Group #11766542 — subgroups

Subgroups — C₉×#1821 #11766542

Order 1152.

474 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 168 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	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₉×#407	C₂	1	2	yes	maximal
3	C₉×#343	C₂	1	4	yes	maximal, complemented
4	C₃×#1821	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₉	C₄	1	2	yes	—
6	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₉×M₁₆	C₂²	1	4	yes	—
8	C₂×C₉×C₄⋊C₄	—	2	2	no	—
9	C₂×C₉×C₄⋊C₄	—	2	2	no	—
10	C₂×C₉×C₄⋊C₄	C₄	1	2	yes	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₉×C₂²⋊C₈	—	2	2	no	—
13	C₉×C₂²⋊C₈	—	2	2	no	—
14	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
15	C₃×#407	C₂×C₃	1	2	yes	—
16	C₃×#343	C₂×C₃	1	4	yes	—
17	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₉	—	2	4	no	—
19	C₂²×C₄×C₉	—	2	2	no	—
20	C₂²×C₄×C₉	—	2	2	no	—
21	C₂²×C₄×C₉	—	4	4	no	—
22	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₉	D₄	1	2	yes	—
25	C₂²×C₄×C₉	D₄	1	2	yes	—
26	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₉×M₁₆	—	2	8	no	—
28	C₂×C₈×C₉	—	2	4	no	—
29	C₉×C₄⋊C₄	—	2	4	no	—
30	C₉×C₄⋊C₄	—	2	4	no	—
31	#1821	C₉	1	1	yes	characteristic, direct product factor, Sylow
32	C₂³×C₃×C₄	C₃×C₄	1	2	yes	—
33	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
34	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
35	C₂×C₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₃×C₄⋊C₄	C₃×C₄	1	2	yes	—
38	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
39	C₃×C₂²⋊C₈	—	2	2	no	—
40	C₃×C₂²⋊C₈	—	2	2	no	—
41	C₂³×C₉	—	2	2	no	—
42	C₂³×C₉	—	2	2	no	—
43	C₂³×C₉	—	2	2	no	—
44	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₉	—	2	2	no	—
48	C₂×C₄×C₉	—	2	4	no	—
49	C₂×C₄×C₉	—	2	4	no	—
50	C₂×C₄×C₉	—	2	4	no	—
51	C₂×C₄×C₉	—	4	4	no	—
52	C₂×C₄×C₉	—	4	4	no	—
53	C₂×C₄×C₉	—	4	4	no	—
54	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₉	#41	1	4	yes	—
57	C₈×C₉	—	4	4	no	—
58	C₂²×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
59	#407	C₂×C₉	1	2	yes	—
60	#343	C₂×C₉	1	4	yes	complemented
61	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₄	—	2	4	no	—
63	C₂²×C₃×C₄	—	2	2	no	—
64	C₂²×C₃×C₄	—	2	2	no	—
65	C₂²×C₃×C₄	—	4	4	no	—
66	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
68	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
69	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
70	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
71	C₃×M₁₆	—	2	8	no	—
72	C₂×C₃×C₈	—	2	4	no	—
73	C₃×C₄⋊C₄	—	2	4	no	—
74	C₃×C₄⋊C₄	—	2	4	no	—
75	C₂²×C₉	—	2	8	no	—
76	C₂²×C₉	—	2	1	no	—
77	C₂²×C₉	—	2	1	no	—
78	C₂²×C₉	—	2	1	no	—
79	C₂²×C₉	—	2	1	no	—
80	C₂²×C₉	—	2	1	no	—
81	C₂²×C₉	—	2	1	no	—
82	C₂²×C₉	#100	1	2	yes	—
83	C₂²×C₉	#101	1	2	yes	—
84	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₉	#117	1	2	yes	—
86	C₄×C₉	—	2	2	no	—
87	C₄×C₉	—	4	4	no	—
88	C₄×C₉	#123	1	4	yes	—
89	C₂³×C₄	C₄×C₉	1	2	yes	—
90	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
91	C₂×M₁₆	C₂²×C₉	1	4	yes	—
92	C₂×C₄⋊C₄	—	2	2	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	C₄×C₉	1	2	yes	—
95	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	2	4	no	—
108	C₂×C₃×C₄	—	4	4	no	—
109	C₂×C₃×C₄	—	4	4	no	—
110	C₂×C₃×C₄	—	4	4	no	—
111	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
112	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
113	C₂×C₃×C₄	C₃×#41	1	4	yes	—
114	C₃×C₈	—	4	4	no	—
115	C₂×C₉	—	2	2	no	—
116	C₂×C₉	—	2	2	no	—
117	C₂×C₉	—	2	2	no	—
118	C₂×C₉	#381	1	1	yes	characteristic
119	C₂×C₉	C₂×#100	1	1	yes	characteristic
120	C₂×C₉	C₂×#101	1	1	yes	characteristic
121	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	4	4	no	—
126	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
128	C₂²×C₄	D₄×C₉	1	2	yes	—
129	C₂²×C₄	D₄×C₉	1	2	yes	—
130	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
131	M₁₆	—	2	8	no	—
132	C₂×C₈	—	2	4	no	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₂²×C₃	—	2	8	no	—
136	C₂²×C₃	—	2	1	no	—
137	C₂²×C₃	—	2	1	no	—
138	C₂²×C₃	—	2	1	no	—
139	C₂²×C₃	—	2	1	no	—
140	C₂²×C₃	—	2	1	no	—
141	C₂²×C₃	—	2	1	no	—
142	C₂²×C₃	C₃×#100	1	2	yes	—
143	C₂²×C₃	C₃×#101	1	2	yes	—
144	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₃	C₃×#117	1	2	yes	—
146	C₃×C₄	—	2	2	no	—
147	C₃×C₄	—	4	4	no	—
148	C₃×C₄	C₃×#123	1	4	yes	—
149	C₉	#1821	1	1	yes	characteristic, direct product factor, Sylow
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	C₉×C₂²⋊C₄	1	1	yes	characteristic
154	C₂³	C₉×C₂²⋊C₄	1	1	yes	characteristic
155	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
164	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
165	C₂×C₄	C₉×#41	1	4	yes	—
166	C₈	—	4	4	no	—
167	C₂×C₃	—	2	2	no	—
168	C₂×C₃	—	2	2	no	—
169	C₂×C₃	—	2	2	no	—
170	C₂×C₃	C₃×#381	1	1	yes	characteristic
171	C₂×C₃	C₂×C₃×#100	1	1	yes	characteristic
172	C₂×C₃	C₂×C₃×#101	1	1	yes	characteristic
173	C₂²	—	2	8	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	C₉×#100	1	2	yes	—
181	C₂²	C₉×#101	1	2	yes	—
182	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
183	C₂²	C₉×#117	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	C₉×#123	1	4	yes	—
187	C₃	C₃×#1821	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	—	2	2	no	—
190	C₂	—	2	2	no	—
191	C₂	C₉×#381	1	1	yes	characteristic
192	C₂	C₂×C₉×#100	1	1	yes	characteristic
193	C₂	C₂×C₉×#101	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₉×C₄⋊C₄

C₂

yes

maximal, characteristic

C₉×#407

C₂

yes

maximal

C₉×#343

C₂

yes

maximal, complemented

C₃×#1821

C₃

yes

maximal, characteristic

C₂³×C₄×C₉

C₄

yes

—

C₂³×C₄×C₉

C₂²

yes

characteristic

C₂×C₉×M₁₆

C₂²

yes

—

C₂×C₉×C₄⋊C₄

—

C₂×C₉×C₄⋊C₄

—

C₂×C₉×C₄⋊C₄

C₄

yes

—

C₂×C₉×C₄⋊C₄

C₂²

yes

—

C₉×C₂²⋊C₈

—

C₉×C₂²⋊C₈

—

C₂²×C₃×C₄⋊C₄

C₂×C₃

yes

characteristic

C₃×#407

C₂×C₃

yes

—

C₃×#343

C₂×C₃

yes

—

C₂⁴×C₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

D₄

yes

—

C₂²×C₄×C₉

D₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₉×M₁₆

—

C₂×C₈×C₉

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

#1821

C₉

yes

characteristic, direct product factor, Sylow

C₂³×C₃×C₄

C₃×C₄

yes

—

C₂³×C₃×C₄

C₂²×C₃

yes

characteristic

C₂×C₃×M₁₆

C₂²×C₃

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₃×C₄

yes

—

C₂×C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₃×C₂²⋊C₈

—

C₃×C₂²⋊C₈

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₂²⋊C₄

yes

characteristic

C₂³×C₉

C₂²⋊C₄

yes

characteristic

C₂³×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²⋊C₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

#41

yes

—

C₈×C₉

—

C₂²×C₄⋊C₄

C₂×C₉

yes

characteristic

#407

C₂×C₉

yes

—

#343

C₂×C₉

yes

complemented

C₂⁴×C₃

C₂×C₃×C₄

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₃×C₄

yes

—

C₂²×C₃×C₄

C₃×D₄

yes

—

C₂²×C₃×C₄

C₃×D₄

yes

—

C₂²×C₃×C₄

C₂³×C₃

yes

characteristic

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

#100

yes

—

C₂²×C₉

#101

yes

—

C₂²×C₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉

#117

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#123

yes

—

C₂³×C₄

C₄×C₉

yes

—

C₂³×C₄

C₂²×C₉

yes

characteristic

C₂×M₁₆

C₂²×C₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₄×C₉

yes

—

C₂×C₄⋊C₄

C₂²×C₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂³×C₃

—

C₂³×C₃

—

100

C₂³×C₃

—

101

C₂³×C₃

C₃×C₂²⋊C₄

yes

characteristic

102

C₂³×C₃

C₃×C₂²⋊C₄

yes

characteristic

103

C₂³×C₃

C₂²×C₃×C₄

yes

characteristic

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

C₃×C₂²⋊C₄

yes

—

112

C₂×C₃×C₄

C₂×C₃×D₄

yes

—

113

C₂×C₃×C₄

C₃×#41

yes

—

114

C₃×C₈

—

115

C₂×C₉

—

116

C₂×C₉

—

117

C₂×C₉

—

118

C₂×C₉

#381

yes

characteristic

119

C₂×C₉

C₂×#100

yes

characteristic

120

C₂×C₉

C₂×#101

yes

characteristic

121

C₂⁴

C₂×C₄×C₉

yes

characteristic

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂×C₄×C₉

yes

characteristic

127

C₂²×C₄

C₂×C₄×C₉

yes

—

128

C₂²×C₄

D₄×C₉

yes

—

129

C₂²×C₄

D₄×C₉

yes

—

130

C₂²×C₄

C₂³×C₉

yes

characteristic

131

M₁₆

—

132

C₂×C₈

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₂²×C₃

—

136

C₂²×C₃

—

137

C₂²×C₃

—

138

C₂²×C₃

—

139

C₂²×C₃

—

140

C₂²×C₃

—

141

C₂²×C₃

—

142

C₂²×C₃

C₃×#100

yes

—

143

C₂²×C₃

C₃×#101

yes

—

144

C₂²×C₃

C₂×C₃×C₂²⋊C₄

yes

characteristic

145

C₂²×C₃

C₃×#117

yes

—

146

C₃×C₄

—

147

C₃×C₄

—

148

C₃×C₄

C₃×#123

yes

—

149

C₉

#1821

yes

characteristic, direct product factor, Sylow

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

C₉×C₂²⋊C₄

yes

characteristic

154

C₂³

C₉×C₂²⋊C₄

yes

characteristic

155

C₂³

C₂²×C₄×C₉

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

C₉×C₂²⋊C₄

yes

—

164

C₂×C₄

C₂×D₄×C₉

yes

—

165

C₂×C₄

C₉×#41

yes

—

166

C₈

—

167

C₂×C₃

—

168

C₂×C₃

—

169

C₂×C₃

—

170

C₂×C₃

C₃×#381

yes

characteristic

171

C₂×C₃

C₂×C₃×#100

yes

characteristic

172

C₂×C₃

C₂×C₃×#101

yes

characteristic

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

C₉×#100

yes

—

181

C₂²

C₉×#101

yes

—

182

C₂²

C₂×C₉×C₂²⋊C₄

yes

characteristic

183

C₂²

C₉×#117

yes

—

184

C₄

—

185

C₄

—

186

C₄

C₉×#123

yes

—

187

C₃

C₃×#1821

yes

characteristic

188

C₂

—

189

C₂

—

190

C₂

—

191

C₂

C₉×#381

yes

characteristic

192

C₂

C₂×C₉×#100

yes

characteristic

193

C₂

C₂×C₉×#101

yes

characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₂²⋊C₄
prime radical	C₂⁴×C₃	C₂×C₃×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂³×C₄×C₉ ◁ G

subgroup

G/H

center

C₂²×C₉

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₉

Frattini, cosocle

C₂²×C₃×C₄

C₂³×C₃

socle

C₂²×C₃

C₂×C₃×C₂²⋊C₄

prime radical

C₂⁴×C₃

C₂×C₃×C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉ ◁ C₂³×C₄×C₉ ◁ G

N	H ≅ G/N	type
3. C₉×#343	C₂	semidirect
31. #1821	C₉	direct (N × H)
60. #343	C₂×C₉	semidirect
149. C₉	#1821	direct (N × H)

H ≅ G/N

type

3. C₉×#343

C₂

semidirect

31. #1821

C₉

direct (N × H)

60. #343

C₂×C₉

semidirect

149. C₉

#1821

direct (N × H)

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

subgroup	index
C₂²×C₉×C₄⋊C₄	2
C₉×#407	2
C₉×#343	2
C₃×#1821	3

subgroup

index

C₂²×C₉×C₄⋊C₄

C₉×#407

C₉×#343

C₃×#1821

Subgroups — C₉×#1821 #11766542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#1821 #11766542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1821 #11766542