Group #34070061 — subgroups

Subgroups — #34070061

Order 19328.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂×#18012739	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#23751975	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₀₂	—	2	1	no	—
6	#18012739	—	2	2	no	—
7	#18012739	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₅₁	—	2	1	no	—
9	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₅₁	—	2	1	no	—
12	C₁₅₁⋊#95	—	2	1	no	—
13	C₂²×C₈×C₁₅₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₅₁	—	2	2	no	—
16	C₂×Dic₃₀₂	—	2	4	no	—
17	C₂×Dic₃₀₂	—	4	2	no	—
18	C₂×Dic₃₀₂	—	4	2	no	—
19	C₂²×Dic₁₅₁	—	2	2	no	—
20	C₂²×Dic₁₅₁	—	2	2	no	—
21	C₄⋊Dic₁₅₁	—	2	4	no	—
22	C₄⋊Dic₁₅₁	—	2	2	no	—
23	C₄⋊Dic₁₅₁	—	2	2	no	—
24	C₄⋊Dic₁₅₁	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₅₁	Q₈	1	4	yes	complemented
26	C₄×Dic₁₅₁	—	4	2	no	—
27	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₅₁	—	2	2	no	—
31	C₂×C₈×C₁₅₁	—	2	4	no	—
32	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
33	C₄²×C₁₅₁	D₄	1	2	yes	—
34	Dic₃₀₂	—	4	8	no	—
35	C₂×Dic₁₅₁	—	2	4	no	—
36	C₂×Dic₁₅₁	—	2	4	no	—
37	C₂×Dic₁₅₁	—	4	2	no	—
38	C₂×Dic₁₅₁	—	4	2	no	—
39	C₂×Dic₁₅₁	—	4	2	no	—
40	C₂×Dic₁₅₁	—	4	2	no	—
41	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	—	2	2	no	—
43	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₅₁	D₈	1	2	yes	—
46	C₂×C₄×C₁₅₁	SD₁₆	1	2	yes	—
47	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₅₁	#41	1	2	yes	—
51	C₈×C₁₅₁	—	4	2	no	—
52	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₁₅₁	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₁₅₁	#137	1	2	yes	—
57	C₄×C₁₅₁	—	2	2	no	—
58	C₄×C₁₅₁	#102	1	4	yes	—
59	C₄×C₁₅₁	C₄×Q₈	1	2	yes	—
60	C₄×C₁₅₁	#124	1	1	yes	characteristic
61	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₁₅₁	—	4	4	no	—
63	Dic₁₅₁	—	4	4	no	—
64	C₂×C₁₅₁	#391	1	1	yes	characteristic
65	C₂×C₁₅₁	C₂×#102	1	1	yes	characteristic
66	C₂×C₁₅₁	#419	1	1	yes	characteristic
67	C₂×C₁₅₁	#474	1	2	yes	—
68	C₂×C₁₅₁	#475	1	2	yes	—
69	C₁₅₁	#1988	1	1	yes	characteristic, complemented, Sylow
70	#1988	—	151	1	no	maximal, Sylow
71	C₂×C₄⋊Q₈	—	151	1	no	—
72	C₂×C₄⋊C₈	D₁₅₁	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	151	1	no	—
74	#340	—	151	4	no	—
75	C₂²×Q₈	—	302	1	no	—
76	C₂²×C₈	C₂×D₁₅₁	1	2	yes	—
77	C₄⋊Q₈	—	302	2	no	—
78	C₄⋊Q₈	—	151	4	no	—
79	C₂×C₄⋊C₄	—	302	1	no	—
80	C₂×C₄⋊C₄	—	151	2	no	—
81	C₂×C₄⋊C₄	—	151	2	no	—
82	C₂×C₄²	—	302	1	no	—
83	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	#95	—	302	1	no	—
86	C₂×Q₈	—	302	4	no	—
87	C₂×Q₈	—	604	2	no	—
88	C₂×Q₈	—	604	2	no	—
89	C₂²×C₄	—	302	2	no	—
90	C₂²×C₄	—	302	2	no	—
91	C₂²×C₄	D₆₀₄	1	1	yes	characteristic
92	C₂²×C₄	D₆₀₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₄⋊C₄	—	302	4	no	—
97	C₄⋊C₄	—	302	2	no	—
98	C₄⋊C₄	—	302	2	no	—
99	C₄⋊C₄	—	151	4	no	—
100	C₄⋊C₄	—	151	4	no	—
101	C₄²	—	604	2	no	—
102	C₄²	C₄×D₁₅₁	1	2	yes	—
103	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
104	C₂³	C₂×D₆₀₄	1	1	yes	characteristic
105	Q₈	—	604	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	302	4	no	—
108	C₂×C₄	—	302	4	no	—
109	C₂×C₄	—	604	2	no	—
110	C₂×C₄	—	604	2	no	—
111	C₂×C₄	—	604	2	no	—
112	C₂×C₄	—	604	2	no	—
113	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	2	yes	—
114	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	2	yes	—
115	C₂×C₄	D₁₂₀₈	1	2	yes	—
116	C₂×C₄	#14558993	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₁₅₁	1	2	yes	—
120	C₂×C₄	#14559003	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₂²	C₂×(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
123	C₂²	#18012719	1	2	yes	—
124	C₂²	C₂×D₁₂₀₈	1	1	yes	characteristic
125	C₂²	C₂×#14558993	1	1	yes	characteristic
126	C₂²	#18012772	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	604	4	no	—
129	C₄	—	604	4	no	—
130	C₄	#18012661	1	4	yes	—
131	C₄	#18012704	1	2	yes	—
132	C₄	#18012726	1	1	yes	characteristic
133	C₄	#18012737	1	1	yes	characteristic
134	C₂	#23752117	1	1	yes	characteristic
135	C₂	C₂×#18012661	1	1	yes	characteristic
136	C₂	#23752222	1	1	yes	characteristic
137	C₂	#23752493	1	2	yes	—
138	C₂	#23752498	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012739

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₅₁

C₂

yes

maximal, characteristic

#23751975

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

—

#18012739

—

#18012739

C₄

yes

complemented

C₂×C₄⋊Dic₁₅₁

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×C₄×Dic₁₅₁

—

C₁₅₁⋊#95

—

C₂²×C₈×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₁

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂²×Dic₁₅₁

—

C₂²×Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂×C₄

yes

—

C₄⋊Dic₁₅₁

Q₈

yes

complemented

C₄×Dic₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

D₄

yes

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

D₈

yes

—

C₂×C₄×C₁₅₁

SD₁₆

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₁

C₂²⋊Q₈

yes

—

C₂²×C₁₅₁

C₂×D₈

yes

characteristic

C₂²×C₁₅₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₅₁

#137

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#102

yes

—

C₄×C₁₅₁

C₄×Q₈

yes

—

C₄×C₁₅₁

#124

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

—

C₂×C₁₅₁

#475

yes

—

C₁₅₁

#1988

yes

characteristic, complemented, Sylow

#1988

—

151

maximal, Sylow

C₂×C₄⋊Q₈

—

151

—

C₂×C₄⋊C₈

D₁₅₁

yes

characteristic

C₄×C₄⋊C₄

—

151

—

#340

—

151

—

C₂²×Q₈

—

302

—

C₂²×C₈

C₂×D₁₅₁

yes

—

C₄⋊Q₈

—

302

—

C₄⋊Q₈

—

151

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄²

—

302

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

#95

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

604

—

C₂×Q₈

—

604

—

C₂²×C₄

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

D₆₀₄

yes

characteristic

C₂²×C₄

D₆₀₄

yes

characteristic

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

151

—

100

C₄⋊C₄

—

151

—

101

C₄²

—

604

—

102

C₄²

C₄×D₁₅₁

yes

—

103

C₄²

C₂²⋊D₁₅₁

yes

—

104

C₂³

C₂×D₆₀₄

yes

characteristic

105

Q₈

—

604

—

106

C₂×C₄

—

107

C₂×C₄

—

302

—

108

C₂×C₄

—

302

—

109

C₂×C₄

—

604

—

110

C₂×C₄

—

604

—

111

C₂×C₄

—

604

—

112

C₂×C₄

—

604

—

113

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

114

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

115

C₂×C₄

D₁₂₀₈

yes

—

116

C₂×C₄

#14558993

yes

—

117

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

119

C₂×C₄

Q₈×D₁₅₁

yes

—

120

C₂×C₄

#14559003

yes

—

121

C₈

—

122

C₂²

C₂×(C₂×D₁₅₁)⋊C₄

yes

characteristic

123

C₂²

#18012719

yes

—

124

C₂²

C₂×D₁₂₀₈

yes

characteristic

125

C₂²

C₂×#14558993

yes

characteristic

126

C₂²

#18012772

yes

—

127

C₄

—

128

C₄

—

604

—

129

C₄

—

604

—

130

C₄

#18012661

yes

—

131

C₄

#18012704

yes

—

132

C₄

#18012726

yes

characteristic

133

C₄

#18012737

yes

characteristic

134

C₂

#23752117

yes

characteristic

135

C₂

C₂×#18012661

yes

characteristic

136

C₂

#23752222

yes

characteristic

137

C₂

#23752493

yes

—

138

C₂

#23752498

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₀₄
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₂×C₄⋊C₈×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅₁
hypocenter	C₁₅₁	#1988
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₆₀₄

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₂×C₄⋊C₈×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₅₁

hypocenter

C₁₅₁

#1988

LCS

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #18012739	C₄	semidirect
25. C₄⋊Dic₁₅₁	Q₈	semidirect
69. C₁₅₁	#1988	semidirect

H ≅ G/N

type

7. #18012739

C₄

semidirect

25. C₄⋊Dic₁₅₁

Q₈

semidirect

69. C₁₅₁

#1988

semidirect

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

subgroup	index
C₂×#18012739	2
C₄×C₄⋊Dic₁₅₁	2
#23751975	2
C₂×C₄⋊C₈×C₁₅₁	2
#1988	151

subgroup

index

C₂×#18012739

C₄×C₄⋊Dic₁₅₁

#23751975

C₂×C₄⋊C₈×C₁₅₁

#1988

151

Subgroups — #34070061

Conjugacy classes of subgroups

Complements

Maximal subgroups