Group #34070098 — subgroups

Subgroups — #34070098

Order 19328.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14558988	C₂	1	1	yes	maximal, characteristic
3	#23751980	C₂	1	4	yes	maximal
4	#423×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₀₂	C₄	1	2	yes	—
6	C₂×#14558988	C₂²	1	2	yes	—
7	#18012739	—	2	1	no	—
8	#18012739	—	2	1	no	—
9	#18012739	—	2	1	no	—
10	#18012739	—	2	1	no	—
11	C₂×C₄⋊Dic₁₅₁	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
13	#18012650	—	2	1	no	—
14	C₄×C₁₅₁⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₁₅₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₅₁	—	2	1	no	—
18	C₄⋊C₈×C₁₅₁	—	2	1	no	—
19	C₂×Dic₃₀₂	—	2	2	no	—
20	C₂×Dic₃₀₂	—	2	2	no	—
21	C₂×Dic₃₀₂	—	4	4	no	—
22	C₂²×Dic₁₅₁	C₂×C₄	1	4	yes	—
23	#14558988	—	2	2	no	—
24	#14558988	Q₈	1	4	yes	complemented
25	C₂×C₁₅₁⋊C₈	—	2	2	no	—
26	C₄⋊Dic₁₅₁	—	2	2	no	—
27	C₄⋊Dic₁₅₁	—	2	2	no	—
28	C₄⋊Dic₁₅₁	—	2	2	no	—
29	C₄⋊Dic₁₅₁	—	2	2	no	—
30	C₂²×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₅₁	—	2	4	no	—
34	C₂×C₈×C₁₅₁	—	2	2	no	—
35	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
36	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
37	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
38	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
39	Dic₃₀₂	—	4	4	no	—
40	Dic₃₀₂	—	4	4	no	—
41	C₂×Dic₁₅₁	—	2	4	no	—
42	C₂×Dic₁₅₁	—	4	4	no	—
43	C₁₅₁⋊C₈	—	2	4	no	—
44	C₂³×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₁	—	2	2	no	—
46	C₂×C₄×C₁₅₁	—	2	2	no	—
47	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₅₁	#41	1	2	yes	—
55	C₈×C₁₅₁	—	4	2	no	—
56	C₂²×C₁₅₁	—	2	1	no	—
57	C₂²×C₁₅₁	—	2	1	no	—
58	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₅₁	C₄×Q₈	1	2	yes	—
60	C₄×C₁₅₁	—	2	1	no	—
61	C₄×C₁₅₁	—	2	1	no	—
62	C₄×C₁₅₁	#100	1	2	yes	—
63	C₄×C₁₅₁	#101	1	2	yes	—
64	C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₅₁	#124	1	1	yes	characteristic
66	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₅₁	—	4	4	no	—
68	C₂×C₁₅₁	—	2	2	no	—
69	C₂×C₁₅₁	#391	1	1	yes	characteristic
70	C₂×C₁₅₁	C₂×#100	1	1	yes	characteristic
71	C₂×C₁₅₁	C₂×#101	1	1	yes	characteristic
72	C₁₅₁	#1997	1	1	yes	characteristic, complemented, Sylow
73	#1997	—	151	1	no	maximal, Sylow
74	C₂×C₄⋊Q₈	—	151	1	no	—
75	#423	D₁₅₁	1	1	yes	characteristic
76	C₄×M₁₆	—	151	1	no	—
77	#343	—	151	4	no	—
78	C₂²×Q₈	—	151	2	no	—
79	C₂×M₁₆	—	151	2	no	—
80	C₂×M₁₆	C₂×D₁₅₁	1	2	yes	—
81	C₄⋊Q₈	—	302	1	no	—
82	C₄⋊Q₈	—	302	1	no	—
83	C₄⋊Q₈	—	302	1	no	—
84	C₄⋊Q₈	—	302	1	no	—
85	C₂×C₄⋊C₄	—	151	2	no	—
86	C₂×C₄⋊C₄	—	151	2	no	—
87	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	302	1	no	—
91	C₄×C₈	—	302	1	no	—
92	C₂×Q₈	—	302	2	no	—
93	C₂×Q₈	—	302	2	no	—
94	C₂×Q₈	—	604	4	no	—
95	C₂²×C₄	—	151	4	no	—
96	C₂²×C₄	C₄×D₁₅₁	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₁₅₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	302	2	no	—
101	M₁₆	—	151	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	302	2	no	—
104	C₄⋊C₄	—	302	2	no	—
105	C₄⋊C₄	—	302	2	no	—
106	C₄⋊C₄	—	302	2	no	—
107	C₄⋊C₄	—	302	2	no	—
108	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
109	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
110	C₄²	D₆₀₄	1	1	yes	characteristic
111	C₄²	D₆₀₄	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₁₅₁	1	1	yes	characteristic
113	Q₈	—	604	4	no	—
114	Q₈	—	604	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	302	4	no	—
118	C₂×C₄	—	604	4	no	—
119	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
124	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₁₅₁	1	2	yes	—
126	C₂×C₄	#14559003	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₈	—	302	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
132	C₂²	#18012704	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	604	4	no	—
136	C₄	#18012655	1	2	yes	—
137	C₄	#18012657	1	2	yes	—
138	C₄	#18012719	1	2	yes	—
139	C₄	#18012726	1	1	yes	characteristic
140	C₄	#18012737	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#23752117	1	1	yes	characteristic
143	C₂	C₂×#18012655	1	1	yes	characteristic
144	C₂	C₂×#18012657	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012739

C₂

yes

maximal, characteristic

C₄×#14558988

C₂

yes

maximal, characteristic

#23751980

C₂

yes

maximal

#423×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

C₄

yes

—

C₂×#14558988

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₁₅₁

C₄

yes

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

#18012650

—

C₄×C₁₅₁⋊C₈

—

C₂×M₁₆×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₁

—

C₄⋊C₈×C₁₅₁

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂²×Dic₁₅₁

C₂×C₄

yes

—

#14558988

—

#14558988

Q₈

yes

complemented

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₄²×C₁₅₁

D₄

yes

characteristic

C₄²×C₁₅₁

D₄

yes

characteristic

C₄²×C₁₅₁

D₄

yes

characteristic

C₄²×C₁₅₁

D₄

yes

characteristic

Dic₃₀₂

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₁

C₄×Q₈

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#100

yes

—

C₄×C₁₅₁

#101

yes

—

C₄×C₁₅₁

C₂²⋊Q₈

yes

—

C₄×C₁₅₁

#124

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

characteristic

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#1997

—

151

maximal, Sylow

C₂×C₄⋊Q₈

—

151

—

#423

D₁₅₁

yes

characteristic

C₄×M₁₆

—

151

—

#343

—

151

—

C₂²×Q₈

—

151

—

C₂×M₁₆

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

—

C₄⋊Q₈

—

302

—

C₄⋊Q₈

—

302

—

C₄⋊Q₈

—

302

—

C₄⋊Q₈

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

302

—

C₄×C₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

604

—

C₂²×C₄

—

151

—

C₂²×C₄

C₄×D₁₅₁

yes

characteristic

C₂²×C₄

C₄×D₁₅₁

yes

characteristic

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

M₁₆

—

100

M₁₆

—

302

—

101

M₁₆

—

151

—

102

C₂×C₈

—

103

C₂×C₈

—

302

—

104

C₄⋊C₄

—

302

—

105

C₄⋊C₄

—

302

—

106

C₄⋊C₄

—

302

—

107

C₄⋊C₄

—

302

—

108

C₄²

C₂²⋊D₁₅₁

yes

characteristic

109

C₄²

C₂²⋊D₁₅₁

yes

characteristic

110

C₄²

D₆₀₄

yes

characteristic

111

C₄²

D₆₀₄

yes

characteristic

112

C₂³

C₂×C₄×D₁₅₁

yes

characteristic

113

Q₈

—

604

—

114

Q₈

—

604

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

302

—

118

C₂×C₄

—

604

—

119

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

124

C₂×C₄

C₂×D₆₀₄

yes

characteristic

125

C₂×C₄

Q₈×D₁₅₁

yes

—

126

C₂×C₄

#14559003

yes

—

127

C₈

—

128

C₈

—

302

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×(C₂×D₁₅₁)⋊C₄

yes

characteristic

132

C₂²

#18012704

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

604

—

136

C₄

#18012655

yes

—

137

C₄

#18012657

yes

—

138

C₄

#18012719

yes

—

139

C₄

#18012726

yes

characteristic

140

C₄

#18012737

yes

characteristic

141

C₂

—

142

C₂

#23752117

yes

characteristic

143

C₂

C₂×#18012655

yes

characteristic

144

C₂

C₂×#18012657

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₅₁)⋊C₄
commutator, prime radical	C₂³×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#423×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#423	D₁₅₁
hypocenter	C₁₅₁	#1997
LCS	G ▷ C₂³×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₅₁)⋊C₄

commutator, prime radical

C₂³×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#423×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂³×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
24. #14558988	Q₈	semidirect
72. C₁₅₁	#1997	semidirect

H ≅ G/N

type

24. #14558988

Q₈

semidirect

72. C₁₅₁

#1997

semidirect

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

subgroup	index
C₂×#18012739	2
C₄×#14558988	2
#23751980	2
#423×C₁₅₁	2
#1997	151

subgroup

index

#423×C₁₅₁

151

Subgroups — #34070098

Conjugacy classes of subgroups

Complements

Maximal subgroups