Group #13142947 — subgroups

Subgroups — C₉×#5067 #13142947

Order 1728.

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

normal subgroups only

This group has 386 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×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#874	C₂	1	1	yes	maximal, characteristic
3	C₉×#809	C₂	1	4	yes	maximal, complemented
4	C₉×#803	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#5067	C₃	1	1	yes	maximal, characteristic
6	C₉×#490	—	3	1	no	maximal
7	C₂×C₉×D₁₂	—	2	2	no	—
8	C₂×C₉×D₁₂	C₂²	1	2	yes	—
9	C₃×C₉×C₄⋊C₄	—	2	2	no	—
10	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃²×#125	C₂×C₃	1	1	yes	characteristic
14	C₃×#874	C₂×C₃	1	1	yes	characteristic
15	C₃×#809	C₂×C₃	1	4	yes	—
16	C₃×#803	C₂×C₃	1	1	yes	characteristic
17	C₉×#127	—	3	1	no	—
18	C₉×#125	—	2	1	no	—
19	C₉×#125	D₃	1	1	yes	characteristic, complemented
20	C₉×#102	—	3	4	no	—
21	C₉×#97	—	3	1	no	—
22	C₂²×D₃×C₉	—	2	2	no	—
23	C₂×C₃×C₄×C₉	—	2	2	no	—
24	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
26	C₉×D₁₂	—	2	4	no	—
27	C₉×D₁₂	—	4	4	no	—
28	C₉×C₃⋊C₈	—	2	2	no	—
29	C₃×#490	—	3	1	no	—
30	#5067	C₉	1	1	yes	characteristic, direct product factor
31	C₂×C₃×D₁₂	—	2	2	no	—
32	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
33	C₃²×C₄⋊C₄	—	2	2	no	—
34	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
35	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
36	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
37	C₂×D₄×C₉	—	3	2	no	—
38	C₂×D₄×C₉	—	6	2	no	—
39	C₂×C₈×C₉	—	3	2	no	—
40	C₉×C₄⋊C₄	—	2	4	no	—
41	C₉×C₄⋊C₄	—	2	2	no	—
42	C₉×C₄⋊C₄	—	2	2	no	—
43	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
44	C₄²×C₉	—	2	1	no	—
45	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
47	C₂×D₃×C₉	—	4	2	no	—
48	C₂×D₃×C₉	—	4	4	no	—
49	C₃×C₄×C₉	—	2	2	no	—
50	C₃×C₄×C₉	—	4	2	no	—
51	C₃×C₄×C₉	#41	1	2	yes	—
52	C₃×#127	—	3	1	no	—
53	C₃×#125	—	2	1	no	—
54	C₃×#125	C₂×C₉	1	1	yes	characteristic, complemented
55	C₃×#125	C₃×D₃	1	1	yes	characteristic
56	#874	C₂×C₉	1	1	yes	characteristic
57	C₃×#102	—	3	4	no	—
58	C₃×#97	—	3	1	no	—
59	#809	C₂×C₉	1	4	yes	complemented
60	#803	C₂×C₉	1	1	yes	characteristic, complemented
61	C₂²×C₃×D₃	—	2	2	no	—
62	C₂×C₃²×C₄	—	2	2	no	—
63	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
64	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
65	C₃×D₁₂	—	2	4	no	—
66	C₃×D₁₂	—	4	4	no	—
67	C₂³×C₉	—	6	2	no	—
68	C₃×C₃⋊C₈	—	2	2	no	—
69	D₄×C₉	—	6	4	no	—
70	D₄×C₉	—	12	4	no	—
71	C₂×C₄×C₉	—	2	4	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	2	1	no	—
75	C₂×C₄×C₉	#67	1	2	yes	—
76	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
77	C₈×C₉	—	6	2	no	—
78	#490	—	3	1	no	Sylow
79	C₂×C₃×C₉	#124	1	1	yes	characteristic
80	C₂×C₃×C₉	AGL(1,ℤ/8)	1	2	yes	—
81	D₃×C₉	—	8	2	no	—
82	C₂×C₃×D₄	—	3	2	no	—
83	C₂×C₃×D₄	—	6	2	no	—
84	C₂×D₁₂	—	2	2	no	—
85	C₂×D₁₂	C₂²×C₉	1	2	yes	—
86	C₂×C₃×C₈	—	3	2	no	—
87	C₃×C₄⋊C₄	—	2	4	no	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	—	2	2	no	—
91	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
92	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
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	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
97	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
98	C₂×C₃×D₃	—	4	2	no	—
99	C₂×C₃×D₃	—	4	4	no	—
100	C₃²×C₄	—	2	2	no	—
101	C₃²×C₄	—	4	2	no	—
102	C₃²×C₄	C₃×#41	1	2	yes	—
103	C₂²×C₉	—	2	1	no	—
104	C₂²×C₉	—	12	2	no	—
105	C₂²×C₉	—	12	4	no	—
106	C₂²×C₉	C₂×#67	1	1	yes	characteristic
107	C₄×C₉	—	2	2	no	—
108	C₄×C₉	—	2	2	no	—
109	C₄×C₉	—	4	2	no	—
110	C₄×C₉	—	4	4	no	—
111	C₄×C₉	—	4	2	no	—
112	C₄×C₉	#239	1	2	yes	—
113	#127	—	3	1	no	—
114	#125	D₃×C₉	1	1	yes	characteristic, complemented
115	#102	—	3	4	no	—
116	#97	—	3	1	no	—
117	C₃×C₉	#490	1	1	yes	characteristic, complemented, Sylow
118	C₂³×C₃	—	6	2	no	—
119	C₂²×D₃	—	2	2	no	—
120	C₃×D₄	—	6	4	no	—
121	C₃×D₄	—	12	4	no	—
122	C₂×C₃×C₄	—	2	4	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
128	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
129	C₂×C₃×C₄	C₃×#67	1	2	yes	—
130	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
131	D₁₂	—	2	4	no	—
132	D₁₂	—	4	4	no	—
133	C₃×C₈	—	6	2	no	—
134	C₃⋊C₈	—	2	2	no	—
135	C₂×C₃²	C₃×#124	1	1	yes	characteristic
136	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	2	yes	—
137	C₃×D₃	—	8	2	no	—
138	C₂×C₉	—	2	1	no	—
139	C₂×C₉	—	2	2	no	—
140	C₂×C₉	—	24	2	no	—
141	C₂×C₉	#947	1	1	yes	characteristic
142	C₂×C₉	#949	1	2	yes	—
143	C₂×D₄	—	3	2	no	—
144	C₂×D₄	—	6	2	no	—
145	C₂×C₈	—	3	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
148	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
149	C₂²×C₃	—	2	1	no	—
150	C₂²×C₃	—	12	2	no	—
151	C₂²×C₃	—	12	4	no	—
152	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
153	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
154	C₂×D₃	—	4	2	no	—
155	C₂×D₃	—	4	4	no	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	4	2	no	—
160	C₃×C₄	—	4	4	no	—
161	C₃×C₄	—	4	2	no	—
162	C₃×C₄	—	4	2	no	—
163	C₃×C₄	C₉×#41	1	2	yes	—
164	C₃×C₄	C₃×#239	1	2	yes	—
165	C₃²	C₃×#490	1	1	yes	characteristic
166	C₉	—	2	1	no	—
167	C₉	#5067	1	1	yes	characteristic, direct product factor
168	C₂³	—	6	2	no	—
169	D₄	—	6	4	no	—
170	D₄	—	12	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	C₉×#67	1	2	yes	—
173	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
174	C₈	—	6	2	no	—
175	C₂×C₃	—	2	1	no	—
176	C₂×C₃	—	2	2	no	—
177	C₂×C₃	—	24	2	no	—
178	C₂×C₃	C₉×#124	1	1	yes	characteristic
179	C₂×C₃	C₉×AGL(1,ℤ/8)	1	2	yes	—
180	C₂×C₃	C₃×#947	1	1	yes	characteristic
181	C₂×C₃	C₃×#949	1	2	yes	—
182	D₃	—	8	2	no	—
183	C₂²	—	12	2	no	—
184	C₂²	—	12	4	no	—
185	C₂²	C₂×C₉×#67	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	C₉×#239	1	2	yes	—
189	C₃	—	2	1	no	—
190	C₃	C₉×#490	1	1	yes	characteristic, complemented
191	C₃	C₃×#5067	1	1	yes	characteristic
192	C₂	—	24	2	no	—
193	C₂	C₉×#947	1	1	yes	characteristic
194	C₂	C₉×#949	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×#125

C₂

yes

maximal, characteristic, complemented

C₉×#874

C₂

yes

maximal, characteristic

C₉×#809

C₂

yes

maximal, complemented

C₉×#803

C₂

yes

maximal, characteristic, complemented

C₃×#5067

C₃

yes

maximal, characteristic

C₉×#490

—

maximal

C₂×C₉×D₁₂

—

C₂×C₉×D₁₂

C₂²

yes

—

C₃×C₉×C₄⋊C₄

—

C₃×C₉×C₄⋊C₄

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₂×C₉×C₃⋊C₈

C₂²

yes

—

C₃²×#125

C₂×C₃

yes

characteristic

C₃×#874

C₂×C₃

yes

characteristic

C₃×#809

C₂×C₃

yes

—

C₃×#803

C₂×C₃

yes

—

—

D₃

yes

characteristic, complemented

C₉×#102

—

C₉×#97

—

C₂²×D₃×C₉

—

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₉×D₁₂

—

C₉×D₁₂

—

C₉×C₃⋊C₈

—

C₃×#490

—

#5067

C₉

yes

characteristic, direct product factor

C₂×C₃×D₁₂

—

C₂×C₃×D₁₂

C₂²×C₃

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

C₂²×C₃

yes

—

C₃²×C₄²

C₂²×C₃

yes

characteristic

C₂×C₃×C₃⋊C₈

C₂²×C₃

yes

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×C₈×C₉

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂×D₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₂×D₃×C₉

—

C₂×D₃×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

#41

yes

—

C₃×#127

—

C₃×#125

—

C₃×#125

C₂×C₉

yes

characteristic, complemented

C₃×#125

C₃×D₃

yes

characteristic

#874

C₂×C₉

yes

characteristic

C₃×#102

—

C₃×#97

—

#809

C₂×C₉

yes

complemented

#803

C₂×C₉

yes

characteristic, complemented

C₂²×C₃×D₃

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₃×D₄

yes

—

C₂×C₃²×C₄

C₂³×C₃

yes

characteristic

C₃×D₁₂

—

C₃×D₁₂

—

C₂³×C₉

—

C₃×C₃⋊C₈

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₈×C₉

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

D₃×C₉

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₂×D₁₂

C₂²×C₉

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂²×C₉

yes

—

C₃×C₄⋊C₄

C₂×C₃×D₃

yes

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₉

yes

characteristic

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₂×C₃⋊C₈

C₂²×C₉

yes

—

C₂²×C₃²

C₂×C₃×D₄

yes

characteristic

C₂×C₃×D₃

—

C₂×C₃×D₃

—

100

C₃²×C₄

—

101

C₃²×C₄

—

102

C₃²×C₄

C₃×#41

yes

—

103

C₂²×C₉

—

104

C₂²×C₉

—

105

C₂²×C₉

—

106

C₂²×C₉

C₂×#67

yes

characteristic

107

C₄×C₉

—

108

C₄×C₉

—

109

C₄×C₉

—

110

C₄×C₉

—

111

C₄×C₉

—

112

C₄×C₉

#239

yes

—

113

#127

—

114

#125

D₃×C₉

yes

characteristic, complemented

115

#102

—

116

#97

—

117

C₃×C₉

#490

yes

characteristic, complemented, Sylow

118

C₂³×C₃

—

119

C₂²×D₃

—

120

C₃×D₄

—

121

C₃×D₄

—

122

C₂×C₃×C₄

—

123

C₂×C₃×C₄

—

124

C₂×C₃×C₄

—

125

C₂×C₃×C₄

—

126

C₂×C₃×C₄

—

127

C₂×C₃×C₄

D₄×C₉

yes

—

128

C₂×C₃×C₄

C₂³×C₉

yes

characteristic

129

C₂×C₃×C₄

C₃×#67

yes

—

130

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

131

D₁₂

—

132

D₁₂

—

133

C₃×C₈

—

134

C₃⋊C₈

—

135

C₂×C₃²

C₃×#124

yes

characteristic

136

C₂×C₃²

C₃×AGL(1,ℤ/8)

yes

—

137

C₃×D₃

—

138

C₂×C₉

—

139

C₂×C₉

—

140

C₂×C₉

—

141

C₂×C₉

#947

yes

characteristic

142

C₂×C₉

#949

yes

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×C₈

—

146

C₄⋊C₄

—

147

C₄⋊C₄

C₂×D₃×C₉

yes

—

148

C₄²

C₂×D₃×C₉

yes

characteristic

149

C₂²×C₃

—

150

C₂²×C₃

—

151

C₂²×C₃

—

152

C₂²×C₃

C₂×D₄×C₉

yes

characteristic

153

C₂²×C₃

C₂×C₃×#67

yes

characteristic

154

C₂×D₃

—

155

C₂×D₃

—

156

C₃×C₄

—

157

C₃×C₄

—

158

C₃×C₄

—

159

C₃×C₄

—

160

C₃×C₄

—

161

C₃×C₄

—

162

C₃×C₄

—

163

C₃×C₄

C₉×#41

yes

—

164

C₃×C₄

C₃×#239

yes

—

165

C₃²

C₃×#490

yes

characteristic

166

C₉

—

167

C₉

#5067

yes

characteristic, direct product factor

168

C₂³

—

169

D₄

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

C₉×#67

yes

—

173

C₂×C₄

C₂²×D₃×C₉

yes

characteristic

174

C₈

—

175

C₂×C₃

—

176

C₂×C₃

—

177

C₂×C₃

—

178

C₂×C₃

C₉×#124

yes

characteristic

179

C₂×C₃

C₉×AGL(1,ℤ/8)

yes

—

180

C₂×C₃

C₃×#947

yes

characteristic

181

C₂×C₃

C₃×#949

yes

—

182

D₃

—

183

C₂²

—

184

C₂²

—

185

C₂²

C₂×C₉×#67

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

C₉×#239

yes

—

189

C₃

—

190

C₃

C₉×#490

yes

characteristic, complemented

191

C₃

C₃×#5067

yes

characteristic

192

C₂

—

193

C₂

C₉×#947

yes

characteristic

194

C₂

C₉×#949

yes

—

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#125	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#874	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#125	D₃
hypocenter	C₃	C₉×#490
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#125

subgroup

G/H

center

C₂²×C₉

C₂×#67

commutator

C₂×C₃×C₄

C₂³×C₉

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

C₃×C₉×#125

C₂

socle

C₂²×C₃²

C₂×C₃×D₄

prime radical

C₃×#874

C₂×C₃

cosocle

C₂×C₃²×C₄

C₂³×C₃

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#125

N	H ≅ G/N	type
1. C₃×C₉×#125	C₂	semidirect
3. C₉×#809	C₂	semidirect
4. C₉×#803	C₂	semidirect
19. C₉×#125	D₃	semidirect
30. #5067	C₉	direct (N × H)
54. C₃×#125	C₂×C₉	semidirect
59. #809	C₂×C₉	semidirect
60. #803	C₂×C₉	semidirect
114. #125	D₃×C₉	semidirect
117. C₃×C₉	#490	semidirect
167. C₉	#5067	direct (N × H)
190. C₃	C₉×#490	semidirect

H ≅ G/N

type

1. C₃×C₉×#125

C₂

semidirect

3. C₉×#809

C₂

semidirect

4. C₉×#803

C₂

semidirect

19. C₉×#125

D₃

semidirect

30. #5067

C₉

direct (N × H)

54. C₃×#125

C₂×C₉

semidirect

59. #809

C₂×C₉

semidirect

60. #803

C₂×C₉

semidirect

114. #125

D₃×C₉

semidirect

117. C₃×C₉

#490

semidirect

167. C₉

#5067

direct (N × H)

190. 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×#125	2
C₉×#874	2
C₉×#809	2
C₉×#803	2
C₃×#5067	3
C₉×#490	3

subgroup

index

C₃×C₉×#125

Subgroups — C₉×#5067 #13142947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#5067 #13142947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5067 #13142947