Group #13091516 — subgroups

Subgroups — #13091516

Order 1458.

512 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 71 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	#10624896	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×#4207	—	3	3	no	maximal
3	C₃×C₃⋊AGL(1,ℤ/9)	—	3	1	no	maximal
4	C₉⋊#272	C₃	1	1	yes	maximal, characteristic, complemented
5	#92304	—	3	6	no	maximal
6	#92302	—	3	1	no	maximal
7	#92300	—	3	1	no	maximal
8	C₃×#4181	—	3	1	no	maximal
9	C₃×#737	D₃	1	3	yes	complemented
10	C₃²×3₋¹⁺²	D₃	1	1	yes	characteristic, complemented
11	#6893	D₃	1	6	yes	complemented
12	C₉⋊H(3,𝔽₃)	—	2	1	no	—
13	C₉⋊H(3,𝔽₃)	—	2	1	no	—
14	C₉⋊H(3,𝔽₃)	—	2	3	no	—
15	C₉⋊H(3,𝔽₃)	—	2	1	no	—
16	C₉⋊H(3,𝔽₃)	—	2	6	no	—
17	#6891	D₃	1	1	yes	characteristic, complemented
18	C₃²⋊H(3,𝔽₃)	D₃	1	1	yes	characteristic
19	C₉×H(3,𝔽₃)	—	2	1	no	—
20	C₉×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×C₃²⋊C₉	D₃	1	1	yes	characteristic, complemented
22	C₃²×C₃⋊D₃	—	9	1	no	—
23	C₃⋊#272	—	3	1	no	—
24	#4207	—	3	9	no	—
25	C₃⋊AGL(1,ℤ/9)	—	3	3	no	—
26	C₃×C₃⋊D₉	—	3	1	no	—
27	C₃×C₃⋊D₉	—	3	3	no	—
28	C₃×AGL(1,ℤ/9)	—	9	3	no	—
29	C₃×AGL(1,ℤ/9)	—	9	2	no	—
30	C₃×AGL(1,ℤ/9)	—	9	1	no	—
31	C₃×(C₃⋊D₃)⋊C₃	—	9	3	no	—
32	C₃²×D₉	—	9	3	no	—
33	#4181	—	3	1	no	—
34	#4181	—	3	1	no	—
35	#4181	—	3	6	no	—
36	C₃⁴	C₃⋊D₃	1	1	yes	characteristic
37	#737	—	3	9	no	—
38	#737	—	6	9	no	—
39	C₃×3₋¹⁺²	—	2	3	no	—
40	C₃×3₋¹⁺²	—	2	2	no	—
41	C₃×3₋¹⁺²	—	2	1	no	—
42	C₃×3₋¹⁺²	—	2	3	no	—
43	C₃×3₋¹⁺²	—	2	3	no	—
44	C₃×3₋¹⁺²	—	3	3	no	—
45	C₃×3₋¹⁺²	—	6	3	no	—
46	C₃×3₋¹⁺²	C₃⋊D₃	1	1	yes	characteristic
47	C₃×3₋¹⁺²	C₃⋊D₃	1	2	yes	—
48	C₃×3₋¹⁺²	C₃⋊D₃	1	3	yes	—
49	C₃×H(3,𝔽₃)	—	2	1	no	—
50	C₃×H(3,𝔽₃)	—	2	3	no	—
51	C₃×H(3,𝔽₃)	C₃×D₃	1	1	yes	characteristic
52	C₃×H(3,𝔽₃)	C₃⋊D₃	1	3	yes	—
53	C₃²×C₉	—	2	1	no	—
54	C₃²×C₉	—	2	3	no	—
55	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
56	C₃²×C₉	C₃×D₃	1	3	yes	complemented
57	C₃²×C₉	C₃⋊D₃	1	3	yes	—
58	C₃²⋊C₉	—	2	1	no	—
59	C₃²⋊C₉	—	2	1	no	—
60	C₃²⋊C₉	—	2	6	no	—
61	C₃²⋊C₉	—	2	3	no	—
62	C₃²⋊C₉	—	2	6	no	—
63	C₃²⋊C₉	—	2	3	no	—
64	C₃²⋊C₉	—	2	6	no	—
65	C₃²⋊C₉	—	2	6	no	—
66	C₃²⋊C₉	C₃×D₃	1	1	yes	characteristic
67	C₃²⋊C₉	C₃×D₃	1	1	yes	characteristic
68	C₃²⋊C₉	C₃×D₃	1	6	yes	complemented
69	C₃×C₃⋊D₃	—	9	3	no	—
70	C₃×C₃⋊D₃	—	9	1	no	—
71	C₃×C₃⋊D₃	—	9	3	no	—
72	C₃²×D₃	—	27	3	no	—
73	C₃²×D₃	—	27	1	no	—
74	#272	—	27	3	no	—
75	C₃⋊D₉	—	3	1	no	—
76	C₃⋊D₉	—	3	3	no	—
77	AGL(1,ℤ/9)	—	9	6	no	—
78	AGL(1,ℤ/9)	—	9	9	no	—
79	AGL(1,ℤ/9)	—	9	3	no	—
80	(C₃⋊D₃)⋊C₃	—	9	9	no	—
81	C₃×D₉	—	9	9	no	—
82	C₃×D₉	—	9	3	no	—
83	C₃×D₉	—	9	3	no	—
84	C₃×D₉	—	9	2	no	—
85	C₃×D₉	—	9	1	no	—
86	C₃³	—	2	3	no	—
87	C₃³	—	2	1	no	—
88	C₃³	—	3	3	no	—
89	C₃³	—	3	3	no	—
90	C₃³	—	18	1	no	—
91	C₃³	C₃×C₃⋊D₃	1	3	yes	—
92	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
93	C₃³	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
94	3₋¹⁺²	—	2	3	no	—
95	3₋¹⁺²	—	3	6	no	—
96	3₋¹⁺²	—	3	9	no	—
97	3₋¹⁺²	—	6	6	no	—
98	3₋¹⁺²	—	6	9	no	—
99	3₋¹⁺²	—	6	9	no	—
100	3₋¹⁺²	—	6	2	no	—
101	3₋¹⁺²	—	6	6	no	—
102	3₋¹⁺²	—	6	1	no	—
103	3₋¹⁺²	—	6	3	no	—
104	3₋¹⁺²	#272	1	3	yes	complemented
105	H(3,𝔽₃)	—	3	3	no	—
106	H(3,𝔽₃)	—	3	9	no	—
107	H(3,𝔽₃)	—	6	3	no	—
108	H(3,𝔽₃)	—	18	3	no	—
109	C₃×C₉	—	2	1	no	—
110	C₃×C₉	—	2	3	no	—
111	C₃×C₉	—	2	2	no	—
112	C₃×C₉	—	2	3	no	—
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	3	no	—
115	C₃×C₉	—	3	9	no	—
116	C₃×C₉	—	6	1	no	—
117	C₃×C₉	—	6	3	no	—
118	C₃×C₉	—	6	9	no	—
119	C₃×C₉	—	6	3	no	—
120	C₃×C₉	C₃×C₃⋊D₃	1	1	yes	characteristic
121	C₃×C₉	C₃×C₃⋊D₃	1	3	yes	—
122	C₃×C₉	C₃×C₃⋊D₃	1	3	yes	—
123	C₃×C₉	C₃×C₃⋊D₃	1	2	yes	—
124	C₂×C₃²	—	81	1	no	—
125	C₃⋊D₃	—	9	1	no	—
126	C₃⋊D₃	—	9	3	no	—
127	C₃×D₃	—	27	9	no	—
128	C₃×D₃	—	27	3	no	—
129	C₃×D₃	—	27	3	no	—
130	C₃×D₃	—	27	1	no	—
131	C₃×D₃	—	81	3	no	—
132	D₉	—	9	3	no	—
133	D₉	—	9	2	no	—
134	D₉	—	9	3	no	—
135	D₉	—	9	1	no	—
136	C₃²	—	3	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	3	no	—
139	C₃²	—	3	3	no	—
140	C₃²	—	6	1	no	—
141	C₃²	—	6	1	no	—
142	C₃²	—	6	3	no	—
143	C₃²	—	6	1	no	—
144	C₃²	—	6	6	no	—
145	C₃²	—	6	1	no	—
146	C₃²	—	6	3	no	—
147	C₃²	—	6	1	no	—
148	C₃²	—	6	3	no	—
149	C₃²	—	6	1	no	—
150	C₃²	—	9	9	no	—
151	C₃²	#4207	1	3	yes	complemented
152	C₃²	C₃⋊#272	1	3	yes	—
153	C₃²	C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
154	C₉	—	2	3	no	—
155	C₉	—	2	1	no	—
156	C₉	—	3	3	no	—
157	C₉	—	3	2	no	—
158	C₉	—	3	3	no	—
159	C₉	—	6	3	no	—
160	C₉	—	6	3	no	—
161	C₉	—	6	2	no	—
162	C₉	C₃×#272	1	1	yes	characteristic
163	C₂×C₃	—	81	3	no	—
164	C₂×C₃	—	81	1	no	—
165	D₃	—	27	3	no	—
166	D₃	—	27	1	no	—
167	D₃	—	81	3	no	—
168	C₃	—	2	1	no	—
169	C₃	—	3	3	no	—
170	C₃	—	3	3	no	—
171	C₃	—	9	3	no	—
172	C₃	—	18	3	no	—
173	C₃	—	18	1	no	—
174	C₃	C₃×C₃⋊#272	1	1	yes	characteristic
175	C₃	C₃⋊#4207	1	1	yes	characteristic
176	C₂	—	81	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624896

C₂

yes

maximal, characteristic, complemented, Sylow

—

—

C₃

yes

maximal, characteristic, complemented

—

—

—

—

D₃

yes

complemented

C₃²×3₋¹⁺²

D₃

yes

characteristic, complemented

#6893

D₃

yes

complemented

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

D₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

D₃

yes

characteristic

C₉×H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

C₂×C₃

yes

characteristic, complemented

C₃×C₃²⋊C₉

D₃

yes

characteristic, complemented

C₃²×C₃⋊D₃

—

C₃⋊#272

—

#4207

—

C₃⋊AGL(1,ℤ/9)

—

C₃×C₃⋊D₉

—

C₃×C₃⋊D₉

—

C₃×AGL(1,ℤ/9)

—

C₃×AGL(1,ℤ/9)

—

C₃×AGL(1,ℤ/9)

—

C₃×(C₃⋊D₃)⋊C₃

—

C₃²×D₉

—

#4181

—

#4181

—

#4181

—

C₃⁴

C₃⋊D₃

yes

characteristic

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃⋊D₃

yes

characteristic

C₃×3₋¹⁺²

C₃⋊D₃

yes

—

C₃×3₋¹⁺²

C₃⋊D₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×D₃

yes

characteristic

C₃×H(3,𝔽₃)

C₃⋊D₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

characteristic

C₃²×C₉

C₃×D₃

yes

complemented

C₃²×C₉

C₃⋊D₃

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×D₃

yes

characteristic

C₃²⋊C₉

C₃×D₃

yes

characteristic

C₃²⋊C₉

C₃×D₃

yes

complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

#272

—

C₃⋊D₉

—

C₃⋊D₉

—

AGL(1,ℤ/9)

—

AGL(1,ℤ/9)

—

AGL(1,ℤ/9)

—

(C₃⋊D₃)⋊C₃

—

C₃×D₉

—

C₃×D₉

—

C₃×D₉

—

C₃×D₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₃⋊D₃

yes

—

C₃³

C₃×C₃⋊D₃

yes

characteristic

C₃³

C₃⋊(C₃⋊D₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

#272

yes

complemented

105

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

H(3,𝔽₃)

—

108

H(3,𝔽₃)

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

C₃×C₃⋊D₃

yes

characteristic

121

C₃×C₉

C₃×C₃⋊D₃

yes

—

122

C₃×C₉

C₃×C₃⋊D₃

yes

—

123

C₃×C₉

C₃×C₃⋊D₃

yes

—

124

C₂×C₃²

—

125

C₃⋊D₃

—

126

C₃⋊D₃

—

127

C₃×D₃

—

128

C₃×D₃

—

129

C₃×D₃

—

130

C₃×D₃

—

131

C₃×D₃

—

132

D₉

—

133

D₉

—

134

D₉

—

135

D₉

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

—

149

C₃²

—

150

C₃²

—

151

C₃²

#4207

yes

complemented

152

C₃²

C₃⋊#272

yes

—

153

C₃²

C₃×C₃⋊(C₃⋊D₃)

yes

characteristic

154

C₉

—

155

C₉

—

156

C₉

—

157

C₉

—

158

C₉

—

159

C₉

—

160

C₉

—

161

C₉

—

162

C₉

C₃×#272

yes

characteristic

163

C₂×C₃

—

164

C₂×C₃

—

165

D₃

—

166

D₃

—

167

D₃

—

168

C₃

—

169

C₃

—

170

C₃

—

171

C₃

—

172

C₃

—

173

C₃

—

174

C₃

C₃×C₃⋊#272

yes

characteristic

175

C₃

C₃⋊#4207

yes

characteristic

176

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₃	C₃⋊#4207
commutator, cosocle, hypocenter	C₉×H(3,𝔽₃)	C₂×C₃
Frattini, socle	C₃²	C₃×C₃⋊(C₃⋊D₃)
Fitting	#10624896	C₂
LCS	G ▷ C₉×H(3,𝔽₃)
UCS	C₁ ◁ C₃

subgroup

G/H

center, hypercenter

C₃

C₃⋊#4207

commutator, cosocle, hypocenter

C₉×H(3,𝔽₃)

C₂×C₃

Frattini, socle

C₃²

C₃×C₃⋊(C₃⋊D₃)

C₂

G ▷ C₉×H(3,𝔽₃)

C₁ ◁ C₃

N	H ≅ G/N	type
1. #10624896	C₂	semidirect
4. C₉⋊#272	C₃	semidirect
9. C₃×#737	D₃	semidirect
10. C₃²×3₋¹⁺²	D₃	semidirect
11. #6893	D₃	semidirect
17. #6891	D₃	semidirect
20. C₉×H(3,𝔽₃)	C₂×C₃	semidirect
21. C₃×C₃²⋊C₉	D₃	semidirect
56. C₃²×C₉	C₃×D₃	semidirect
68. C₃²⋊C₉	C₃×D₃	semidirect
104. 3₋¹⁺²	#272	semidirect
151. C₃²	#4207	semidirect

H ≅ G/N

type

1. #10624896

C₂

semidirect

4. C₉⋊#272

C₃

semidirect

9. C₃×#737

D₃

semidirect

10. C₃²×3₋¹⁺²

D₃

semidirect

11. #6893

D₃

semidirect

17. #6891

D₃

semidirect

20. C₉×H(3,𝔽₃)

C₂×C₃

semidirect

21. C₃×C₃²⋊C₉

D₃

semidirect

56. C₃²×C₉

C₃×D₃

semidirect

68. C₃²⋊C₉

C₃×D₃

semidirect

104. 3₋¹⁺²

#272

semidirect

151. C₃²

#4207

semidirect

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

subgroup	index
#10624896	2
C₃×#4207	3
C₃×C₃⋊AGL(1,ℤ/9)	3
C₉⋊#272	3
#92304	3
#92302	3
#92300	3
C₃×#4181	3

subgroup

index

Subgroups — #13091516

Conjugacy classes of subgroups

Complements

Maximal subgroups