₄

#18012769

yes

characteristic

137

C₄

#18012772

yes

characteristic

138

C₂

#23752422

yes

characteristic

139

C₂

#23752458

yes

characteristic

140

C₂

#23752475

yes

characteristic

	subgroup	G/H
center	C₂²	#18012713
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#446×C₁₅₁	C₂
socle, prime radical	C₂²×C₁₅₁	#121
hypercenter	#446	D₁₅₁
hypocenter	C₁₅₁	#1692
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#18012713

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#446×C₁₅₁

C₂

C₂²×C₁₅₁

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
56. C₈×C₁₅₁	Q₁₆	semidirect
70. C₁₅₁	#1692	semidirect
127. C₈	#14558994	semidirect

H ≅ G/N

type

56. C₈×C₁₅₁

Q₁₆

semidirect

70. C₁₅₁

#1692

semidirect

127. C₈

#14558994

semidirect

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

subgroup	index
#23752610	2
#23752507	2
#23752422	2
#23751965	2
#23751960	2
#23751957	2
#446×C₁₅₁	2
#1692	151

subgroup

index

#446×C₁₅₁

151

Subgroups — #34068616

Conjugacy classes of subgroups

Complements

Maximal subgroups