Group #34079633 — subgroups

Subgroups — #34079633

Order 19328.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#23752765	C₂	1	2	yes	maximal
2	#23752620	C₂	1	4	yes	maximal, complemented
3	#23752619	C₂	1	4	yes	maximal, complemented
4	#23752250	C₂	1	1	yes	maximal, characteristic
5	#23752248	C₂	1	2	yes	maximal
6	C₂×#18012649	C₂	1	1	yes	maximal, characteristic
7	#427×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
9	#18012738	C₂²	1	4	yes	—
10	#18012730	C₂²	1	4	yes	—
11	#18012716	—	2	4	no	—
12	C₄×Dic₃₀₂	—	2	4	no	—
13	#18012698	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
15	#18012673	C₂²	1	2	yes	—
16	#18012672	C₂²	1	4	yes	—
17	#18012671	C₂²	1	2	yes	—
18	#18012670	C₂²	1	4	yes	—
19	#18012649	C₂²	1	2	yes	—
20	#18012649	C₂²	1	2	yes	—
21	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
22	#117×C₁₅₁	C₂²	1	2	yes	—
23	#107×C₁₅₁	C₂²	1	2	yes	—
24	#106×C₁₅₁	C₂²	1	2	yes	—
25	C₂×Dic₃₀₂	—	2	4	no	—
26	C₂²×Dic₁₅₁	D₄	1	2	yes	—
27	C₂×C₁₅₁⋊C₈	Q₈	1	4	yes	complemented
28	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₁₅₁	—	2	4	no	—
30	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
31	Dic₁₅₁⋊C₄	—	2	4	no	—
32	Dic₁₅₁⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₁₅₁	—	2	2	no	—
34	C₄×Dic₁₅₁	—	2	2	no	—
35	C₄×Dic₁₅₁	D₄	1	2	yes	—
36	C₄×Dic₁₅₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₅₁	Q₈	1	4	yes	complemented
39	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
43	C₄²×C₁₅₁	—	2	2	no	—
44	Dic₃₀₂	—	4	8	no	—
45	C₂×Dic₁₅₁	—	2	4	no	—
46	C₂×Dic₁₅₁	—	4	2	no	—
47	C₂×Dic₁₅₁	C₂×D₄	1	4	yes	—
48	C₁₅₁⋊C₈	C₂×Q₈	1	4	yes	—
49	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₁	—	2	4	no	—
51	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
55	C₈×C₁₅₁	C₂×Q₈	1	2	yes	—
56	C₈×C₁₅₁	C₂×Q₈	1	2	yes	—
57	C₂²×C₁₅₁	—	2	2	no	—
58	C₂²×C₁₅₁	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₁₅₁	—	4	4	no	—
61	C₄×C₁₅₁	C₄⋊Q₈	1	2	yes	—
62	C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
64	Dic₁₅₁	—	2	4	no	—
65	Dic₁₅₁	—	4	4	no	—
66	C₂×C₁₅₁	—	2	2	no	—
67	C₂×C₁₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₁₅₁	#575	1	2	yes	—
69	C₁₅₁	#3162	1	1	yes	characteristic, complemented, Sylow
70	#3162	—	151	1	no	maximal, Sylow
71	#533	—	151	2	no	—
72	#501	—	151	4	no	—
73	#501	—	151	4	no	—
74	#427	—	151	1	no	—
75	#427	—	151	2	no	—
76	#427	D₁₅₁	1	1	yes	characteristic
77	C₂×#97	—	151	1	no	—
78	C₂²×C₈	—	151	1	no	—
79	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
80	C₄⋊Q₈	—	151	4	no	—
81	#125	—	151	4	no	—
82	C₂²⋊Q₈	—	302	4	no	—
83	C₄×Q₈	—	302	4	no	—
84	#117	—	151	2	no	—
85	#117	C₂×D₁₅₁	1	2	yes	—
86	C₂×C₄²	—	151	1	no	—
87	#107	—	151	2	no	—
88	#107	—	151	4	no	—
89	#107	C₂×D₁₅₁	1	2	yes	—
90	#106	—	151	2	no	—
91	#106	—	151	4	no	—
92	#106	C₂×D₁₅₁	1	2	yes	—
93	#97	—	151	2	no	—
94	#97	—	151	2	no	—
95	C₂×Q₈	—	302	4	no	—
96	C₂²×C₄	—	151	2	no	—
97	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
98	C₂×C₈	—	151	4	no	—
99	C₂×C₈	—	151	2	no	—
100	C₂×C₈	Dic₃₀₂	1	4	yes	complemented
101	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
103	C₄⋊C₄	—	302	4	no	—
104	C₄⋊C₄	—	302	4	no	—
105	C₄⋊C₄	—	302	4	no	—
106	C₄⋊C₄	—	151	4	no	—
107	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	302	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	302	2	no	—
112	C₄²	—	151	2	no	—
113	C₄²	—	151	2	no	—
114	C₂³	D₄×D₁₅₁	1	1	yes	characteristic
115	Q₈	—	604	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	302	4	no	—
118	C₂×C₄	—	604	2	no	—
119	C₂×C₄	—	151	4	no	—
120	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₁₅₁	1	2	yes	—
122	C₂×C₄	C₂×Dic₃₀₂	1	2	yes	—
123	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
124	C₈	—	151	4	no	—
125	C₈	C₂×Dic₃₀₂	1	2	yes	—
126	C₈	C₂×Dic₃₀₂	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#18012738	1	2	yes	—
129	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	302	4	no	—
132	C₄	—	604	4	no	—
133	C₄	#18012738	1	2	yes	—
134	C₄	C₂×Q₈×D₁₅₁	1	1	yes	characteristic
135	C₄	C₂²×Dic₃₀₂	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#18012738	1	1	yes	characteristic
138	C₂	#23753009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752765

C₂

yes

maximal

#23752620

C₂

yes

maximal, complemented

#23752619

C₂

yes

maximal, complemented

#23752250

C₂

yes

maximal, characteristic

#23752248

C₂

yes

maximal

C₂×#18012649

C₂

yes

maximal, characteristic

#427×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

characteristic

#18012738

C₂²

yes

—

#18012730

C₂²

yes

—

#18012716

—

C₄×Dic₃₀₂

—

#18012698

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

#18012673

C₂²

yes

—

#18012672

C₂²

yes

—

#18012671

C₂²

yes

—

#18012670

C₂²

yes

—

#18012649

C₂²

yes

—

#18012649

C₂²

yes

—

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

#117×C₁₅₁

C₂²

yes

—

#107×C₁₅₁

C₂²

yes

—

#106×C₁₅₁

C₂²

yes

—

C₂×Dic₃₀₂

—

C₂²×Dic₁₅₁

D₄

yes

—

C₂×C₁₅₁⋊C₈

Q₈

yes

complemented

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂³

yes

—

Dic₁₅₁⋊C₄

—

Dic₁₅₁⋊C₄

—

C₂²⋊Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

D₄

yes

—

C₄×Dic₁₅₁

C₂³

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

Q₈

yes

complemented

C₂×C₈×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×D₄

yes

—

C₁₅₁⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₈×C₁₅₁

C₂×Q₈

yes

—

C₈×C₁₅₁

C₂×Q₈

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₄⋊Q₈

yes

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

C₄⋊Q₈

yes

—

C₄×C₁₅₁

C₂²×Q₈

yes

characteristic

C₄×C₁₅₁

C₂²×Q₈

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₁

#575

yes

—

C₁₅₁

#3162

yes

characteristic, complemented, Sylow

#3162

—

151

maximal, Sylow

#533

—

151

—

#501

—

151

—

#501

—

151

—

#427

—

151

—

#427

—

151

—

#427

D₁₅₁

yes

characteristic

C₂×#97

—

151

—

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

C₄⋊Q₈

—

151

—

#125

—

151

—

C₂²⋊Q₈

—

302

—

C₄×Q₈

—

302

—

#117

—

151

—

#117

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

#107

—

151

—

#107

—

151

—

#107

C₂×D₁₅₁

yes

—

#106

—

151

—

#106

—

151

—

#106

C₂×D₁₅₁

yes

—

#97

—

151

—

#97

—

151

—

C₂×Q₈

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

151

—

C₂×C₈

—

151

—

100

C₂×C₈

Dic₃₀₂

yes

complemented

101

C₂×C₈

C₂²×D₁₅₁

yes

characteristic

102

C₂×C₈

C₂²×D₁₅₁

yes

characteristic

103

C₄⋊C₄

—

302

—

104

C₄⋊C₄

—

302

—

105

C₄⋊C₄

—

302

—

106

C₄⋊C₄

—

151

—

107

C₄⋊C₄

C₂²×D₁₅₁

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

302

—

110

C₄²

—

111

C₄²

—

302

—

112

C₄²

—

151

—

113

C₄²

—

151

—

114

C₂³

D₄×D₁₅₁

yes

characteristic

115

Q₈

—

604

—

116

C₂×C₄

—

117

C₂×C₄

—

302

—

118

C₂×C₄

—

604

—

119

C₂×C₄

—

151

—

120

C₂×C₄

D₄×D₁₅₁

yes

characteristic

121

C₂×C₄

Q₈×D₁₅₁

yes

—

122

C₂×C₄

C₂×Dic₃₀₂

yes

—

123

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

124

C₈

—

151

—

125

C₈

C₂×Dic₃₀₂

yes

—

126

C₈

C₂×Dic₃₀₂

yes

—

127

C₂²

—

128

C₂²

#18012738

yes

—

129

C₂²

C₂×D₄×D₁₅₁

yes

characteristic

130

C₄

—

131

C₄

—

302

—

132

C₄

—

604

—

133

C₄

#18012738

yes

—

134

C₄

C₂×Q₈×D₁₅₁

yes

characteristic

135

C₄

C₂²×Dic₃₀₂

yes

characteristic

136

C₂

—

137

C₂

C₂×#18012738

yes

characteristic

138

C₂

#23753009

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#427×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂³×C₁₅₁	C₂×D₄
hypercenter	#427	D₁₅₁
hypocenter	C₁₅₁	#3162
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

#427×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂²×D₄

prime radical

C₂³×C₁₅₁

C₂×D₄

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #23752620	C₂	semidirect
3. #23752619	C₂	semidirect
27. C₂×C₁₅₁⋊C₈	Q₈	semidirect
38. C₂×C₈×C₁₅₁	Q₈	semidirect
69. C₁₅₁	#3162	semidirect
100. C₂×C₈	Dic₃₀₂	semidirect

H ≅ G/N

type

2. #23752620

C₂

semidirect

3. #23752619

C₂

semidirect

27. C₂×C₁₅₁⋊C₈

Q₈

semidirect

38. C₂×C₈×C₁₅₁

Q₈

semidirect

69. C₁₅₁

#3162

semidirect

100. C₂×C₈

Dic₃₀₂

semidirect

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

subgroup	index
#23752765	2
#23752620	2
#23752619	2
#23752250	2
#23752248	2
C₂×#18012649	2
#427×C₁₅₁	2
#3162	151

subgroup

index

#427×C₁₅₁

151

Subgroups — #34079633

Conjugacy classes of subgroups

Complements

Maximal subgroups