Group #11783833 — subgroups

Subgroups — #11783833

Order 1152.

426 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 152 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	C₂×#63510	C₂	1	1	yes	maximal, characteristic
2	C₉×#413	C₂	1	1	yes	maximal, characteristic
3	#10588920	C₂	1	1	yes	maximal, characteristic
4	#10588797	C₂	1	2	yes	maximal, complemented
5	#10588232	C₂	1	2	yes	maximal, complemented
6	#70133	—	3	1	no	maximal
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₂×C₉×M₁₆	C₂²	1	2	yes	—
9	#63510	—	2	2	no	—
10	#63510	—	2	2	no	—
11	#63488	C₂²	1	2	yes	—
12	C₂×#3667	C₄	1	2	yes	complemented
13	C₂×#3667	C₂²	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₄	1	2	yes	complemented
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₉×#101	C₄	1	2	yes	complemented
17	C₉×#100	C₄	1	2	yes	complemented
18	#63393	C₄	1	4	yes	—
19	#63375	—	2	2	no	—
20	C₈×Dic₉	—	2	2	no	—
21	C₂×#948	—	3	1	no	—
22	C₃×#413	D₃	1	1	yes	characteristic
23	#5240	—	3	1	no	—
24	#5122	—	3	2	no	—
25	#4559	—	3	2	no	—
26	C₉×#41	—	2	2	no	—
27	C₉×#41	—	2	2	no	—
28	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₉×M₁₆	—	2	2	no	—
34	C₉×M₁₆	C₂×C₄	1	4	yes	—
35	C₂×C₈×C₉	—	2	2	no	—
36	#3676	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₉	—	2	2	no	—
38	C₄×Dic₉	—	2	2	no	—
39	#3667	—	2	2	no	—
40	#3667	—	2	4	no	—
41	C₂×C₉⋊C₈	—	2	2	no	—
42	C₂×C₉⋊C₈	—	2	2	no	—
43	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
44	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₉⋊C₈	D₄	1	2	yes	—
47	C₂×C₉⋊C₈	D₄	1	2	yes	—
48	#1757	—	9	1	no	Sylow
49	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
51	#948	—	6	2	no	—
52	#948	—	6	2	no	—
53	#926	—	3	2	no	—
54	C₂×#208	—	3	2	no	—
55	C₂×#208	—	3	1	no	—
56	C₂²×C₃⋊C₈	—	3	1	no	—
57	C₂²×C₃⋊C₈	—	3	2	no	—
58	C₃×#101	Dic₃	1	2	yes	—
59	C₃×#100	Dic₃	1	2	yes	—
60	#834	—	3	4	no	—
61	#815	—	6	2	no	—
62	C₈×Dic₃	—	6	2	no	—
63	C₂³×C₉	C₄²	1	2	yes	—
64	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
65	Q₈×C₉	—	2	2	no	—
66	D₄×C₉	—	2	4	no	—
67	D₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	C₄²	1	2	yes	—
71	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₉	#41	1	2	yes	—
75	C₂×Dic₉	—	2	4	no	—
76	C₈×C₉	—	2	4	no	—
77	C₉⋊C₈	—	2	2	no	—
78	C₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₉⋊C₈	C₂²⋊C₄	1	2	yes	—
81	C₂×#131	—	9	1	no	—
82	#413	D₉	1	1	yes	characteristic
83	#410	—	9	1	no	—
84	#405	—	9	2	no	—
85	#344	—	9	2	no	—
86	C₃×#41	—	2	2	no	—
87	C₃×#41	—	2	2	no	—
88	C₂×C₃×Q₈	C₂×Dic₃	1	1	yes	characteristic
89	C₂×C₃×D₄	C₄×D₃	1	2	yes	—
90	C₂×C₃×D₄	C₂×Dic₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
92	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₃×M₁₆	—	2	2	no	—
94	C₃×M₁₆	C₂×Dic₃	1	4	yes	—
95	C₂×C₃×C₈	—	2	2	no	—
96	#217	—	3	4	no	—
97	C₄⋊Dic₃	—	6	2	no	—
98	C₄×Dic₃	—	6	2	no	—
99	#208	—	6	4	no	—
100	#208	—	6	2	no	—
101	C₂×C₃⋊C₈	—	3	1	no	—
102	C₂×C₃⋊C₈	—	3	1	no	—
103	C₂×C₃⋊C₈	—	3	2	no	—
104	C₂×C₃⋊C₈	—	3	2	no	—
105	C₂×C₃⋊C₈	—	3	2	no	—
106	C₂×C₃⋊C₈	—	6	2	no	—
107	C₂×C₃⋊C₈	—	6	2	no	—
108	C₂²×C₉	—	2	2	no	—
109	C₂²×C₉	—	4	1	no	—
110	C₂²×C₉	—	4	2	no	—
111	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
112	C₂²×C₉	C₄×D₄	1	2	yes	—
113	C₄×C₉	—	2	2	no	—
114	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₄×C₉	#117	1	1	yes	characteristic
116	C₄×C₉	C₄×D₄	1	2	yes	—
117	Dic₉	—	4	4	no	—
118	C₂×#41	C₂×D₉	1	1	yes	characteristic
119	#131	—	18	2	no	—
120	#131	—	18	2	no	—
121	C₂×M₁₆	—	9	2	no	—
122	C₂×M₁₆	—	9	1	no	—
123	C₂×M₁₆	C₂×D₉	1	2	yes	—
124	C₂²×C₈	—	9	1	no	—
125	C₂²×C₈	—	9	2	no	—
126	#117	—	9	2	no	—
127	#101	Dic₉	1	2	yes	complemented
128	#100	Dic₉	1	2	yes	complemented
129	#99	—	9	4	no	—
130	#97	—	18	2	no	—
131	C₄×C₈	—	18	2	no	—
132	C₂³×C₃	C₄×Dic₃	1	2	yes	—
133	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
134	C₃×Q₈	—	2	2	no	—
135	C₃×D₄	—	2	4	no	—
136	C₃×D₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
142	C₂×C₃×C₄	#237	1	2	yes	—
143	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
144	C₂×Dic₃	—	6	4	no	—
145	C₃×C₈	—	2	4	no	—
146	C₃⋊C₈	—	3	1	no	—
147	C₃⋊C₈	—	3	1	no	—
148	C₃⋊C₈	—	3	2	no	—
149	C₃⋊C₈	—	6	2	no	—
150	C₂×C₉	—	2	2	no	—
151	C₂×C₉	—	2	1	no	—
152	C₂×C₉	—	4	2	no	—
153	C₂×C₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	C₂×Q₈	C₂×Dic₉	1	1	yes	characteristic
157	C₂×D₄	C₄×D₉	1	2	yes	—
158	C₂×D₄	C₂×Dic₉	1	1	yes	characteristic
159	C₂²×C₄	C₄×D₉	1	2	yes	—
160	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	18	4	no	—
163	M₁₆	—	18	2	no	—
164	M₁₆	C₂×Dic₉	1	4	yes	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	9	1	no	—
167	C₂×C₈	—	9	1	no	—
168	C₂×C₈	—	9	2	no	—
169	C₂×C₈	—	9	2	no	—
170	C₂×C₈	—	9	2	no	—
171	C₂×C₈	—	18	2	no	—
172	C₂×C₈	—	18	2	no	—
173	C₄⋊C₄	—	18	2	no	—
174	C₂²⋊C₄	—	9	4	no	—
175	C₄²	—	18	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	4	1	no	—
178	C₂²×C₃	—	4	2	no	—
179	C₂²×C₃	#67⋊C₄	1	2	yes	—
180	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	#877	1	1	yes	characteristic
183	C₃×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
184	C₃×C₄	D₄×Dic₃	1	2	yes	—
185	Dic₃	—	12	4	no	—
186	C₉	#1757	1	1	yes	characteristic, complemented, Sylow
187	C₂³	C₄×Dic₉	1	2	yes	—
188	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	D₄	—	2	4	no	—
191	D₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	18	4	no	—
195	C₂×C₄	C₄×Dic₉	1	2	yes	—
196	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
197	C₂×C₄	D₄×D₉	1	2	yes	—
198	C₂×C₄	#3699	1	2	yes	—
199	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
200	C₈	—	2	4	no	—
201	C₈	—	9	1	no	—
202	C₈	—	9	1	no	—
203	C₈	—	9	2	no	—
204	C₈	—	18	2	no	—
205	C₂×C₃	—	2	2	no	—
206	C₂×C₃	—	2	1	no	—
207	C₂×C₃	—	4	2	no	—
208	C₂×C₃	Dic₃×C₂²⋊C₄	1	1	yes	characteristic
209	C₂²	—	2	2	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	2	no	—
212	C₂²	#611⋊C₄	1	2	yes	—
213	C₂²	C₂×C₄×Dic₉	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	36	4	no	—
216	C₄	#63439	1	1	yes	characteristic
217	C₄	C₂²⋊C₄×D₉	1	1	yes	characteristic
218	C₄	D₄×Dic₉	1	2	yes	—
219	C₃	#70133	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	—	2	1	no	—
222	C₂	—	4	2	no	—
223	C₂	C₂²⋊C₄×Dic₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63510

C₂

yes

maximal, characteristic

C₉×#413

C₂

yes

maximal, characteristic

#10588920

C₂

yes

maximal, characteristic

#10588797

C₂

yes

maximal, complemented

#10588232

C₂

yes

maximal, complemented

#70133

—

maximal

C₂×C₉×#41

C₂²

yes

characteristic

C₂×C₉×M₁₆

C₂²

yes

—

#63510

—

#63510

—

#63488

C₂²

yes

—

C₂×#3667

C₄

yes

complemented

C₂×#3667

C₂²

yes

characteristic

C₂²×C₉⋊C₈

C₄

yes

complemented

C₂²×C₉⋊C₈

C₂²

yes

characteristic

C₉×#101

C₄

yes

complemented

C₉×#100

C₄

yes

complemented

#63393

C₄

yes

—

#63375

—

C₈×Dic₉

—

C₂×#948

—

C₃×#413

D₃

yes

—

—

—

C₉×#41

—

C₉×#41

—

C₂×Q₈×C₉

C₂×C₄

yes

characteristic

C₂×D₄×C₉

C₂×C₄

yes

—

C₂×D₄×C₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₉×M₁₆

—

C₉×M₁₆

C₂×C₄

yes

—

C₂×C₈×C₉

—

#3676

C₂×C₄

yes

—

C₄⋊Dic₉

—

C₄×Dic₉

—

#3667

—

#3667

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

C₂×C₄

yes

—

C₂×C₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₉⋊C₈

D₄

yes

—

C₂×C₉⋊C₈

D₄

yes

—

#1757

—

Sylow

C₂×C₃×#41

C₂×D₃

yes

characteristic

C₂×C₃×M₁₆

C₂×D₃

yes

—

—

—

—

—

—

C₂²×C₃⋊C₈

—

C₂²×C₃⋊C₈

—

C₃×#101

Dic₃

yes

—

C₃×#100

Dic₃

yes

—

#834

—

#815

—

C₈×Dic₃

—

C₂³×C₉

C₄²

yes

—

C₂³×C₉

C₂²×C₄

yes

characteristic

Q₈×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₄²

yes

—

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

#41

yes

—

C₂×Dic₉

—

C₈×C₉

—

C₉⋊C₈

—

C₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₉⋊C₈

C₂²⋊C₄

yes

—

C₂×#131

—

#413

D₉

yes

—

—

—

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

C₂×Dic₃

yes

characteristic

C₂×C₃×D₄

C₄×D₃

yes

—

C₂×C₃×D₄

C₂×Dic₃

yes

characteristic

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

C₂×Dic₃

yes

—

C₂×C₃×C₈

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

#208

—

100

#208

—

101

C₂×C₃⋊C₈

—

102

C₂×C₃⋊C₈

—

103

C₂×C₃⋊C₈

—

104

C₂×C₃⋊C₈

—

105

C₂×C₃⋊C₈

—

106

C₂×C₃⋊C₈

—

107

C₂×C₃⋊C₈

—

108

C₂²×C₉

—

109

C₂²×C₉

—

110

C₂²×C₉

—

111

C₂²×C₉

C₂×C₄²

yes

characteristic

112

C₂²×C₉

C₄×D₄

yes

—

113

C₄×C₉

—

114

C₄×C₉

C₂×C₂²⋊C₄

yes

characteristic

115

C₄×C₉

#117

yes

characteristic

116

C₄×C₉

C₄×D₄

yes

—

117

Dic₉

—

118

C₂×#41

C₂×D₉

yes

characteristic

119

#131

—

120

#131

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂×M₁₆

C₂×D₉

yes

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

#117

—

127

#101

Dic₉

yes

complemented

128

#100

Dic₉

yes

complemented

129

#99

—

130

#97

—

131

C₄×C₈

—

132

C₂³×C₃

C₄×Dic₃

yes

—

133

C₂³×C₃

C₂×C₄×D₃

yes

characteristic

134

C₃×Q₈

—

135

C₃×D₄

—

136

C₃×D₄

—

137

C₂×C₃×C₄

—

138

C₂×C₃×C₄

—

139

C₂×C₃×C₄

C₄×Dic₃

yes

—

140

C₂×C₃×C₄

C₂×C₄×D₃

yes

characteristic

141

C₂×C₃×C₄

D₃×D₄

yes

—

142

C₂×C₃×C₄

#237

yes

—

143

C₂×C₃×C₄

C₂²×Dic₃

yes

characteristic

144

C₂×Dic₃

—

145

C₃×C₈

—

146

C₃⋊C₈

—

147

C₃⋊C₈

—

148

C₃⋊C₈

—

149

C₃⋊C₈

—

150

C₂×C₉

—

151

C₂×C₉

—

152

C₂×C₉

—

153

C₂×C₉

C₄×C₂²⋊C₄

yes

characteristic

154

#41

—

155

#41

—

156

C₂×Q₈

C₂×Dic₉

yes

characteristic

157

C₂×D₄

C₄×D₉

yes

—

158

C₂×D₄

C₂×Dic₉

yes

characteristic

159

C₂²×C₄

C₄×D₉

yes

—

160

C₂²×C₄

C₂²×D₉

yes

characteristic

161

M₁₆

—

162

M₁₆

—

163

M₁₆

—

164

M₁₆

C₂×Dic₉

yes

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

—

176

C₂²×C₃

—

177

C₂²×C₃

—

178

C₂²×C₃

—

179

C₂²×C₃

#67⋊C₄

yes

—

180

C₂²×C₃

C₂×C₄×Dic₃

yes

characteristic

181

C₃×C₄

—

182

C₃×C₄

#877

yes

characteristic

183

C₃×C₄

D₃×C₂²⋊C₄

yes

characteristic

184

C₃×C₄

D₄×Dic₃

yes

—

185

Dic₃

—

186

C₉

#1757

yes

characteristic, complemented, Sylow

187

C₂³

C₄×Dic₉

yes

—

188

C₂³

C₂×C₄×D₉

yes

characteristic

189

Q₈

—

190

D₄

—

191

D₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

C₄×Dic₉

yes

—

196

C₂×C₄

C₂×C₄×D₉

yes

characteristic

197

C₂×C₄

D₄×D₉

yes

—

198

C₂×C₄

#3699

yes

—

199

C₂×C₄

C₂²×Dic₉

yes

characteristic

200

C₈

—

201

C₈

—

202

C₈

—

203

C₈

—

204

C₈

—

205

C₂×C₃

—

206

C₂×C₃

—

207

C₂×C₃

—

208

C₂×C₃

Dic₃×C₂²⋊C₄

yes

characteristic

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#611⋊C₄

yes

—

213

C₂²

C₂×C₄×Dic₉

yes

characteristic

214

C₄

—

215

C₄

—

216

C₄

#63439

yes

characteristic

217

C₄

C₂²⋊C₄×D₉

yes

characteristic

218

C₄

D₄×Dic₉

yes

—

219

C₃

#70133

yes

characteristic

220

C₂

—

221

C₂

—

222

C₂

—

223

C₂

C₂²⋊C₄×Dic₉

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₉
commutator	C₂²×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#413	C₂
socle	C₂×C₃	Dic₃×C₂²⋊C₄
prime radical	C₂×C₃×#41	C₂×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#413	D₉
hypocenter	C₉	#1757
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

C₂²⋊C₄×D₉

commutator

C₂²×C₉

C₂×C₄²

Frattini

C₂²×C₃×C₄

C₂²×D₃

Fitting

C₉×#413

C₂

socle

C₂×C₃

Dic₃×C₂²⋊C₄

prime radical

C₂×C₃×#41

C₂×D₃

cosocle

C₂²×C₄×C₉

C₂³

D₉

C₉

G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
4. #10588797	C₂	semidirect
5. #10588232	C₂	semidirect
12. C₂×#3667	C₄	semidirect
14. C₂²×C₉⋊C₈	C₄	semidirect
16. C₉×#101	C₄	semidirect
17. C₉×#100	C₄	semidirect
127. #101	Dic₉	semidirect
128. #100	Dic₉	semidirect
186. C₉	#1757	semidirect

H ≅ G/N

type

4. #10588797

C₂

semidirect

5. #10588232

C₂

semidirect

12. C₂×#3667

C₄

semidirect

14. C₂²×C₉⋊C₈

C₄

semidirect

16. C₉×#101

C₄

semidirect

17. C₉×#100

C₄

semidirect

127. #101

Dic₉

semidirect

128. #100

Dic₉

semidirect

186. C₉

#1757

semidirect

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

subgroup	index
C₂×#63510	2
C₉×#413	2
#10588920	2
#10588797	2
#10588232	2
#70133	3

subgroup

index

Subgroups — #11783833

Conjugacy classes of subgroups

Complements

Maximal subgroups