Group #11766768 — subgroups

Subgroups — C₉×#2047 #11766768

Order 1152.

384 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 126 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×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₉×#424	C₂	1	1	yes	maximal, characteristic
3	C₉×#419	C₂	1	2	yes	maximal
4	C₂×C₉×#101	C₂	1	2	yes	maximal
5	C₉×#341	C₂	1	1	yes	maximal, characteristic
6	C₃×#2047	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₂×C₉×Q₁₆	—	2	4	no	—
9	C₂×C₉×Q₁₆	—	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	1	yes	characteristic
14	C₉×C₄⋊C₈	—	2	2	no	—
15	C₉×#103	—	2	4	no	—
16	C₉×#101	—	2	4	no	—
17	C₂²×C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic
18	C₃×#424	C₂×C₃	1	1	yes	characteristic
19	C₃×#419	C₂×C₃	1	2	yes	—
20	C₂×C₃×#101	C₂×C₃	1	2	yes	—
21	C₃×#341	C₂×C₃	1	1	yes	characteristic
22	C₂×Q₈×C₉	—	2	2	no	—
23	C₂×Q₈×C₉	—	2	2	no	—
24	C₂×Q₈×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	—	2	2	no	—
26	C₂×Q₈×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	D₄	1	4	yes	—
28	C₂²×C₄×C₉	—	2	2	no	—
29	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₉×Q₁₆	—	2	8	no	—
31	C₉×M₁₆	—	2	2	no	—
32	C₉×M₁₆	—	2	2	no	—
33	C₉×M₁₆	—	2	2	no	—
34	C₂×C₈×C₉	—	2	2	no	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₂×C₈×C₉	—	2	1	no	—
37	C₂×C₈×C₉	D₄	1	2	yes	—
38	C₉×C₄⋊C₄	D₄	1	2	yes	—
39	C₉×C₂²⋊C₄	—	2	1	no	—
40	C₄²×C₉	—	2	1	no	—
41	#2047	C₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
43	C₂×C₃×Q₁₆	—	2	4	no	—
44	C₂×C₃×Q₁₆	—	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	1	yes	characteristic
49	C₃×C₄⋊C₈	—	2	2	no	—
50	C₃×#103	—	2	4	no	—
51	C₃×#101	—	2	4	no	—
52	C₂³×C₉	#41	1	1	yes	characteristic
53	Q₈×C₉	—	2	4	no	—
54	Q₈×C₉	—	2	4	no	—
55	Q₈×C₉	—	4	2	no	—
56	Q₈×C₉	—	4	2	no	—
57	C₂×C₄×C₉	—	2	2	no	—
58	C₂×C₄×C₉	—	2	4	no	—
59	C₂×C₄×C₉	—	4	2	no	—
60	C₂×C₄×C₉	—	4	2	no	—
61	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉	#41	1	2	yes	—
65	C₈×C₉	—	2	2	no	—
66	C₈×C₉	—	4	1	no	—
67	C₈×C₉	—	4	2	no	—
68	C₂²×Q₁₆	C₂×C₉	1	1	yes	characteristic
69	#424	C₂×C₉	1	1	yes	characteristic
70	#419	C₂×C₉	1	2	yes	—
71	C₂×#101	C₂×C₉	1	2	yes	—
72	#341	C₂×C₉	1	1	yes	characteristic
73	C₂×C₃×Q₈	—	2	2	no	—
74	C₂×C₃×Q₈	—	2	2	no	—
75	C₂×C₃×Q₈	—	2	2	no	—
76	C₂×C₃×Q₈	—	2	2	no	—
77	C₂×C₃×Q₈	—	2	2	no	—
78	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
79	C₂²×C₃×C₄	—	2	2	no	—
80	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
81	C₃×Q₁₆	—	2	8	no	—
82	C₃×M₁₆	—	2	2	no	—
83	C₃×M₁₆	—	2	2	no	—
84	C₃×M₁₆	—	2	2	no	—
85	C₂×C₃×C₈	—	2	2	no	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
89	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₄²	—	2	1	no	—
92	C₂²×C₉	—	2	2	no	—
93	C₂²×C₉	#121	1	1	yes	characteristic
94	C₂²×C₉	#123	1	2	yes	—
95	C₄×C₉	—	4	4	no	—
96	C₄×C₉	—	4	2	no	—
97	C₄×C₉	#120	1	1	yes	characteristic
98	C₄×C₉	#121	1	2	yes	—
99	C₄×C₉	#124	1	1	yes	characteristic
100	C₂²×Q₈	C₂²×C₉	1	2	yes	—
101	C₂×Q₁₆	—	2	4	no	—
102	C₂×Q₁₆	—	2	2	no	—
103	C₂×M₁₆	C₂²×C₉	1	2	yes	—
104	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
106	#117	C₂²×C₉	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	#103	—	2	4	no	—
109	#101	—	2	4	no	—
110	C₂³×C₃	C₃×#41	1	1	yes	characteristic
111	C₃×Q₈	—	2	4	no	—
112	C₃×Q₈	—	2	4	no	—
113	C₃×Q₈	—	4	2	no	—
114	C₃×Q₈	—	4	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	4	2	no	—
118	C₂×C₃×C₄	—	4	2	no	—
119	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
120	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₃×#41	1	2	yes	—
123	C₃×C₈	—	2	2	no	—
124	C₃×C₈	—	4	1	no	—
125	C₃×C₈	—	4	2	no	—
126	C₂×C₉	—	2	2	no	—
127	C₂×C₉	#394	1	1	yes	characteristic
128	C₂×C₉	#473	1	2	yes	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	D₄×C₉	1	4	yes	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
137	Q₁₆	—	2	8	no	—
138	M₁₆	—	2	2	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	D₄×C₉	1	2	yes	—
145	C₄⋊C₄	D₄×C₉	1	2	yes	—
146	C₂²⋊C₄	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	C₃×#121	1	1	yes	characteristic
150	C₂²×C₃	C₃×#123	1	2	yes	—
151	C₃×C₄	—	4	4	no	—
152	C₃×C₄	—	4	2	no	—
153	C₃×C₄	C₃×#120	1	1	yes	characteristic
154	C₃×C₄	C₃×#121	1	2	yes	—
155	C₃×C₄	C₃×#124	1	1	yes	characteristic
156	C₉	#2047	1	1	yes	characteristic, direct product factor, Sylow
157	C₂³	C₉×#41	1	1	yes	characteristic
158	Q₈	—	2	4	no	—
159	Q₈	—	2	4	no	—
160	Q₈	—	4	2	no	—
161	Q₈	—	4	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
167	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
168	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
169	C₂×C₄	C₉×#41	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	—	4	1	no	—
172	C₈	—	4	2	no	—
173	C₂×C₃	—	2	2	no	—
174	C₂×C₃	C₃×#394	1	1	yes	characteristic
175	C₂×C₃	C₃×#473	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	C₉×#121	1	1	yes	characteristic
178	C₂²	C₉×#123	1	2	yes	—
179	C₄	—	4	4	no	—
180	C₄	—	4	2	no	—
181	C₄	C₉×#120	1	1	yes	characteristic
182	C₄	C₉×#121	1	2	yes	—
183	C₄	C₉×#124	1	1	yes	characteristic
184	C₃	C₃×#2047	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₉×#394	1	1	yes	characteristic
187	C₂	C₉×#473	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₉×Q₁₆

C₂

yes

maximal, characteristic

C₉×#424

C₂

yes

maximal, characteristic

C₉×#419

C₂

yes

maximal

C₂×C₉×#101

C₂

yes

maximal

C₉×#341

C₂

yes

maximal, characteristic

C₃×#2047

C₃

yes

maximal, characteristic

C₂²×Q₈×C₉

C₂²

yes

—

C₂×C₉×Q₁₆

—

C₂×C₉×Q₁₆

—

C₂×C₉×M₁₆

C₂²

yes

characteristic

C₂×C₉×M₁₆

C₂²

yes

—

C₂²×C₈×C₉

C₂²

yes

characteristic

C₉×#117

C₂²

yes

characteristic

C₉×C₄⋊C₈

—

C₉×#103

—

C₉×#101

—

C₂²×C₃×Q₁₆

C₂×C₃

yes

characteristic

C₃×#424

C₂×C₃

yes

characteristic

C₃×#419

C₂×C₃

yes

—

C₂×C₃×#101

C₂×C₃

yes

—

C₃×#341

C₂×C₃

yes

characteristic

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

D₄

yes

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₉×Q₁₆

—

C₉×M₁₆

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

D₄

yes

—

C₉×C₄⋊C₄

D₄

yes

—

C₉×C₂²⋊C₄

—

C₄²×C₉

—

#2047

C₉

yes

characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₂²×C₃

yes

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

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

characteristic

C₃×C₄⋊C₈

—

C₃×#103

—

C₃×#101

—

C₂³×C₉

#41

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

—

C₂²×Q₁₆

C₂×C₉

yes

characteristic

#424

C₂×C₉

yes

characteristic

#419

C₂×C₉

yes

—

C₂×#101

C₂×C₉

yes

—

#341

C₂×C₉

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃×D₄

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³×C₃

yes

characteristic

C₃×Q₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×D₄

yes

—

C₃×C₄⋊C₄

C₃×D₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₂²×C₉

—

C₂²×C₉

#121

yes

characteristic

C₂²×C₉

#123

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#120

yes

characteristic

C₄×C₉

#121

yes

—

C₄×C₉

#124

yes

characteristic

100

C₂²×Q₈

C₂²×C₉

yes

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂×M₁₆

C₂²×C₉

yes

—

104

C₂×M₁₆

C₂²×C₉

yes

characteristic

105

C₂²×C₈

C₂²×C₉

yes

characteristic

106

#117

C₂²×C₉

yes

characteristic

107

C₄⋊C₈

—

108

#103

—

109

#101

—

110

C₂³×C₃

C₃×#41

yes

characteristic

111

C₃×Q₈

—

112

C₃×Q₈

—

113

C₃×Q₈

—

114

C₃×Q₈

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

—

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

C₂×C₃×D₄

yes

—

120

C₂×C₃×C₄

C₂×C₃×D₄

yes

characteristic

121

C₂×C₃×C₄

C₂×C₃×D₄

yes

characteristic

122

C₂×C₃×C₄

C₃×#41

yes

—

123

C₃×C₈

—

124

C₃×C₈

—

125

C₃×C₈

—

126

C₂×C₉

—

127

C₂×C₉

#394

yes

characteristic

128

C₂×C₉

#473

yes

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

D₄×C₉

yes

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂³×C₉

yes

characteristic

137

Q₁₆

—

138

M₁₆

—

139

M₁₆

—

140

M₁₆

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

D₄×C₉

yes

—

145

C₄⋊C₄

D₄×C₉

yes

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₂²×C₃

—

149

C₂²×C₃

C₃×#121

yes

characteristic

150

C₂²×C₃

C₃×#123

yes

—

151

C₃×C₄

—

152

C₃×C₄

—

153

C₃×C₄

C₃×#120

yes

characteristic

154

C₃×C₄

C₃×#121

yes

—

155

C₃×C₄

C₃×#124

yes

characteristic

156

C₉

#2047

yes

characteristic, direct product factor, Sylow

157

C₂³

C₉×#41

yes

characteristic

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

Q₈

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

C₂×D₄×C₉

yes

—

167

C₂×C₄

C₂×D₄×C₉

yes

characteristic

168

C₂×C₄

C₂×D₄×C₉

yes

characteristic

169

C₂×C₄

C₉×#41

yes

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₂×C₃

—

174

C₂×C₃

C₃×#394

yes

characteristic

175

C₂×C₃

C₃×#473

yes

—

176

C₂²

—

177

C₂²

C₉×#121

yes

characteristic

178

C₂²

C₉×#123

yes

—

179

C₄

—

180

C₄

—

181

C₄

C₉×#120

yes

characteristic

182

C₄

C₉×#121

yes

—

183

C₄

C₉×#124

yes

characteristic

184

C₃

C₃×#2047

yes

characteristic

185

C₂

—

186

C₂

C₉×#394

yes

characteristic

187

C₂

C₉×#473

yes

—

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#121
prime radical	C₂³×C₃	C₃×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

subgroup

G/H

center

C₂²×C₉

#121

commutator

C₂²×C₄

C₂³×C₉

Frattini, cosocle

C₂²×C₃×C₄

C₂³×C₃

socle

C₂²×C₃

C₃×#121

prime radical

C₂³×C₃

C₃×#41

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

N	H ≅ G/N	type
41. #2047	C₉	direct (N × H)
156. C₉	#2047	direct (N × H)

H ≅ G/N

type

41. #2047

C₉

direct (N × H)

156. C₉

#2047

direct (N × H)

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

subgroup	index
C₂²×C₉×Q₁₆	2
C₉×#424	2
C₉×#419	2
C₂×C₉×#101	2
C₉×#341	2
C₃×#2047	3

subgroup

index

C₂²×C₉×Q₁₆

C₂×C₉×#101

Subgroups — C₉×#2047 #11766768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#2047 #11766768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2047 #11766768