Group #11766589 — subgroups

Subgroups — C₉×#1868 #11766589

Order 1152.

390 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 162 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	C₉×#568	C₂	1	1	yes	maximal, characteristic
2	C₉×#424	C₂	1	2	yes	maximal, complemented
3	C₉×#410	C₂	1	1	yes	maximal, characteristic
4	C₉×#408	C₂	1	1	yes	maximal, characteristic
5	C₉×#344	C₂	1	2	yes	maximal, complemented
6	C₃×#1868	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₉×#131	—	2	4	no	—
9	C₂×C₉×M₁₆	—	2	2	no	—
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₉×#117	C₂²	1	2	yes	—
14	C₉×C₄⋊C₈	—	2	4	no	—
15	C₉×#99	C₄	1	4	yes	—
16	C₉×C₂²⋊C₈	—	2	2	no	—
17	C₃×#568	C₂×C₃	1	1	yes	characteristic
18	C₃×#424	C₂×C₃	1	2	yes	—
19	C₃×#410	C₂×C₃	1	1	yes	characteristic
20	C₃×#408	C₂×C₃	1	1	yes	characteristic
21	C₃×#344	C₂×C₃	1	2	yes	—
22	C₉×#41	—	2	2	no	—
23	C₉×#41	—	2	2	no	—
24	C₂×Q₈×C₉	Q₈	1	1	yes	characteristic
25	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
26	C₂×D₄×C₉	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₉	D₄	1	2	yes	—
28	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₉×M₁₆	—	2	4	no	—
30	C₉×M₁₆	—	2	2	no	—
31	C₉×M₁₆	—	2	2	no	—
32	C₉×M₁₆	—	2	2	no	—
33	C₂×C₈×C₉	—	2	4	no	—
34	C₂×C₈×C₉	—	2	1	no	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₂×C₈×C₉	—	2	1	no	—
37	C₂×C₈×C₉	D₄	1	4	yes	—
38	C₉×C₄⋊C₄	—	2	2	no	—
39	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
40	C₄²×C₉	—	2	2	no	—
41	#1868	C₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
43	C₃×#131	—	2	4	no	—
44	C₂×C₃×M₁₆	—	2	2	no	—
45	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
47	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₃×#117	C₂²×C₃	1	2	yes	—
49	C₃×C₄⋊C₈	—	2	4	no	—
50	C₃×#99	C₃×C₄	1	4	yes	—
51	C₃×C₂²⋊C₈	—	2	2	no	—
52	C₂³×C₉	C₄⋊C₄	1	2	yes	—
53	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
54	Q₈×C₉	—	2	2	no	—
55	D₄×C₉	—	2	4	no	—
56	D₄×C₉	—	2	2	no	—
57	C₂×C₄×C₉	—	2	2	no	—
58	C₂×C₄×C₉	—	2	2	no	—
59	C₂×C₄×C₉	—	2	4	no	—
60	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₉	#41	1	2	yes	—
65	C₈×C₉	—	2	4	no	—
66	C₈×C₉	—	4	1	no	—
67	C₈×C₉	—	4	1	no	—
68	#568	C₂×C₉	1	1	yes	characteristic
69	#424	C₂×C₉	1	2	yes	complemented
70	#410	C₂×C₉	1	1	yes	characteristic
71	#408	C₂×C₉	1	1	yes	characteristic
72	#344	C₂×C₉	1	2	yes	complemented
73	C₃×#41	—	2	2	no	—
74	C₃×#41	—	2	2	no	—
75	C₂×C₃×Q₈	C₃×Q₈	1	1	yes	characteristic
76	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
77	C₂×C₃×D₄	C₃×Q₈	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
79	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
80	C₃×M₁₆	—	2	4	no	—
81	C₃×M₁₆	—	2	2	no	—
82	C₃×M₁₆	—	2	2	no	—
83	C₃×M₁₆	—	2	2	no	—
84	C₂×C₃×C₈	—	2	4	no	—
85	C₂×C₃×C₈	—	2	1	no	—
86	C₂×C₃×C₈	—	2	1	no	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
91	C₃×C₄²	—	2	2	no	—
92	C₂²×C₉	—	2	2	no	—
93	C₂²×C₉	—	4	1	no	—
94	C₂²×C₉	—	4	2	no	—
95	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₉	C₄×D₄	1	2	yes	—
97	C₄×C₉	—	2	2	no	—
98	C₄×C₉	—	4	4	no	—
99	C₄×C₉	#120	1	1	yes	characteristic
100	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
101	C₄×C₉	#123	1	1	yes	characteristic
102	C₂×#41	C₂²×C₉	1	1	yes	characteristic
103	#131	—	2	4	no	—
104	C₂×M₁₆	—	2	2	no	—
105	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
106	C₂×M₁₆	C₂²×C₉	1	2	yes	—
107	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
108	#117	C₂²×C₉	1	2	yes	—
109	C₄⋊C₈	—	2	4	no	—
110	#99	C₄×C₉	1	4	yes	—
111	C₂²⋊C₈	—	2	2	no	—
112	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
113	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
114	C₃×Q₈	—	2	2	no	—
115	C₃×D₄	—	2	4	no	—
116	C₃×D₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	4	no	—
120	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
121	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
122	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
124	C₂×C₃×C₄	C₃×#41	1	2	yes	—
125	C₃×C₈	—	2	4	no	—
126	C₃×C₈	—	4	1	no	—
127	C₃×C₈	—	4	1	no	—
128	C₂×C₉	—	2	2	no	—
129	C₂×C₉	—	2	1	no	—
130	C₂×C₉	—	4	2	no	—
131	C₂×C₉	#385	1	1	yes	characteristic
132	#41	—	2	2	no	—
133	#41	—	2	2	no	—
134	C₂×Q₈	Q₈×C₉	1	1	yes	characteristic
135	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
136	C₂×D₄	Q₈×C₉	1	1	yes	characteristic
137	C₂²×C₄	D₄×C₉	1	2	yes	—
138	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	—	2	2	no	—
142	M₁₆	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	D₄×C₉	1	4	yes	—
148	C₄⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	—	4	1	no	—
153	C₂²×C₃	—	4	2	no	—
154	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	4	4	no	—
158	C₃×C₄	C₃×#120	1	1	yes	characteristic
159	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
160	C₃×C₄	C₃×#123	1	1	yes	characteristic
161	C₉	#1868	1	1	yes	characteristic, direct product factor, Sylow
162	C₂³	C₉×C₄⋊C₄	1	2	yes	—
163	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
164	Q₈	—	2	2	no	—
165	D₄	—	2	4	no	—
166	D₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	—
171	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
172	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
174	C₂×C₄	C₉×#41	1	2	yes	—
175	C₈	—	2	4	no	—
176	C₈	—	4	1	no	—
177	C₈	—	4	1	no	—
178	C₂×C₃	—	2	2	no	—
179	C₂×C₃	—	2	1	no	—
180	C₂×C₃	—	4	2	no	—
181	C₂×C₃	C₃×#385	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	—	4	1	no	—
184	C₂²	—	4	2	no	—
185	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
186	C₂²	C₄×D₄×C₉	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	4	4	no	—
189	C₄	C₉×#120	1	1	yes	characteristic
190	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
191	C₄	C₉×#123	1	1	yes	characteristic
192	C₃	C₃×#1868	1	1	yes	characteristic
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	C₉×#385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#568

C₂

yes

maximal, characteristic

C₉×#424

C₂

yes

maximal, complemented

C₉×#410

C₂

yes

maximal, characteristic

C₉×#408

C₂

yes

maximal, characteristic

C₉×#344

C₂

yes

maximal, complemented

C₃×#1868

C₃

yes

maximal, characteristic

C₂×C₉×#41

C₂²

yes

characteristic

C₉×#131

—

C₂×C₉×M₁₆

—

C₂×C₉×M₁₆

C₂²

yes

characteristic

C₂×C₉×M₁₆

C₂²

yes

—

C₂²×C₈×C₉

C₂²

yes

characteristic

C₉×#117

C₂²

yes

—

C₉×C₄⋊C₈

—

C₉×#99

C₄

yes

—

C₉×C₂²⋊C₈

—

C₃×#568

C₂×C₃

yes

characteristic

C₃×#424

C₂×C₃

yes

—

C₃×#410

C₂×C₃

yes

characteristic

C₃×#408

C₂×C₃

yes

characteristic

C₃×#344

C₂×C₃

yes

—

C₉×#41

—

C₉×#41

—

C₂×Q₈×C₉

Q₈

yes

characteristic

C₂×D₄×C₉

C₂×C₄

yes

—

C₂×D₄×C₉

Q₈

yes

characteristic

C₂²×C₄×C₉

D₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₉×M₁₆

—

C₉×M₁₆

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

D₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₉

—

#1868

C₉

yes

characteristic, direct product factor, Sylow

C₂×C₃×#41

C₂²×C₃

yes

characteristic

C₃×#131

—

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

C₂²×C₃

yes

characteristic

C₂×C₃×M₁₆

C₂²×C₃

yes

—

C₂²×C₃×C₈

C₂²×C₃

yes

characteristic

C₃×#117

C₂²×C₃

yes

—

C₃×C₄⋊C₈

—

C₃×#99

C₃×C₄

yes

—

C₃×C₂²⋊C₈

—

C₂³×C₉

C₄⋊C₄

yes

—

C₂³×C₉

C₂²×C₄

yes

characteristic

Q₈×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₄⋊C₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

—

#568

C₂×C₉

yes

characteristic

#424

C₂×C₉

yes

complemented

#410

C₂×C₉

yes

characteristic

#408

C₂×C₉

yes

characteristic

#344

C₂×C₉

yes

complemented

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

C₃×Q₈

yes

characteristic

C₂×C₃×D₄

C₂×C₃×C₄

yes

—

C₂×C₃×D₄

C₃×Q₈

yes

characteristic

C₂²×C₃×C₄

C₃×D₄

yes

—

C₂²×C₃×C₄

C₂³×C₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

C₂×C₃×C₄

yes

—

C₃×C₄²

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉

C₄×D₄

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#120

yes

characteristic

100

C₄×C₉

C₂²⋊Q₈

yes

—

101

C₄×C₉

#123

yes

characteristic

102

C₂×#41

C₂²×C₉

yes

characteristic

103

#131

—

104

C₂×M₁₆

—

105

C₂×M₁₆

C₂²×C₉

yes

characteristic

106

C₂×M₁₆

C₂²×C₉

yes

—

107

C₂²×C₈

C₂²×C₉

yes

characteristic

108

#117

C₂²×C₉

yes

—

109

C₄⋊C₈

—

110

#99

C₄×C₉

yes

—

111

C₂²⋊C₈

—

112

C₂³×C₃

C₃×C₄⋊C₄

yes

—

113

C₂³×C₃

C₂²×C₃×C₄

yes

characteristic

114

C₃×Q₈

—

115

C₃×D₄

—

116

C₃×D₄

—

117

C₂×C₃×C₄

—

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

—

120

C₂×C₃×C₄

C₃×C₄⋊C₄

yes

—

121

C₂×C₃×C₄

C₂×C₃×D₄

yes

—

122

C₂×C₃×C₄

C₂×C₃×D₄

yes

characteristic

123

C₂×C₃×C₄

C₂×C₃×Q₈

yes

characteristic

124

C₂×C₃×C₄

C₃×#41

yes

—

125

C₃×C₈

—

126

C₃×C₈

—

127

C₃×C₈

—

128

C₂×C₉

—

129

C₂×C₉

—

130

C₂×C₉

—

131

C₂×C₉

#385

yes

characteristic

132

#41

—

133

#41

—

134

C₂×Q₈

Q₈×C₉

yes

characteristic

135

C₂×D₄

C₂×C₄×C₉

yes

—

136

C₂×D₄

Q₈×C₉

yes

characteristic

137

C₂²×C₄

D₄×C₉

yes

—

138

C₂²×C₄

C₂³×C₉

yes

characteristic

139

M₁₆

—

140

M₁₆

—

141

M₁₆

—

142

M₁₆

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

D₄×C₉

yes

—

148

C₄⋊C₄

—

149

C₂²⋊C₄

C₂×C₄×C₉

yes

—

150

C₄²

—

151

C₂²×C₃

—

152

C₂²×C₃

—

153

C₂²×C₃

—

154

C₂²×C₃

C₂×C₃×C₄⋊C₄

yes

characteristic

155

C₂²×C₃

C₃×C₄×D₄

yes

—

156

C₃×C₄

—

157

C₃×C₄

—

158

C₃×C₄

C₃×#120

yes

characteristic

159

C₃×C₄

C₃×C₂²⋊Q₈

yes

—

160

C₃×C₄

C₃×#123

yes

characteristic

161

C₉

#1868

yes

characteristic, direct product factor, Sylow

162

C₂³

C₉×C₄⋊C₄

yes

—

163

C₂³

C₂²×C₄×C₉

yes

characteristic

164

Q₈

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

C₉×C₄⋊C₄

yes

—

171

C₂×C₄

C₂×D₄×C₉

yes

—

172

C₂×C₄

C₂×D₄×C₉

yes

characteristic

173

C₂×C₄

C₂×Q₈×C₉

yes

characteristic

174

C₂×C₄

C₉×#41

yes

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₂×C₃

—

179

C₂×C₃

—

180

C₂×C₃

—

181

C₂×C₃

C₃×#385

yes

characteristic

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

C₂×C₉×C₄⋊C₄

yes

characteristic

186

C₂²

C₄×D₄×C₉

yes

—

187

C₄

—

188

C₄

—

189

C₄

C₉×#120

yes

characteristic

190

C₄

C₉×C₂²⋊Q₈

yes

—

191

C₄

C₉×#123

yes

characteristic

192

C₃

C₃×#1868

yes

characteristic

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

C₉×#385

yes

characteristic

	subgroup	G/H
center	C₄×C₉	#120
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#385
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

subgroup

G/H

center

C₄×C₉

#120

commutator

C₂³

C₂²×C₄×C₉

Frattini, cosocle

C₂²×C₃×C₄

C₂³×C₃

socle

C₂×C₃

C₃×#385

prime radical

C₂×C₃×#41

C₂²×C₃

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

N	H ≅ G/N	type
2. C₉×#424	C₂	semidirect
5. C₉×#344	C₂	semidirect
41. #1868	C₉	direct (N × H)
69. #424	C₂×C₉	semidirect
72. #344	C₂×C₉	semidirect
161. C₉	#1868	direct (N × H)

H ≅ G/N

type

2. C₉×#424

C₂

semidirect

5. C₉×#344

C₂

semidirect

41. #1868

C₉

direct (N × H)

69. #424

C₂×C₉

semidirect

72. #344

C₂×C₉

semidirect

161. C₉

#1868

direct (N × H)

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

subgroup	index
C₉×#568	2
C₉×#424	2
C₉×#410	2
C₉×#408	2
C₉×#344	2
C₃×#1868	3

subgroup

index

Subgroups — C₉×#1868 #11766589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#1868 #11766589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1868 #11766589