Group #13141395 — subgroups

Subgroups — #13141395

Order 1728.

596 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 156 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×(C₂×AGL(1,ℤ/9))⋊C₄	C₂	1	3	yes	maximal
2	C₂×C₄⋊(3₋¹⁺²⋊C₄)	C₂	1	3	yes	maximal, complemented
3	3₋¹⁺²×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4677	—	3	1	no	maximal
5	#10588357	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂³×AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
7	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
8	C₂²×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
9	(C₂×AGL(1,ℤ/9))⋊C₄	—	2	6	no	—
10	C₄⋊(3₋¹⁺²⋊C₄)	—	2	6	no	—
11	C₂×C₃×(C₂×D₃)⋊C₄	—	3	3	no	—
12	C₂×C₃×C₄⋊Dic₃	—	3	3	no	—
13	C₃²×#95	D₃	1	1	yes	characteristic
14	C₂×(C₂×D₉)⋊C₄	C₂×C₃	1	3	yes	—
15	C₂×C₄⋊Dic₉	C₂×C₃	1	3	yes	complemented
16	C₉×#95	—	2	1	no	—
17	C₉×#95	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
19	C₂²×AGL(1,ℤ/9)	—	2	3	no	—
20	C₂²×AGL(1,ℤ/9)	—	2	3	no	—
21	C₂²×AGL(1,ℤ/9)	Q₈	1	2	yes	—
22	C₂×C₄×3₋¹⁺²	—	2	6	no	—
23	C₂×C₄×3₋¹⁺²	D₄	1	6	yes	—
24	C₂×3₋¹⁺²⋊C₄	—	2	6	no	—
25	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
26	C₃×#399	—	9	1	no	—
27	#4677	—	3	1	no	—
28	C₂³×C₃×D₃	—	3	1	no	—
29	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
30	C₂²×C₃×Dic₃	—	3	3	no	—
31	C₂³×D₉	C₂²×C₃	1	1	yes	characteristic
32	C₃×(C₂×D₃)⋊C₄	—	6	6	no	—
33	C₃×C₄⋊Dic₃	—	6	6	no	—
34	C₂²×C₄×C₉	—	2	3	no	—
35	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
36	C₂²×Dic₉	C₂²×C₃	1	3	yes	—
37	(C₂×D₉)⋊C₄	—	2	6	no	—
38	C₄⋊Dic₉	—	2	6	no	—
39	C₂²×3₋¹⁺²	C₂×D₄	1	3	yes	—
40	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
41	C₂²×3₋¹⁺²	#41	1	3	yes	—
42	C₂×AGL(1,ℤ/9)	—	2	6	no	—
43	C₂×AGL(1,ℤ/9)	—	4	3	no	—
44	C₂×AGL(1,ℤ/9)	—	4	1	no	—
45	C₄×3₋¹⁺²	—	2	6	no	—
46	3₋¹⁺²⋊C₄	—	4	3	no	—
47	C₂×C₃×C₄⋊C₄	—	9	3	no	—
48	C₂×C₃×C₂²⋊C₄	—	9	3	no	—
49	C₂×(C₂×D₃)⋊C₄	—	3	3	no	—
50	C₂×C₄⋊Dic₃	—	3	3	no	—
51	C₃×#95	—	3	1	no	—
52	C₃×#95	C₃×D₃	1	1	yes	characteristic
53	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
54	C₂²×C₃×D₃	—	3	2	no	—
55	C₂²×C₃×D₃	—	6	3	no	—
56	C₂²×C₃×D₃	—	6	3	no	—
57	C₂×C₃²×C₄	—	2	6	no	—
58	C₂×C₃²×C₄	D₁₂	1	6	yes	—
59	C₂×C₃×Dic₃	—	6	6	no	—
60	C₂×C₃×Dic₃	—	6	3	no	—
61	C₂³×C₉	—	2	1	no	—
62	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
63	C₂²×D₉	—	2	3	no	—
64	C₂²×D₉	—	2	3	no	—
65	C₂²×D₉	C₃×Q₈	1	2	yes	—
66	C₂×C₄×C₉	—	2	6	no	—
67	C₂×C₄×C₉	—	2	6	no	—
68	C₂×C₄×C₉	—	4	6	no	—
69	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
70	C₂×Dic₉	—	2	6	no	—
71	C₂×Dic₉	—	2	3	no	—
72	#399	—	9	1	no	Sylow
73	C₂×3₋¹⁺²	C₂²⋊Q₈	1	3	yes	—
74	C₂×3₋¹⁺²	#123	1	3	yes	—
75	C₂×3₋¹⁺²	#127	1	1	yes	characteristic
76	AGL(1,ℤ/9)	—	4	2	no	—
77	C₂⁴×C₃	—	9	1	no	—
78	C₂³×D₃	—	3	1	no	—
79	C₂²×C₃×C₄	—	3	3	no	—
80	C₂²×C₃×C₄	—	9	3	no	—
81	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
82	C₂²×Dic₃	—	3	3	no	—
83	C₃×C₄⋊C₄	—	18	6	no	—
84	C₃×C₂²⋊C₄	—	18	6	no	—
85	(C₂×D₃)⋊C₄	—	6	6	no	—
86	C₄⋊Dic₃	—	6	6	no	—
87	C₂²×C₃²	C₂×D₁₂	1	3	yes	—
88	C₂²×C₃²	#237	1	3	yes	—
89	C₂²×C₃²	D₃×Q₈	1	1	yes	characteristic
90	C₂×C₃×D₃	—	6	6	no	—
91	C₂×C₃×D₃	—	12	1	no	—
92	C₂×C₃×D₃	—	12	3	no	—
93	C₃²×C₄	—	2	6	no	—
94	C₃×Dic₃	—	12	3	no	—
95	C₂²×C₉	—	2	3	no	—
96	C₂²×C₉	—	2	3	no	—
97	C₂²×C₉	—	2	1	no	—
98	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
99	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
100	C₂²×C₉	C₃×#41	1	3	yes	—
101	C₂×D₉	—	2	6	no	—
102	C₂×D₉	—	4	3	no	—
103	C₂×D₉	—	4	1	no	—
104	C₄×C₉	—	2	6	no	—
105	C₄×C₉	—	4	6	no	—
106	Dic₉	—	4	3	no	—
107	C₂×C₄⋊C₄	—	9	3	no	—
108	C₂×C₂²⋊C₄	—	9	3	no	—
109	#95	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
110	3₋¹⁺²	#399	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₃	—	3	1	no	—
112	C₂³×C₃	—	9	2	no	—
113	C₂³×C₃	—	18	3	no	—
114	C₂³×C₃	—	18	3	no	—
115	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
116	C₂²×D₃	—	3	2	no	—
117	C₂²×D₃	—	6	3	no	—
118	C₂²×D₃	—	6	3	no	—
119	C₂×C₃×C₄	—	2	6	no	—
120	C₂×C₃×C₄	—	3	6	no	—
121	C₂×C₃×C₄	—	6	6	no	—
122	C₂×C₃×C₄	—	18	3	no	—
123	C₂×C₃×C₄	—	18	6	no	—
124	C₂×C₃×C₄	C₃×D₁₂	1	6	yes	—
125	C₂×Dic₃	—	6	6	no	—
126	C₂×Dic₃	—	6	3	no	—
127	C₂×C₃²	#874	1	1	yes	characteristic
128	C₂×C₃²	#886	1	3	yes	—
129	C₂×C₃²	#897	1	3	yes	—
130	C₃×D₃	—	12	2	no	—
131	C₂×C₉	—	2	3	no	—
132	C₂×C₉	—	2	3	no	—
133	C₂×C₉	—	2	1	no	—
134	C₂×C₉	C₃×C₂²⋊Q₈	1	3	yes	—
135	C₂×C₉	C₃×#123	1	3	yes	—
136	C₂×C₉	C₃×#127	1	1	yes	characteristic
137	D₉	—	4	2	no	—
138	C₂⁴	—	9	1	no	—
139	C₂²×C₄	—	9	3	no	—
140	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
141	C₄⋊C₄	—	18	6	no	—
142	C₂²⋊C₄	—	18	6	no	—
143	C₂²×C₃	—	3	3	no	—
144	C₂²×C₃	—	3	3	no	—
145	C₂²×C₃	—	3	1	no	—
146	C₂²×C₃	—	18	6	no	—
147	C₂²×C₃	—	36	1	no	—
148	C₂²×C₃	—	36	3	no	—
149	C₂²×C₃	C₂×C₃×D₁₂	1	3	yes	—
150	C₂²×C₃	C₃×#237	1	3	yes	—
151	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
152	C₂×D₃	—	6	6	no	—
153	C₂×D₃	—	12	3	no	—
154	C₂×D₃	—	12	1	no	—
155	C₃×C₄	—	2	6	no	—
156	C₃×C₄	—	6	6	no	—
157	C₃×C₄	—	36	3	no	—
158	Dic₃	—	12	3	no	—
159	C₃²	#4677	1	1	yes	characteristic
160	C₉	—	2	1	no	—
161	C₉	C₃×#399	1	1	yes	characteristic, complemented
162	C₂³	—	9	2	no	—
163	C₂³	—	18	3	no	—
164	C₂³	—	18	3	no	—
165	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
166	C₂×C₄	—	2	6	no	—
167	C₂×C₄	—	18	3	no	—
168	C₂×C₄	—	18	6	no	—
169	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	6	yes	—
170	C₂×C₃	—	3	3	no	—
171	C₂×C₃	—	3	1	no	—
172	C₂×C₃	—	3	3	no	—
173	C₂×C₃	—	36	2	no	—
174	C₂×C₃	C₃×#874	1	1	yes	characteristic
175	C₂×C₃	C₃×#886	1	3	yes	—
176	C₂×C₃	C₃×#897	1	3	yes	—
177	D₃	—	12	2	no	—
178	C₂²	—	18	6	no	—
179	C₂²	—	36	1	no	—
180	C₂²	—	36	3	no	—
181	C₂²	C₂×C₄⋊AGL(1,ℤ/9)	1	3	yes	—
182	C₂²	#3699⋊C₃	1	3	yes	—
183	C₂²	Q₈×AGL(1,ℤ/9)	1	1	yes	characteristic
184	C₄	—	2	6	no	—
185	C₄	—	36	3	no	—
186	C₃	—	3	1	no	—
187	C₃	C₃×#4677	1	1	yes	characteristic
188	C₂	—	36	2	no	—
189	C₂	#11721394	1	1	yes	characteristic
190	C₂	#63448⋊C₃	1	3	yes	—
191	C₂	#63459⋊C₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×(C₂×AGL(1,ℤ/9))⋊C₄

C₂

yes

maximal

C₂×C₄⋊(3₋¹⁺²⋊C₄)

C₂

yes

maximal, complemented

3₋¹⁺²×#95

C₂

yes

maximal, characteristic, complemented

C₃×#4677

—

maximal

#10588357

C₃

yes

maximal, characteristic, complemented

C₂³×AGL(1,ℤ/9)

C₂²

yes

characteristic

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

—

(C₂×AGL(1,ℤ/9))⋊C₄

—

C₄⋊(3₋¹⁺²⋊C₄)

—

C₂×C₃×(C₂×D₃)⋊C₄

—

C₂×C₃×C₄⋊Dic₃

—

C₃²×#95

D₃

yes

characteristic

C₂×(C₂×D₉)⋊C₄

C₂×C₃

yes

—

C₂×C₄⋊Dic₉

C₂×C₃

yes

complemented

C₉×#95

—

C₉×#95

C₂×C₃

yes

characteristic, complemented

C₂³×3₋¹⁺²

C₂³

yes

—

—

Q₈

yes

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

—

C₃×#399

—

#4677

—

C₂³×C₃×D₃

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₂³×D₉

C₂²×C₃

yes

characteristic

C₃×(C₂×D₃)⋊C₄

—

C₃×C₄⋊Dic₃

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×Dic₉

C₂²×C₃

yes

—

(C₂×D₉)⋊C₄

—

C₄⋊Dic₉

—

C₂²×3₋¹⁺²

C₂×D₄

yes

—

C₂²×3₋¹⁺²

C₂×Q₈

yes

characteristic

C₂²×3₋¹⁺²

#41

yes

—

C₂×AGL(1,ℤ/9)

—

C₂×AGL(1,ℤ/9)

—

C₂×AGL(1,ℤ/9)

—

C₄×3₋¹⁺²

—

3₋¹⁺²⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₂²⋊C₄

—

C₂×(C₂×D₃)⋊C₄

—

C₂×C₄⋊Dic₃

—

C₃×#95

—

C₃×#95

C₃×D₃

yes

characteristic

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₂³×C₃

yes

characteristic

C₂²×D₉

—

C₂²×D₉

—

C₂²×D₉

C₃×Q₈

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

#399

—

Sylow

C₂×3₋¹⁺²

C₂²⋊Q₈

yes

—

C₂×3₋¹⁺²

#123

yes

—

C₂×3₋¹⁺²

#127

yes

characteristic

AGL(1,ℤ/9)

—

C₂⁴×C₃

—

C₂³×D₃

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₂²×C₃²

#237

yes

—

C₂²×C₃²

D₃×Q₈

yes

characteristic

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×D₄

yes

—

C₂²×C₉

C₂×C₃×Q₈

yes

characteristic

100

C₂²×C₉

C₃×#41

yes

—

101

C₂×D₉

—

102

C₂×D₉

—

103

C₂×D₉

—

104

C₄×C₉

—

105

C₄×C₉

—

106

Dic₉

—

107

C₂×C₄⋊C₄

—

108

C₂×C₂²⋊C₄

—

109

#95

AGL(1,ℤ/9)

yes

characteristic, complemented

110

3₋¹⁺²

#399

yes

characteristic, complemented, Sylow

111

C₂³×C₃

—

112

C₂³×C₃

—

113

C₂³×C₃

—

114

C₂³×C₃

—

115

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

116

C₂²×D₃

—

117

C₂²×D₃

—

118

C₂²×D₃

—

119

C₂×C₃×C₄

—

120

C₂×C₃×C₄

—

121

C₂×C₃×C₄

—

122

C₂×C₃×C₄

—

123

C₂×C₃×C₄

—

124

C₂×C₃×C₄

C₃×D₁₂

yes

—

125

C₂×Dic₃

—

126

C₂×Dic₃

—

127

C₂×C₃²

#874

yes

characteristic

128

C₂×C₃²

#886

yes

—

129

C₂×C₃²

#897

yes

—

130

C₃×D₃

—

131

C₂×C₉

—

132

C₂×C₉

—

133

C₂×C₉

—

134

C₂×C₉

C₃×C₂²⋊Q₈

yes

—

135

C₂×C₉

C₃×#123

yes

—

136

C₂×C₉

C₃×#127

yes

characteristic

137

D₉

—

138

C₂⁴

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

141

C₄⋊C₄

—

142

C₂²⋊C₄

—

143

C₂²×C₃

—

144

C₂²×C₃

—

145

C₂²×C₃

—

146

C₂²×C₃

—

147

C₂²×C₃

—

148

C₂²×C₃

—

149

C₂²×C₃

C₂×C₃×D₁₂

yes

—

150

C₂²×C₃

C₃×#237

yes

—

151

C₂²×C₃

C₃×D₃×Q₈

yes

characteristic

152

C₂×D₃

—

153

C₂×D₃

—

154

C₂×D₃

—

155

C₃×C₄

—

156

C₃×C₄

—

157

C₃×C₄

—

158

Dic₃

—

159

C₃²

#4677

yes

characteristic

160

C₉

—

161

C₉

C₃×#399

yes

characteristic, complemented

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₄⋊AGL(1,ℤ/9)

yes

—

170

C₂×C₃

—

171

C₂×C₃

—

172

C₂×C₃

—

173

C₂×C₃

—

174

C₂×C₃

C₃×#874

yes

characteristic

175

C₂×C₃

C₃×#886

yes

—

176

C₂×C₃

C₃×#897

yes

—

177

D₃

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

yes

—

182

C₂²

#3699⋊C₃

yes

—

183

C₂²

Q₈×AGL(1,ℤ/9)

yes

characteristic

184

C₄

—

185

C₄

—

186

C₃

—

187

C₃

C₃×#4677

yes

characteristic

188

C₂

—

189

C₂

#11721394

yes

characteristic

190

C₂

#63448⋊C₃

yes

—

191

C₂

#63459⋊C₃

yes

—

	subgroup	G/H
center	C₂³	C₂²×AGL(1,ℤ/9)
commutator, cosocle	C₂³×C₉	C₂³×C₃
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#95	C₂
prime radical	C₂³×AGL(1,ℤ/9)	C₂²
hypercenter	#95	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#399
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center

C₂³

C₂²×AGL(1,ℤ/9)

commutator, cosocle

C₂³×C₉

C₂³×C₃

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

3₋¹⁺²×#95

C₂

C₂²

C₉

G ▷ C₂³×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
2. C₂×C₄⋊(3₋¹⁺²⋊C₄)	C₂	semidirect
3. 3₋¹⁺²×#95	C₂	semidirect
5. #10588357	C₃	semidirect
15. C₂×C₄⋊Dic₉	C₂×C₃	semidirect
17. C₉×#95	C₂×C₃	semidirect
109. #95	AGL(1,ℤ/9)	semidirect
110. 3₋¹⁺²	#399	semidirect
161. C₉	C₃×#399	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(3₋¹⁺²⋊C₄)

C₂

semidirect

3. 3₋¹⁺²×#95

C₂

semidirect

5. #10588357

C₃

semidirect

15. C₂×C₄⋊Dic₉

C₂×C₃

semidirect

17. C₉×#95

C₂×C₃

semidirect

109. #95

AGL(1,ℤ/9)

semidirect

110. 3₋¹⁺²

#399

semidirect

161. C₉

C₃×#399

semidirect

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

subgroup	index
C₂×(C₂×AGL(1,ℤ/9))⋊C₄	2
C₂×C₄⋊(3₋¹⁺²⋊C₄)	2
3₋¹⁺²×#95	2
C₃×#4677	3
#10588357	3

subgroup

index

C₂×(C₂×AGL(1,ℤ/9))⋊C₄

C₂×C₄⋊(3₋¹⁺²⋊C₄)

3₋¹⁺²×#95

C₃×#4677

#10588357

Subgroups — #13141395

Conjugacy classes of subgroups

Complements

Maximal subgroups