Group #14518081 — subgroups

Subgroups — #14518081

Order 1944.

237 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 37 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	#11757844	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)	—	3	1	no	maximal
3	#10619980	—	3	1	no	maximal
4	C₉⋊((C₂²⋊C₉)⋊C₂)	C₃	1	1	yes	maximal, characteristic, complemented
5	#10619746	—	3	2	no	maximal
6	#92309	—	4	1	no	maximal
7	C₂²×C₃×3₋¹⁺²	D₃	1	1	yes	characteristic
8	C₂×C₃⋊AGL(1,ℤ/9)	—	3	1	no	—
9	C₃×#1094	D₃	1	1	yes	characteristic
10	C₃⋊(3₋¹⁺²⋊C₄)	—	3	1	no	—
11	C₃²⋊(C₂²⋊C₉)	D₃	1	2	yes	—
12	C₉⋊(C₂²⋊C₉)	—	2	1	no	—
13	(C₂²⋊C₉)⋊C₉	—	2	1	no	—
14	(C₂²⋊C₉)⋊C₉	—	2	2	no	—
15	C₉×C₂²⋊C₉	C₂×C₃	1	1	yes	characteristic, complemented
16	#6896	—	4	1	no	Sylow
17	C₃×#638	—	9	1	no	—
18	C₃⋊((C₂²⋊C₉)⋊C₂)	—	3	1	no	—
19	C₃⋊((C₂²⋊C₉)⋊C₂)	—	3	2	no	—
20	((C₂²⋊C₉)⋊C₂)⋊C₃	—	9	3	no	—
21	#6233	—	3	1	no	—
22	#611⋊C₃	—	9	3	no	—
23	C₂×C₃×3₋¹⁺²	—	3	1	no	—
24	C₃⋊AGL(1,ℤ/9)	—	6	1	no	—
25	C₃⋊AGL(1,ℤ/9)	—	12	1	no	—
26	C₉⋊D₉	—	4	1	no	—
27	#4169	—	12	2	no	—
28	C₂²×C₃³	C₃⋊D₃	1	1	yes	characteristic
29	C₂×C₃×C₃⋊D₃	—	9	1	no	—
30	C₃×C₃⋊Dic₃	—	9	1	no	—
31	C₂²×3₋¹⁺²	—	3	3	no	—
32	C₂²×C₃×C₉	—	2	1	no	—
33	C₂²×C₃×C₉	C₃×D₃	1	1	yes	characteristic
34	C₂×C₃⋊D₉	—	3	1	no	—
35	C₂×AGL(1,ℤ/9)	—	9	3	no	—
36	#1094	—	3	3	no	—
37	C₃×C₂²⋊C₉	—	2	2	no	—
38	C₃×C₂²⋊C₉	—	2	1	no	—
39	C₃×C₂²⋊C₉	C₃×D₃	1	2	yes	—
40	C₃×C₂²⋊C₉	C₃×D₃	1	1	yes	characteristic
41	C₃⋊Dic₉	—	3	1	no	—
42	3₋¹⁺²⋊C₄	—	9	3	no	—
43	C₃×3₋¹⁺²	—	4	1	no	—
44	C₃×3₋¹⁺²	S₄	1	1	yes	characteristic
45	C₉⋊C₉	—	8	1	no	—
46	C₉⋊C₉	—	8	2	no	—
47	C₉⋊C₉	—	8	1	no	—
48	C₃²⋊C₉	—	4	2	no	—
49	C₉²	—	4	1	no	—
50	#638	—	9	1	no	—
51	C₃×#67	—	27	1	no	—
52	C₃×#67	—	27	1	no	—
53	C₃×#67	—	27	2	no	—
54	(C₂²⋊C₉)⋊C₂	—	9	3	no	—
55	(C₂²⋊C₉)⋊C₂	—	27	2	no	—
56	#611	—	9	3	no	—
57	C₂×C₃³	—	3	1	no	—
58	C₃×C₃⋊D₃	—	18	1	no	—
59	C₂×3₋¹⁺²	—	3	3	no	—
60	C₂×3₋¹⁺²	—	6	3	no	—
61	C₂×C₃×C₉	—	3	1	no	—
62	C₂×C₃×C₉	—	6	1	no	—
63	C₃⋊D₉	—	6	1	no	—
64	C₃⋊D₉	—	12	1	no	—
65	C₃⋊D₉	—	12	2	no	—
66	AGL(1,ℤ/9)	—	18	3	no	—
67	AGL(1,ℤ/9)	—	36	3	no	—
68	C₂²×C₃²	—	3	1	no	—
69	C₂²×C₃²	—	3	1	no	—
70	C₂²×C₃²	—	3	2	no	—
71	C₂²×C₃²	C₃×C₃⋊D₃	1	1	yes	characteristic
72	C₂×C₃⋊D₃	—	9	1	no	—
73	C₂×C₃×D₃	—	27	1	no	—
74	C₂×C₃×D₃	—	27	1	no	—
75	C₂×C₃×D₃	—	27	2	no	—
76	C₃⋊Dic₃	—	9	1	no	—
77	C₃×Dic₃	—	27	1	no	—
78	C₃×Dic₃	—	27	1	no	—
79	C₃×Dic₃	—	27	2	no	—
80	C₂²×C₉	—	6	1	no	—
81	C₂²×C₉	AGL(1,ℤ/9)	1	3	yes	complemented
82	C₂×D₉	—	9	3	no	—
83	C₂²⋊C₉	—	3	2	no	—
84	C₂²⋊C₉	—	6	1	no	—
85	C₂²⋊C₉	—	6	2	no	—
86	C₂²⋊C₉	AGL(1,ℤ/9)	1	3	yes	complemented
87	Dic₉	—	9	3	no	—
88	C₃³	C₃⋊S₄	1	1	yes	characteristic
89	3₋¹⁺²	—	3	3	no	—
90	3₋¹⁺²	—	12	3	no	—
91	C₃×C₉	—	2	1	no	—
92	C₃×C₉	—	4	1	no	—
93	C₃×C₉	—	4	2	no	—
94	C₃×C₉	—	8	2	no	—
95	C₃×C₉	—	8	1	no	—
96	C₃×C₉	C₃×S₄	1	1	yes	characteristic
97	C₃×D₄	—	81	1	no	—
98	#67	—	27	1	no	—
99	#67	—	27	1	no	—
100	#67	—	27	2	no	—
101	C₂×C₃²	—	3	1	no	—
102	C₂×C₃²	—	3	1	no	—
103	C₂×C₃²	—	6	1	no	—
104	C₂×C₃²	—	9	2	no	—
105	C₂×C₃²	—	9	1	no	—
106	C₃⋊D₃	—	18	1	no	—
107	C₃×D₃	—	54	1	no	—
108	C₃×D₃	—	54	1	no	—
109	C₃×D₃	—	54	2	no	—
110	C₂×C₉	—	3	3	no	—
111	C₂×C₉	—	6	3	no	—
112	D₉	—	18	3	no	—
113	D₉	—	36	3	no	—
114	D₉	—	108	2	no	—
115	C₂²×C₃	—	9	1	no	—
116	C₂²×C₃	—	81	1	no	—
117	C₂²×C₃	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
118	C₂²×C₃	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
119	C₂²×C₃	#4207	1	2	yes	—
120	C₂×D₃	—	27	1	no	—
121	C₂×D₃	—	27	2	no	—
122	C₂×D₃	—	27	1	no	—
123	C₃×C₄	—	81	1	no	—
124	Dic₃	—	27	1	no	—
125	Dic₃	—	27	2	no	—
126	Dic₃	—	27	1	no	—
127	C₃²	—	3	2	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	C₃×C₃⋊S₄	1	1	yes	characteristic
131	C₉	—	4	3	no	—
132	C₉	—	6	1	no	—
133	C₉	—	12	2	no	—
134	C₉	—	24	2	no	—
135	C₉	—	24	1	no	—
136	C₉	((C₂²⋊C₉)⋊C₂)⋊C₃	1	3	yes	complemented
137	D₄	—	81	1	no	Sylow
138	C₂×C₃	—	3	2	no	—
139	C₂×C₃	—	3	1	no	—
140	C₂×C₃	—	3	1	no	—
141	C₂×C₃	—	9	1	no	—
142	C₂×C₃	—	18	1	no	—
143	C₂×C₃	—	162	1	no	—
144	D₃	—	54	2	no	—
145	D₃	—	54	1	no	—
146	D₃	—	54	1	no	—
147	C₂²	—	81	1	no	—
148	C₂²	#92309	1	1	yes	characteristic, complemented
149	C₄	—	81	1	no	—
150	C₃	—	9	1	no	—
151	C₃	#66440⋊C₂	1	2	yes	—
152	C₃	A₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
153	C₃	C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)	1	1	yes	characteristic
154	C₂	—	3	1	no	—
155	C₂	—	162	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11757844

C₂

yes

maximal, characteristic, complemented

C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)

—

maximal

#10619980

—

maximal

C₉⋊((C₂²⋊C₉)⋊C₂)

C₃

yes

maximal, characteristic, complemented

—

—

C₂²×C₃×3₋¹⁺²

D₃

yes

characteristic

C₂×C₃⋊AGL(1,ℤ/9)

—

C₃×#1094

D₃

yes

characteristic

C₃⋊(3₋¹⁺²⋊C₄)

—

C₃²⋊(C₂²⋊C₉)

D₃

yes

—

C₉⋊(C₂²⋊C₉)

—

(C₂²⋊C₉)⋊C₉

—

(C₂²⋊C₉)⋊C₉

—

C₉×C₂²⋊C₉

C₂×C₃

yes

characteristic, complemented

#6896

—

Sylow

C₃×#638

—

C₃⋊((C₂²⋊C₉)⋊C₂)

—

C₃⋊((C₂²⋊C₉)⋊C₂)

—

((C₂²⋊C₉)⋊C₂)⋊C₃

—

#6233

—

#611⋊C₃

—

C₂×C₃×3₋¹⁺²

—

C₃⋊AGL(1,ℤ/9)

—

C₃⋊AGL(1,ℤ/9)

—

C₉⋊D₉

—

#4169

—

C₂²×C₃³

C₃⋊D₃

yes

characteristic

C₂×C₃×C₃⋊D₃

—

C₃×C₃⋊Dic₃

—

C₂²×3₋¹⁺²

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₃×D₃

yes

characteristic

C₂×C₃⋊D₉

—

C₂×AGL(1,ℤ/9)

—

#1094

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

C₃×D₃

yes

—

C₃×C₂²⋊C₉

C₃×D₃

yes

characteristic

C₃⋊Dic₉

—

3₋¹⁺²⋊C₄

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

S₄

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

#638

—

C₃×#67

—

C₃×#67

—

C₃×#67

—

(C₂²⋊C₉)⋊C₂

—

(C₂²⋊C₉)⋊C₂

—

#611

—

C₂×C₃³

—

C₃×C₃⋊D₃

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₃⋊D₉

—

C₃⋊D₉

—

C₃⋊D₉

—

AGL(1,ℤ/9)

—

AGL(1,ℤ/9)

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×C₃⋊D₃

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

AGL(1,ℤ/9)

yes

complemented

C₂×D₉

—

C₂²⋊C₉

—

C₂²⋊C₉

—

C₂²⋊C₉

—

C₂²⋊C₉

AGL(1,ℤ/9)

yes

complemented

Dic₉

—

C₃³

C₃⋊S₄

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×S₄

yes

characteristic

C₃×D₄

—

#67

—

#67

—

100

#67

—

101

C₂×C₃²

—

102

C₂×C₃²

—

103

C₂×C₃²

—

104

C₂×C₃²

—

105

C₂×C₃²

—

106

C₃⋊D₃

—

107

C₃×D₃

—

108

C₃×D₃

—

109

C₃×D₃

—

110

C₂×C₉

—

111

C₂×C₉

—

112

D₉

—

113

D₉

—

114

D₉

—

108

—

115

C₂²×C₃

—

116

C₂²×C₃

—

117

C₂²×C₃

C₃⋊AGL(1,ℤ/9)

yes

characteristic

118

C₂²×C₃

C₃⋊AGL(1,ℤ/9)

yes

characteristic

119

C₂²×C₃

#4207

yes

—

120

C₂×D₃

—

121

C₂×D₃

—

122

C₂×D₃

—

123

C₃×C₄

—

124

Dic₃

—

125

Dic₃

—

126

Dic₃

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

C₃×C₃⋊S₄

yes

characteristic

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

((C₂²⋊C₉)⋊C₂)⋊C₃

yes

complemented

137

D₄

—

Sylow

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

—

143

C₂×C₃

—

162

—

144

D₃

—

145

D₃

—

146

D₃

—

147

C₂²

—

148

C₂²

#92309

yes

characteristic, complemented

149

C₄

—

150

C₃

—

151

C₃

#66440⋊C₂

yes

—

152

C₃

A₄⋊AGL(1,ℤ/9)

yes

characteristic

153

C₃

C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)

yes

characteristic

154

C₂

—

155

C₂

—

162

—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉×C₂²⋊C₉	C₂×C₃
Frattini	C₃²	C₃×C₃⋊S₄
Fitting	C₂²×C₃×3₋¹⁺²	D₃
socle	C₂²×C₃²	C₃×C₃⋊D₃
LCS	G ▷ C₉×C₂²⋊C₉
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, cosocle, hypocenter

C₉×C₂²⋊C₉

C₂×C₃

Frattini

C₃²

C₃×C₃⋊S₄

Fitting

C₂²×C₃×3₋¹⁺²

D₃

socle

C₂²×C₃²

C₃×C₃⋊D₃

LCS

G ▷ C₉×C₂²⋊C₉

UCS

C₁

N	H ≅ G/N	type
1. #11757844	C₂	semidirect
4. C₉⋊((C₂²⋊C₉)⋊C₂)	C₃	semidirect
15. C₉×C₂²⋊C₉	C₂×C₃	semidirect
81. C₂²×C₉	AGL(1,ℤ/9)	semidirect
86. C₂²⋊C₉	AGL(1,ℤ/9)	semidirect
136. C₉	((C₂²⋊C₉)⋊C₂)⋊C₃	semidirect
148. C₂²	#92309	semidirect

H ≅ G/N

type

1. #11757844

C₂

semidirect

4. C₉⋊((C₂²⋊C₉)⋊C₂)

C₃

semidirect

15. C₉×C₂²⋊C₉

C₂×C₃

semidirect

81. C₂²×C₉

AGL(1,ℤ/9)

semidirect

86. C₂²⋊C₉

AGL(1,ℤ/9)

semidirect

136. C₉

((C₂²⋊C₉)⋊C₂)⋊C₃

semidirect

148. C₂²

#92309

semidirect

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

subgroup	index
#11757844	2
C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)	3
#10619980	3
C₉⋊((C₂²⋊C₉)⋊C₂)	3
#10619746	3
#92309	4

subgroup

index

#11757844

C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)

#10619980

C₉⋊((C₂²⋊C₉)⋊C₂)

#10619746

#92309

Subgroups — #14518081

Conjugacy classes of subgroups

Complements

Maximal subgroups