Group #14518007 — subgroups

Subgroups — #14518007

Order 1944.

217 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 34 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	#11757769	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)	—	3	2	no	maximal
3	C₉⋊((C₂²⋊C₉)⋊C₂)	C₃	1	1	yes	maximal, characteristic, complemented
4	#10619746	—	3	1	no	maximal
5	#10619630	—	3	1	no	maximal
6	#92309	—	4	1	no	maximal
7	C₃×#1094	D₃	1	2	yes	—
8	C₂²×C₃²⋊C₉	D₃	1	1	yes	characteristic
9	C₂×#4169	—	3	1	no	—
10	C₃²⋊(C₂²⋊C₉)	D₃	1	1	yes	characteristic
11	C₉⋊(C₂²⋊C₉)	—	2	1	no	—
12	(C₂²⋊C₉)⋊C₉	—	2	1	no	—
13	(C₂²⋊C₉)⋊C₉	—	2	2	no	—
14	C₉×C₂²⋊C₉	C₂×C₃	1	1	yes	characteristic, complemented
15	#66316	—	3	1	no	—
16	#6896	—	4	1	no	Sylow
17	C₃×#638	—	9	1	no	—
18	C₃⋊((C₂²⋊C₉)⋊C₂)	—	3	2	no	—
19	C₃⋊((C₂²⋊C₉)⋊C₂)	—	3	1	no	—
20	((C₂²⋊C₉)⋊C₂)⋊C₃	—	9	6	no	—
21	#6233	—	3	1	no	—
22	C₃⋊AGL(1,ℤ/9)	—	12	2	no	—
23	C₂×C₃²⋊C₉	—	3	1	no	—
24	C₉⋊D₉	—	4	1	no	—
25	#4169	—	6	1	no	—
26	#4169	—	12	1	no	—
27	C₂²×C₃³	C₃⋊D₃	1	1	yes	characteristic
28	C₂×C₃×C₃⋊D₃	—	9	1	no	—
29	C₃×C₃⋊Dic₃	—	9	1	no	—
30	C₂²×C₃×C₉	—	2	1	no	—
31	C₂²×C₃×C₉	C₃×D₃	1	1	yes	characteristic
32	C₂×C₃⋊D₉	—	3	1	no	—
33	#1094	—	3	6	no	—
34	C₃×C₂²⋊C₉	—	2	1	no	—
35	C₃×C₂²⋊C₉	—	2	2	no	—
36	C₃×C₂²⋊C₉	C₃×D₃	1	1	yes	characteristic
37	C₃×C₂²⋊C₉	C₃×D₃	1	2	yes	—
38	C₃⋊Dic₉	—	3	1	no	—
39	C₃×3₋¹⁺²	—	4	2	no	—
40	C₉⋊C₉	—	8	2	no	—
41	C₉⋊C₉	—	8	1	no	—
42	C₉⋊C₉	—	8	1	no	—
43	C₃²⋊C₉	—	4	1	no	—
44	C₃²⋊C₉	S₄	1	1	yes	characteristic
45	C₉²	—	4	1	no	—
46	#638	—	9	1	no	—
47	C₃×#67	—	27	2	no	—
48	C₃×#67	—	27	1	no	—
49	C₃×#67	—	27	1	no	—
50	(C₂²⋊C₉)⋊C₂	—	9	6	no	—
51	(C₂²⋊C₉)⋊C₂	—	27	1	no	—
52	#611	—	27	1	no	—
53	C₂×C₃³	—	3	1	no	—
54	C₃×C₃⋊D₃	—	18	1	no	—
55	C₂×C₃×C₉	—	3	1	no	—
56	C₂×C₃×C₉	—	6	1	no	—
57	C₃⋊D₉	—	6	1	no	—
58	C₃⋊D₉	—	12	2	no	—
59	C₃⋊D₉	—	12	1	no	—
60	AGL(1,ℤ/9)	—	36	6	no	—
61	C₂²×C₃²	—	3	2	no	—
62	C₂²×C₃²	—	3	1	no	—
63	C₂²×C₃²	—	3	1	no	—
64	C₂²×C₃²	C₃×C₃⋊D₃	1	1	yes	characteristic
65	C₂×C₃⋊D₃	—	9	1	no	—
66	C₂×C₃×D₃	—	27	1	no	—
67	C₂×C₃×D₃	—	27	1	no	—
68	C₂×C₃×D₃	—	27	2	no	—
69	C₃⋊Dic₃	—	9	1	no	—
70	C₃×Dic₃	—	27	1	no	—
71	C₃×Dic₃	—	27	1	no	—
72	C₃×Dic₃	—	27	2	no	—
73	C₂²×C₉	—	3	1	no	—
74	C₂²×C₉	—	6	1	no	—
75	C₂×D₉	—	27	1	no	—
76	C₂²⋊C₉	—	3	1	no	—
77	C₂²⋊C₉	—	6	2	no	—
78	C₂²⋊C₉	—	6	1	no	—
79	C₂²⋊C₉	AGL(1,ℤ/9)	1	6	yes	complemented
80	Dic₉	—	27	1	no	—
81	C₃³	C₃⋊S₄	1	1	yes	characteristic
82	3₋¹⁺²	—	12	6	no	—
83	C₃×C₉	—	2	1	no	—
84	C₃×C₉	—	4	2	no	—
85	C₃×C₉	—	4	1	no	—
86	C₃×C₉	—	8	1	no	—
87	C₃×C₉	—	8	2	no	—
88	C₃×C₉	C₃×S₄	1	1	yes	characteristic
89	C₃×D₄	—	81	1	no	—
90	#67	—	27	2	no	—
91	#67	—	27	1	no	—
92	#67	—	27	1	no	—
93	C₂×C₃²	—	3	1	no	—
94	C₂×C₃²	—	3	1	no	—
95	C₂×C₃²	—	6	1	no	—
96	C₂×C₃²	—	9	1	no	—
97	C₂×C₃²	—	9	2	no	—
98	C₃⋊D₃	—	18	1	no	—
99	C₃×D₃	—	54	2	no	—
100	C₃×D₃	—	54	1	no	—
101	C₃×D₃	—	54	1	no	—
102	C₂×C₉	—	9	1	no	—
103	C₂×C₉	—	18	1	no	—
104	D₉	—	36	6	no	—
105	D₉	—	54	1	no	—
106	D₉	—	108	1	no	—
107	C₂²×C₃	—	9	1	no	—
108	C₂²×C₃	—	81	1	no	—
109	C₂²×C₃	C₃⋊AGL(1,ℤ/9)	1	2	yes	—
110	C₂²×C₃	#4207	1	1	yes	characteristic
111	C₂²×C₃	#4207	1	1	yes	characteristic
112	C₂×D₃	—	27	1	no	—
113	C₂×D₃	—	27	1	no	—
114	C₂×D₃	—	27	2	no	—
115	C₃×C₄	—	81	1	no	—
116	Dic₃	—	27	1	no	—
117	Dic₃	—	27	1	no	—
118	Dic₃	—	27	2	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	2	no	—
122	C₃²	C₃×C₃⋊S₄	1	1	yes	characteristic
123	C₉	—	3	1	no	—
124	C₉	—	4	6	no	—
125	C₉	—	6	1	no	—
126	C₉	—	12	1	no	—
127	C₉	—	24	1	no	—
128	C₉	—	24	2	no	—
129	D₄	—	81	1	no	Sylow
130	C₂×C₃	—	3	1	no	—
131	C₂×C₃	—	3	1	no	—
132	C₂×C₃	—	3	2	no	—
133	C₂×C₃	—	9	1	no	—
134	C₂×C₃	—	18	1	no	—
135	C₂×C₃	—	162	1	no	—
136	D₃	—	54	1	no	—
137	D₃	—	54	1	no	—
138	D₃	—	54	2	no	—
139	C₂²	—	81	1	no	—
140	C₂²	#92309	1	1	yes	characteristic, complemented
141	C₄	—	81	1	no	—
142	C₃	—	9	1	no	—
143	C₃	#10620045	1	1	yes	characteristic
144	C₃	#10620047	1	1	yes	characteristic
145	C₃	A₄⋊AGL(1,ℤ/9)	1	2	yes	—
146	C₂	—	3	1	no	—
147	C₂	—	162	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11757769

C₂

yes

maximal, characteristic, complemented

C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)

—

maximal

C₉⋊((C₂²⋊C₉)⋊C₂)

C₃

yes

maximal, characteristic, complemented

—

—

—

D₃

yes

—

C₂²×C₃²⋊C₉

D₃

yes

characteristic

C₂×#4169

—

C₃²⋊(C₂²⋊C₉)

D₃

yes

characteristic

C₉⋊(C₂²⋊C₉)

—

(C₂²⋊C₉)⋊C₉

—

(C₂²⋊C₉)⋊C₉

—

C₉×C₂²⋊C₉

C₂×C₃

yes

characteristic, complemented

—

—

—

C₃⋊((C₂²⋊C₉)⋊C₂)

—

C₃⋊((C₂²⋊C₉)⋊C₂)

—

((C₂²⋊C₉)⋊C₂)⋊C₃

—

#6233

—

C₃⋊AGL(1,ℤ/9)

—

C₂×C₃²⋊C₉

—

C₉⋊D₉

—

#4169

—

#4169

—

C₂²×C₃³

C₃⋊D₃

yes

characteristic

C₂×C₃×C₃⋊D₃

—

C₃×C₃⋊Dic₃

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₃×D₃

yes

characteristic

C₂×C₃⋊D₉

—

#1094

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

C₃×D₃

yes

characteristic

C₃×C₂²⋊C₉

C₃×D₃

yes

—

C₃⋊Dic₉

—

C₃×3₋¹⁺²

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

S₄

yes

characteristic

C₉²

—

#638

—

C₃×#67

—

C₃×#67

—

C₃×#67

—

(C₂²⋊C₉)⋊C₂

—

(C₂²⋊C₉)⋊C₂

—

#611

—

C₂×C₃³

—

C₃×C₃⋊D₃

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₃⋊D₉

—

C₃⋊D₉

—

C₃⋊D₉

—

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

—

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

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×S₄

yes

characteristic

C₃×D₄

—

#67

—

#67

—

#67

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

100

C₃×D₃

—

101

C₃×D₃

—

102

C₂×C₉

—

103

C₂×C₉

—

104

D₉

—

105

D₉

—

106

D₉

—

108

—

107

C₂²×C₃

—

108

C₂²×C₃

—

109

C₂²×C₃

C₃⋊AGL(1,ℤ/9)

yes

—

110

C₂²×C₃

#4207

yes

characteristic

111

C₂²×C₃

#4207

yes

characteristic

112

C₂×D₃

—

113

C₂×D₃

—

114

C₂×D₃

—

115

C₃×C₄

—

116

Dic₃

—

117

Dic₃

—

118

Dic₃

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

C₃×C₃⋊S₄

yes

characteristic

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

—

129

D₄

—

Sylow

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂×C₃

—

133

C₂×C₃

—

134

C₂×C₃

—

135

C₂×C₃

—

162

—

136

D₃

—

137

D₃

—

138

D₃

—

139

C₂²

—

140

C₂²

#92309

yes

characteristic, complemented

141

C₄

—

142

C₃

—

143

C₃

#10620045

yes

characteristic

144

C₃

#10620047

yes

characteristic

145

C₃

A₄⋊AGL(1,ℤ/9)

yes

—

146

C₂

—

147

C₂

—

162

—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉×C₂²⋊C₉	C₂×C₃
Frattini	C₃²	C₃×C₃⋊S₄
Fitting	C₂²×C₃²⋊C₉	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×C₃²⋊C₉

D₃

socle

C₂²×C₃²

C₃×C₃⋊D₃

LCS

G ▷ C₉×C₂²⋊C₉

UCS

C₁

N	H ≅ G/N	type
1. #11757769	C₂	semidirect
3. C₉⋊((C₂²⋊C₉)⋊C₂)	C₃	semidirect
14. C₉×C₂²⋊C₉	C₂×C₃	semidirect
79. C₂²⋊C₉	AGL(1,ℤ/9)	semidirect
140. C₂²	#92309	semidirect

H ≅ G/N

type

1. #11757769

C₂

semidirect

3. C₉⋊((C₂²⋊C₉)⋊C₂)

C₃

semidirect

14. C₉×C₂²⋊C₉

C₂×C₃

semidirect

79. C₂²⋊C₉

AGL(1,ℤ/9)

semidirect

140. C₂²

#92309

semidirect

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

subgroup	index
#11757769	2
C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)	3
C₉⋊((C₂²⋊C₉)⋊C₂)	3
#10619746	3
#10619630	3
#92309	4

subgroup

index

#11757769

C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)

C₉⋊((C₂²⋊C₉)⋊C₂)

#10619746

#10619630

#92309

Subgroups — #14518007

Conjugacy classes of subgroups

Complements

Maximal subgroups