Group #13142970 — subgroups

Subgroups — C₉×#5071 #13142970

Order 1728.

438 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₉×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#832	C₂	1	4	yes	maximal, complemented
4	C₉×#803	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#5071	C₃	1	1	yes	maximal, characteristic
6	C₉×#490	—	3	1	no	maximal
7	C₂×C₃×D₄×C₉	—	2	2	no	—
8	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
9	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₉×Dic₃⋊C₄	—	2	2	no	—
12	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃²×#127	C₂×C₃	1	1	yes	characteristic
14	C₃×#870	C₂×C₃	1	1	yes	characteristic
15	C₃×#832	C₂×C₃	1	4	yes	—
16	C₃×#803	C₂×C₃	1	1	yes	characteristic
17	C₉×#127	—	2	1	no	—
18	C₉×#127	D₃	1	1	yes	characteristic
19	C₉×#125	—	3	1	no	—
20	C₉×#102	—	3	4	no	—
21	C₉×#97	—	3	1	no	—
22	C₂³×C₃×C₉	—	2	2	no	—
23	C₃×D₄×C₉	—	2	4	no	—
24	C₃×D₄×C₉	—	4	4	no	—
25	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
26	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₂×C₉×Dic₃	—	2	2	no	—
28	C₉×C₃⋊C₈	—	2	2	no	—
29	C₃×#490	—	3	1	no	—
30	#5071	C₉	1	1	yes	characteristic, direct product factor
31	C₂×C₃²×D₄	—	2	2	no	—
32	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
33	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
34	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
35	C₃×Dic₃⋊C₄	—	2	2	no	—
36	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
37	C₂×D₄×C₉	—	2	4	no	—
38	C₂×D₄×C₉	—	2	2	no	—
39	C₂×D₄×C₉	—	2	2	no	—
40	C₂×D₄×C₉	C₂×D₃	1	2	yes	—
41	C₂×C₈×C₉	—	3	2	no	—
42	C₉×C₄⋊C₄	—	3	2	no	—
43	C₉×C₄⋊C₄	—	6	2	no	—
44	C₄²×C₉	—	2	1	no	—
45	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃×C₉	—	4	2	no	—
47	C₂²×C₃×C₉	—	4	4	no	—
48	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
49	C₃×C₄×C₉	—	2	2	no	—
50	C₃×C₄×C₉	#41	1	2	yes	—
51	C₉×Dic₃	—	4	2	no	—
52	C₃×#127	—	2	1	no	—
53	C₃×#127	C₂×C₉	1	1	yes	characteristic
54	C₃×#127	C₃×D₃	1	1	yes	characteristic
55	C₃×#125	—	3	1	no	—
56	#870	C₂×C₉	1	1	yes	characteristic, complemented
57	C₃×#102	—	3	4	no	—
58	C₃×#97	—	3	1	no	—
59	#832	C₂×C₉	1	4	yes	complemented
60	#803	C₂×C₉	1	1	yes	characteristic, complemented
61	C₂³×C₃²	—	2	2	no	—
62	C₃²×D₄	—	2	4	no	—
63	C₃²×D₄	—	4	4	no	—
64	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
65	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
66	C₂×C₃×Dic₃	—	2	2	no	—
67	C₂³×C₉	—	2	4	no	—
68	C₂³×C₉	—	2	2	no	—
69	C₃×C₃⋊C₈	—	2	2	no	—
70	D₄×C₉	—	2	4	no	—
71	D₄×C₉	—	4	4	no	—
72	D₄×C₉	—	4	8	no	—
73	D₄×C₉	—	4	4	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	1	no	—
76	C₂×C₄×C₉	—	6	2	no	—
77	C₂×C₄×C₉	#67	1	2	yes	—
78	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
79	C₈×C₉	—	6	2	no	—
80	#490	—	3	1	no	Sylow
81	C₂×C₃×C₉	—	8	2	no	—
82	C₂×C₃×C₉	#124	1	1	yes	characteristic
83	C₂×C₃×C₉	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₃×D₄	—	2	4	no	—
85	C₂×C₃×D₄	—	2	2	no	—
86	C₂×C₃×D₄	—	2	2	no	—
87	C₂×C₃×D₄	—	2	2	no	—
88	C₂×C₃×D₄	C₂²×C₉	1	2	yes	—
89	C₂×C₃×D₄	C₂×C₃×D₃	1	2	yes	—
90	C₂×C₃×C₈	—	3	2	no	—
91	C₃×C₄⋊C₄	—	3	2	no	—
92	C₃×C₄⋊C₄	—	6	2	no	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
95	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
96	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
97	Dic₃⋊C₄	—	2	2	no	—
98	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
99	C₂²×C₃²	—	4	2	no	—
100	C₂²×C₃²	—	4	4	no	—
101	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
102	C₃²×C₄	—	2	2	no	—
103	C₃²×C₄	C₃×#41	1	2	yes	—
104	C₃×Dic₃	—	4	2	no	—
105	C₂²×C₉	—	2	1	no	—
106	C₂²×C₉	—	4	8	no	—
107	C₂²×C₉	—	4	4	no	—
108	C₂²×C₉	—	4	4	no	—
109	C₂²×C₉	—	4	2	no	—
110	C₂²×C₉	C₂×#67	1	1	yes	characteristic
111	C₄×C₉	—	2	4	no	—
112	C₄×C₉	—	2	2	no	—
113	C₄×C₉	—	2	2	no	—
114	C₄×C₉	—	12	2	no	—
115	C₄×C₉	#237	1	2	yes	—
116	#127	D₃×C₉	1	1	yes	characteristic
117	#125	—	3	1	no	—
118	#102	—	3	4	no	—
119	#97	—	3	1	no	—
120	C₃×C₉	#490	1	1	yes	characteristic, complemented, Sylow
121	C₂³×C₃	—	2	4	no	—
122	C₂³×C₃	—	2	2	no	—
123	C₂³×C₃	—	2	2	no	—
124	C₃×D₄	—	2	4	no	—
125	C₃×D₄	—	2	4	no	—
126	C₃×D₄	—	4	4	no	—
127	C₃×D₄	—	4	4	no	—
128	C₃×D₄	—	4	8	no	—
129	C₃×D₄	—	4	4	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	6	2	no	—
133	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
134	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
135	C₂×C₃×C₄	C₃×#67	1	2	yes	—
136	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
137	C₂×Dic₃	—	2	2	no	—
138	C₃×C₈	—	6	2	no	—
139	C₃⋊C₈	—	2	2	no	—
140	C₂×C₃²	—	8	2	no	—
141	C₂×C₃²	C₃×#124	1	1	yes	characteristic
142	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₉	—	2	2	no	—
144	C₂×C₉	—	2	1	no	—
145	C₂×C₉	—	8	4	no	—
146	C₂×C₉	—	8	2	no	—
147	C₂×C₉	#932	1	2	yes	—
148	C₂×C₉	#936	1	1	yes	characteristic
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	C₂×D₃×C₉	1	2	yes	—
151	C₂×C₈	—	3	2	no	—
152	C₄⋊C₄	—	3	2	no	—
153	C₄⋊C₄	—	6	2	no	—
154	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	4	8	no	—
157	C₂²×C₃	—	4	4	no	—
158	C₂²×C₃	—	4	2	no	—
159	C₂²×C₃	—	4	4	no	—
160	C₂²×C₃	—	4	4	no	—
161	C₂²×C₃	—	4	2	no	—
162	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
163	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
164	C₃×C₄	—	2	4	no	—
165	C₃×C₄	—	2	2	no	—
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	—	12	2	no	—
169	C₃×C₄	C₉×#41	1	2	yes	—
170	C₃×C₄	C₃×#237	1	2	yes	—
171	Dic₃	—	4	2	no	—
172	C₃²	C₃×#490	1	1	yes	characteristic
173	C₉	—	2	1	no	—
174	C₉	#5071	1	1	yes	characteristic, direct product factor
175	C₂³	—	2	2	no	—
176	D₄	—	2	4	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	6	2	no	—
179	C₂×C₄	C₉×#67	1	2	yes	—
180	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
181	C₈	—	6	2	no	—
182	C₂×C₃	—	2	2	no	—
183	C₂×C₃	—	2	1	no	—
184	C₂×C₃	—	8	4	no	—
185	C₂×C₃	—	8	2	no	—
186	C₂×C₃	—	8	2	no	—
187	C₂×C₃	C₉×#124	1	1	yes	characteristic
188	C₂×C₃	C₉×AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₃	C₃×#932	1	2	yes	—
190	C₂×C₃	C₃×#936	1	1	yes	characteristic
191	C₂²	—	4	4	no	—
192	C₂²	—	4	2	no	—
193	C₂²	C₂×C₉×#67	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	12	2	no	—
196	C₄	C₉×#237	1	2	yes	—
197	C₃	—	2	1	no	—
198	C₃	C₉×#490	1	1	yes	characteristic, complemented
199	C₃	C₃×#5071	1	1	yes	characteristic
200	C₂	—	8	2	no	—
201	C₂	C₉×#932	1	2	yes	—
202	C₂	C₉×#936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×#127

C₂

yes

maximal, characteristic

C₉×#870

C₂

yes

maximal, characteristic, complemented

C₉×#832

C₂

yes

maximal, complemented

C₉×#803

C₂

yes

maximal, characteristic, complemented

C₃×#5071

C₃

yes

maximal, characteristic

C₉×#490

—

maximal

C₂×C₃×D₄×C₉

—

C₂×C₃×D₄×C₉

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₉×C₄⋊Dic₃

C₂²

yes

—

C₉×Dic₃⋊C₄

—

C₂×C₉×C₃⋊C₈

C₂²

yes

—

C₃²×#127

C₂×C₃

yes

characteristic

C₃×#870

C₂×C₃

yes

characteristic

C₃×#832

C₂×C₃

yes

—

C₃×#803

C₂×C₃

yes

characteristic

C₉×#127

—

C₉×#127

D₃

yes

characteristic

C₉×#125

—

C₉×#102

—

C₉×#97

—

C₂³×C₃×C₉

—

C₃×D₄×C₉

—

C₃×D₄×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₉×Dic₃

—

C₉×C₃⋊C₈

—

C₃×#490

—

#5071

C₉

yes

characteristic, direct product factor

C₂×C₃²×D₄

—

C₂×C₃²×D₄

C₂²×C₃

yes

—

C₃²×C₄²

C₂²×C₃

yes

characteristic

C₃×C₄⋊Dic₃

C₂²×C₃

yes

—

C₃×Dic₃⋊C₄

—

C₂×C₃×C₃⋊C₈

C₂²×C₃

yes

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

C₂×D₃

yes

—

C₂×C₈×C₉

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

#41

yes

—

C₉×Dic₃

—

C₃×#127

—

C₃×#127

C₂×C₉

yes

characteristic

C₃×#127

C₃×D₃

yes

characteristic

C₃×#125

—

#870

C₂×C₉

yes

characteristic, complemented

C₃×#102

—

C₃×#97

—

#832

C₂×C₉

yes

complemented

#803

C₂×C₉

yes

characteristic, complemented

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

C₃×D₄

yes

—

C₂×C₃²×C₄

C₂³×C₃

yes

characteristic

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

—

C₃×C₃⋊C₈

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₈×C₉

—

#490

—

Sylow

C₂×C₃×C₉

—

C₂×C₃×C₉

#124

yes

characteristic

C₂×C₃×C₉

AGL(1,ℤ/8)

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂²×C₉

yes

—

C₂×C₃×D₄

C₂×C₃×D₃

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₉

yes

characteristic

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₄⋊Dic₃

C₂²×C₉

yes

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

C₂²×C₉

yes

—

C₂²×C₃²

—

100

C₂²×C₃²

—

101

C₂²×C₃²

C₂×C₃×D₄

yes

characteristic

102

C₃²×C₄

—

103

C₃²×C₄

C₃×#41

yes

—

104

C₃×Dic₃

—

105

C₂²×C₉

—

106

C₂²×C₉

—

107

C₂²×C₉

—

108

C₂²×C₉

—

109

C₂²×C₉

—

110

C₂²×C₉

C₂×#67

yes

characteristic

111

C₄×C₉

—

112

C₄×C₉

—

113

C₄×C₉

—

114

C₄×C₉

—

115

C₄×C₉

#237

yes

—

116

#127

D₃×C₉

yes

characteristic

117

#125

—

118

#102

—

119

#97

—

120

C₃×C₉

#490

yes

characteristic, complemented, Sylow

121

C₂³×C₃

—

122

C₂³×C₃

—

123

C₂³×C₃

—

124

C₃×D₄

—

125

C₃×D₄

—

126

C₃×D₄

—

127

C₃×D₄

—

128

C₃×D₄

—

129

C₃×D₄

—

130

C₂×C₃×C₄

—

131

C₂×C₃×C₄

—

132

C₂×C₃×C₄

—

133

C₂×C₃×C₄

D₄×C₉

yes

—

134

C₂×C₃×C₄

C₂³×C₉

yes

characteristic

135

C₂×C₃×C₄

C₃×#67

yes

—

136

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

137

C₂×Dic₃

—

138

C₃×C₈

—

139

C₃⋊C₈

—

140

C₂×C₃²

—

141

C₂×C₃²

C₃×#124

yes

characteristic

142

C₂×C₃²

C₃×AGL(1,ℤ/8)

yes

—

143

C₂×C₉

—

144

C₂×C₉

—

145

C₂×C₉

—

146

C₂×C₉

—

147

C₂×C₉

#932

yes

—

148

C₂×C₉

#936

yes

characteristic

149

C₂×D₄

—

150

C₂×D₄

C₂×D₃×C₉

yes

—

151

C₂×C₈

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

C₂×D₃×C₉

yes

characteristic

155

C₂²×C₃

—

156

C₂²×C₃

—

157

C₂²×C₃

—

158

C₂²×C₃

—

159

C₂²×C₃

—

160

C₂²×C₃

—

161

C₂²×C₃

—

162

C₂²×C₃

C₂×D₄×C₉

yes

characteristic

163

C₂²×C₃

C₂×C₃×#67

yes

characteristic

164

C₃×C₄

—

165

C₃×C₄

—

166

C₃×C₄

—

167

C₃×C₄

—

168

C₃×C₄

—

169

C₃×C₄

C₉×#41

yes

—

170

C₃×C₄

C₃×#237

yes

—

171

Dic₃

—

172

C₃²

C₃×#490

yes

characteristic

173

C₉

—

174

C₉

#5071

yes

characteristic, direct product factor

175

C₂³

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

C₉×#67

yes

—

180

C₂×C₄

C₂²×D₃×C₉

yes

characteristic

181

C₈

—

182

C₂×C₃

—

183

C₂×C₃

—

184

C₂×C₃

—

185

C₂×C₃

—

186

C₂×C₃

—

187

C₂×C₃

C₉×#124

yes

characteristic

188

C₂×C₃

C₉×AGL(1,ℤ/8)

yes

—

189

C₂×C₃

C₃×#932

yes

—

190

C₂×C₃

C₃×#936

yes

characteristic

191

C₂²

—

192

C₂²

—

193

C₂²

C₂×C₉×#67

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

C₉×#237

yes

—

197

C₃

—

198

C₃

C₉×#490

yes

characteristic, complemented

199

C₃

C₃×#5071

yes

characteristic

200

C₂

—

201

C₂

C₉×#932

yes

—

202

C₂

C₉×#936

yes

characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#127	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃²×#127	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#127	D₃
hypocenter	C₃	C₉×#490
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#127

subgroup

G/H

center

C₂²×C₉

C₂×#67

commutator

C₂×C₃×C₄

C₂³×C₉

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

C₃×C₉×#127

C₂

socle

C₂²×C₃²

C₂×C₃×D₄

prime radical

C₃²×#127

C₂×C₃

cosocle

C₂×C₃²×C₄

C₂³×C₃

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#127

N	H ≅ G/N	type
2. C₉×#870	C₂	semidirect
3. C₉×#832	C₂	semidirect
4. C₉×#803	C₂	semidirect
30. #5071	C₉	direct (N × H)
56. #870	C₂×C₉	semidirect
59. #832	C₂×C₉	semidirect
60. #803	C₂×C₉	semidirect
120. C₃×C₉	#490	semidirect
174. C₉	#5071	direct (N × H)
198. C₃	C₉×#490	semidirect

H ≅ G/N

type

2. C₉×#870

C₂

semidirect

3. C₉×#832

C₂

semidirect

4. C₉×#803

C₂

semidirect

30. #5071

C₉

direct (N × H)

56. #870

C₂×C₉

semidirect

59. #832

C₂×C₉

semidirect

60. #803

C₂×C₉

semidirect

120. C₃×C₉

#490

semidirect

174. C₉

#5071

direct (N × H)

198. C₃

C₉×#490

semidirect

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

subgroup	index
C₃×C₉×#127	2
C₉×#870	2
C₉×#832	2
C₉×#803	2
C₃×#5071	3
C₉×#490	3

subgroup

index

C₃×C₉×#127

Subgroups — C₉×#5071 #13142970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#5071 #13142970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5071 #13142970