Group #11788911 — subgroups

Subgroups — #11788911

Order 1152.

462 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 132 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×#536	C₂	1	1	yes	maximal, characteristic
2	#10589921	C₂	1	1	yes	maximal, characteristic
3	#10588915	C₂	1	1	yes	maximal, characteristic
4	C₂×#63394	C₂	1	2	yes	maximal, complemented
5	#10588209	C₂	1	2	yes	maximal, complemented
6	#71866	—	3	1	no	maximal
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₉×C₄⋊Q₈	—	2	1	no	—
10	C₉×#124	—	2	1	no	—
11	C₉×#123	C₄	1	4	yes	—
12	C₉×C₂²⋊Q₈	—	2	2	no	—
13	C₉×#117	C₂²	1	1	yes	characteristic
14	#63510	—	2	4	no	—
15	C₂×#3667	—	2	2	no	—
16	C₂×#3667	C₂²	1	2	yes	—
17	C₂×#3667	C₂²	1	2	yes	—
18	#63394	—	2	1	no	—
19	#63394	—	2	4	no	—
20	#63391	—	2	1	no	—
21	C₃×#536	D₃	1	1	yes	characteristic
22	#5819	—	3	1	no	—
23	#5237	—	3	1	no	—
24	C₂×#836	—	3	2	no	—
25	#4533	—	3	2	no	—
26	C₉×#41	—	2	2	no	—
27	C₉×#41	—	2	2	no	—
28	C₂×Q₈×C₉	—	2	1	no	—
29	C₂×Q₈×C₉	—	2	1	no	—
30	C₂×Q₈×C₉	—	2	2	no	—
31	C₂×Q₈×C₉	—	2	2	no	—
32	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
33	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₉	D₄	1	2	yes	—
37	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₉×C₄⋊C₄	—	2	4	no	—
39	C₉×C₄⋊C₄	—	2	1	no	—
40	C₉×C₂²⋊C₄	—	2	4	no	—
41	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
42	C₄²×C₉	—	2	1	no	—
43	#3667	—	2	4	no	—
44	#3667	—	2	2	no	—
45	#3667	—	2	2	no	—
46	#3667	—	2	2	no	—
47	#3667	D₄	1	4	yes	—
48	C₂×C₉⋊C₈	—	2	4	no	—
49	C₂×C₉⋊C₈	—	2	2	no	—
50	C₂×C₉⋊C₈	—	2	2	no	—
51	#1915	—	9	1	no	Sylow
52	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
54	C₃×C₄⋊Q₈	—	2	1	no	—
55	C₃×#124	—	2	1	no	—
56	C₃×#123	Dic₃	1	4	yes	—
57	C₃×C₂²⋊Q₈	—	2	2	no	—
58	C₃×#117	C₂×D₃	1	1	yes	characteristic
59	#948	—	6	4	no	—
60	C₂×#208	—	3	2	no	—
61	C₂×#208	—	3	2	no	—
62	C₂×#208	—	6	2	no	—
63	#836	—	6	4	no	—
64	#836	—	6	1	no	—
65	#833	—	6	1	no	—
66	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
67	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
68	Q₈×C₉	—	2	2	no	—
69	Q₈×C₉	—	4	1	no	—
70	Q₈×C₉	—	4	2	no	—
71	Q₈×C₉	—	4	1	no	—
72	D₄×C₉	—	2	4	no	—
73	D₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	4	no	—
79	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₉	#41	1	2	yes	—
84	C₉⋊C₈	—	2	4	no	—
85	C₉⋊C₈	—	4	2	no	—
86	#568	—	9	1	no	—
87	#536	D₉	1	1	yes	characteristic
88	#413	—	9	1	no	—
89	C₂×#101	—	9	2	no	—
90	#344	—	9	2	no	—
91	C₃×#41	—	2	2	no	—
92	C₃×#41	—	2	2	no	—
93	C₂×C₃×Q₈	—	2	2	no	—
94	C₂×C₃×Q₈	—	2	2	no	—
95	C₂×C₃×Q₈	—	2	1	no	—
96	C₂×C₃×Q₈	—	2	1	no	—
97	C₂×C₃×Q₈	#67	1	1	yes	characteristic
98	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
99	C₂×C₃×D₄	#67	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
101	C₂²×C₃×C₄	#67	1	2	yes	—
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₃×C₄⋊C₄	—	2	4	no	—
104	C₃×C₄⋊C₄	—	2	1	no	—
105	C₃×C₂²⋊C₄	—	2	4	no	—
106	C₃×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
107	C₃×C₄²	—	2	1	no	—
108	#208	—	3	4	no	—
109	#208	—	6	4	no	—
110	#208	—	6	2	no	—
111	#208	—	6	2	no	—
112	#208	—	6	2	no	—
113	C₂×C₃⋊C₈	—	6	4	no	—
114	C₂×C₃⋊C₈	—	6	2	no	—
115	C₂×C₃⋊C₈	—	6	2	no	—
116	C₂²×C₉	—	2	2	no	—
117	C₂²×C₉	—	4	1	no	—
118	C₂²×C₉	—	4	2	no	—
119	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₉	C₄×D₄	1	2	yes	—
121	C₄×C₉	—	2	2	no	—
122	C₄×C₉	—	4	2	no	—
123	C₄×C₉	—	4	2	no	—
124	C₄×C₉	#120	1	1	yes	characteristic
125	C₄×C₉	#121	1	2	yes	—
126	C₄×C₉	#123	1	1	yes	characteristic
127	C₂×#41	C₂×D₉	1	1	yes	characteristic
128	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
129	#131	—	18	4	no	—
130	C₂×M₁₆	—	9	2	no	—
131	C₂×M₁₆	—	9	2	no	—
132	C₂×M₁₆	—	18	2	no	—
133	C₄⋊Q₈	—	2	1	no	—
134	#124	—	2	1	no	—
135	#123	Dic₉	1	4	yes	—
136	C₂²⋊Q₈	—	2	2	no	—
137	#117	C₂×D₉	1	1	yes	characteristic
138	#101	—	18	4	no	—
139	#101	—	18	1	no	—
140	#100	—	18	1	no	—
141	C₂³×C₃	#217	1	2	yes	—
142	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
143	C₃×Q₈	—	2	2	no	—
144	C₃×Q₈	—	4	1	no	—
145	C₃×Q₈	—	4	1	no	—
146	C₃×Q₈	—	4	2	no	—
147	C₃×D₄	—	2	4	no	—
148	C₃×D₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	#217	1	2	yes	—
155	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
156	C₂×C₃×C₄	#237	1	2	yes	—
157	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
159	C₃⋊C₈	—	6	4	no	—
160	C₃⋊C₈	—	12	2	no	—
161	C₂×C₉	—	2	1	no	—
162	C₂×C₉	—	2	2	no	—
163	C₂×C₉	—	4	2	no	—
164	C₂×C₉	#386	1	1	yes	characteristic
165	#41	—	2	2	no	—
166	#41	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	#611	1	1	yes	characteristic
172	C₂×D₄	C₂×Dic₉	1	2	yes	—
173	C₂×D₄	#611	1	1	yes	characteristic
174	C₂²×C₄	C₂×Dic₉	1	2	yes	—
175	C₂²×C₄	#611	1	2	yes	—
176	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
177	M₁₆	—	9	4	no	—
178	M₁₆	—	18	4	no	—
179	M₁₆	—	18	2	no	—
180	M₁₆	—	18	2	no	—
181	M₁₆	—	18	2	no	—
182	C₂×C₈	—	18	4	no	—
183	C₂×C₈	—	18	2	no	—
184	C₂×C₈	—	18	2	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	C₂×Dic₉	1	2	yes	—
189	C₄²	—	2	1	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	4	1	no	—
192	C₂²×C₃	—	4	2	no	—
193	C₂²×C₃	D₄×Dic₃	1	2	yes	—
194	C₂²×C₃	C₂×#217	1	1	yes	characteristic
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	#935	1	1	yes	characteristic
199	C₃×C₄	#937	1	1	yes	characteristic
200	C₃×C₄	#939	1	2	yes	—
201	C₉	#1915	1	1	yes	characteristic, complemented, Sylow
202	C₂³	#3676	1	2	yes	—
203	C₂³	C₂²×Dic₉	1	1	yes	characteristic
204	Q₈	—	2	2	no	—
205	Q₈	—	4	1	no	—
206	Q₈	—	4	1	no	—
207	Q₈	—	4	2	no	—
208	D₄	—	2	4	no	—
209	D₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	#3676	1	2	yes	—
216	C₂×C₄	D₄×D₉	1	2	yes	—
217	C₂×C₄	#3699	1	2	yes	—
218	C₂×C₄	C₂×#611	1	1	yes	characteristic
219	C₂×C₄	C₂×#611	1	1	yes	characteristic
220	C₈	—	18	4	no	—
221	C₈	—	36	2	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	2	no	—
224	C₂×C₃	—	4	2	no	—
225	C₂×C₃	#5220	1	1	yes	characteristic
226	C₂²	—	2	2	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	2	no	—
229	C₂²	D₄×Dic₉	1	2	yes	—
230	C₂²	C₂×#3676	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	#63497	1	1	yes	characteristic
235	C₄	#63499	1	1	yes	characteristic
236	C₄	#63501	1	2	yes	—
237	C₃	#71866	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	2	no	—
240	C₂	—	4	2	no	—
241	C₂	#10588900	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#536

C₂

yes

maximal, characteristic

#10589921

C₂

yes

maximal, characteristic

#10588915

C₂

yes

maximal, characteristic

C₂×#63394

C₂

yes

maximal, complemented

#10588209

C₂

yes

maximal, complemented

#71866

—

maximal

C₂×C₉×#41

C₂²

yes

characteristic

C₂²×Q₈×C₉

C₂²

yes

characteristic

C₉×C₄⋊Q₈

—

C₉×#124

—

C₉×#123

C₄

yes

—

C₉×C₂²⋊Q₈

—

C₉×#117

C₂²

yes

—

—

C₂²

yes

—

C₂×#3667

C₂²

yes

—

—

—

—

D₃

yes

—

—

—

—

C₉×#41

—

C₉×#41

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

D₄

yes

characteristic

C₂×D₄×C₉

C₂×C₄

yes

—

C₂×D₄×C₉

D₄

yes

characteristic

C₂²×C₄×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

D₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₂²⋊C₄

—

C₉×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₉

—

—

—

—

—

D₄

yes

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

#1915

—

Sylow

C₂×C₃×#41

C₂×D₃

yes

characteristic

C₂²×C₃×Q₈

C₂×D₃

yes

characteristic

C₃×C₄⋊Q₈

—

C₃×#124

—

C₃×#123

Dic₃

yes

—

C₃×C₂²⋊Q₈

—

C₃×#117

C₂×D₃

yes

—

—

—

—

—

—

—

C₂³×C₉

C₂²⋊C₄

yes

—

C₂³×C₉

C₂²×C₄

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

D₄×C₉

—

D₄×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

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₉⋊C₈

—

C₉⋊C₈

—

#568

—

#536

D₉

yes

—

—

—

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

#67

yes

characteristic

C₂×C₃×D₄

C₂×Dic₃

yes

—

C₂×C₃×D₄

#67

yes

characteristic

100

C₂²×C₃×C₄

C₂×Dic₃

yes

—

101

C₂²×C₃×C₄

#67

yes

—

102

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

103

C₃×C₄⋊C₄

—

104

C₃×C₄⋊C₄

—

105

C₃×C₂²⋊C₄

—

106

C₃×C₂²⋊C₄

C₂×Dic₃

yes

—

107

C₃×C₄²

—

108

#208

—

109

#208

—

110

#208

—

111

#208

—

112

#208

—

113

C₂×C₃⋊C₈

—

114

C₂×C₃⋊C₈

—

115

C₂×C₃⋊C₈

—

116

C₂²×C₉

—

117

C₂²×C₉

—

118

C₂²×C₉

—

119

C₂²×C₉

C₂×C₂²⋊C₄

yes

characteristic

120

C₂²×C₉

C₄×D₄

yes

—

121

C₄×C₉

—

122

C₄×C₉

—

123

C₄×C₉

—

124

C₄×C₉

#120

yes

characteristic

125

C₄×C₉

#121

yes

—

126

C₄×C₉

#123

yes

characteristic

127

C₂×#41

C₂×D₉

yes

characteristic

128

C₂²×Q₈

C₂×D₉

yes

characteristic

129

#131

—

130

C₂×M₁₆

—

131

C₂×M₁₆

—

132

C₂×M₁₆

—

133

C₄⋊Q₈

—

134

#124

—

135

#123

Dic₉

yes

—

136

C₂²⋊Q₈

—

137

#117

C₂×D₉

yes

characteristic

138

#101

—

139

#101

—

140

#100

—

141

C₂³×C₃

#217

yes

—

142

C₂³×C₃

C₂²×Dic₃

yes

characteristic

143

C₃×Q₈

—

144

C₃×Q₈

—

145

C₃×Q₈

—

146

C₃×Q₈

—

147

C₃×D₄

—

148

C₃×D₄

—

149

C₂×C₃×C₄

—

150

C₂×C₃×C₄

—

151

C₂×C₃×C₄

—

152

C₂×C₃×C₄

—

153

C₂×C₃×C₄

—

154

C₂×C₃×C₄

#217

yes

—

155

C₂×C₃×C₄

D₃×D₄

yes

—

156

C₂×C₃×C₄

#237

yes

—

157

C₂×C₃×C₄

C₂×#67

yes

characteristic

158

C₂×C₃×C₄

C₂×#67

yes

characteristic

159

C₃⋊C₈

—

160

C₃⋊C₈

—

161

C₂×C₉

—

162

C₂×C₉

—

163

C₂×C₉

—

164

C₂×C₉

#386

yes

characteristic

165

#41

—

166

#41

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

#611

yes

characteristic

172

C₂×D₄

C₂×Dic₉

yes

—

173

C₂×D₄

#611

yes

characteristic

174

C₂²×C₄

C₂×Dic₉

yes

—

175

C₂²×C₄

#611

yes

—

176

C₂²×C₄

C₂²×D₉

yes

characteristic

177

M₁₆

—

178

M₁₆

—

179

M₁₆

—

180

M₁₆

—

181

M₁₆

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₂²⋊C₄

—

188

C₂²⋊C₄

C₂×Dic₉

yes

—

189

C₄²

—

190

C₂²×C₃

—

191

C₂²×C₃

—

192

C₂²×C₃

—

193

C₂²×C₃

D₄×Dic₃

yes

—

194

C₂²×C₃

C₂×#217

yes

characteristic

195

C₃×C₄

—

196

C₃×C₄

—

197

C₃×C₄

—

198

C₃×C₄

#935

yes

characteristic

199

C₃×C₄

#937

yes

characteristic

200

C₃×C₄

#939

yes

—

201

C₉

#1915

yes

characteristic, complemented, Sylow

202

C₂³

#3676

yes

—

203

C₂³

C₂²×Dic₉

yes

characteristic

204

Q₈

—

205

Q₈

—

206

Q₈

—

207

Q₈

—

208

D₄

—

209

D₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

#3676

yes

—

216

C₂×C₄

D₄×D₉

yes

—

217

C₂×C₄

#3699

yes

—

218

C₂×C₄

C₂×#611

yes

characteristic

219

C₂×C₄

C₂×#611

yes

characteristic

220

C₈

—

221

C₈

—

222

C₂×C₃

—

223

C₂×C₃

—

224

C₂×C₃

—

225

C₂×C₃

#5220

yes

characteristic

226

C₂²

—

227

C₂²

—

228

C₂²

—

229

C₂²

D₄×Dic₉

yes

—

230

C₂²

C₂×#3676

yes

characteristic

231

C₄

—

232

C₄

—

233

C₄

—

234

C₄

#63497

yes

characteristic

235

C₄

#63499

yes

characteristic

236

C₄

#63501

yes

—

237

C₃

#71866

yes

characteristic

238

C₂

—

239

C₂

—

240

C₂

—

241

C₂

#10588900

yes

characteristic

	subgroup	G/H
center	C₂	#10588900
commutator	C₂³×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#536	C₂
socle	C₂×C₃	#5220
prime radical	C₂×C₃×#41	C₂×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#536	D₉
hypocenter	C₉	#1915
LCS	G ▷ C₂³×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#10588900

commutator

C₂³×C₉

C₂²×C₄

Frattini

C₂²×C₃×C₄

C₂²×D₃

C₂

C₂×C₃

C₂×C₃×#41

C₂×D₃

cosocle

C₂²×C₄×C₉

C₂³

D₉

C₉

G ▷ C₂³×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
4. C₂×#63394	C₂	semidirect
5. #10588209	C₂	semidirect
201. C₉	#1915	semidirect

H ≅ G/N

type

4. C₂×#63394

C₂

semidirect

5. #10588209

C₂

semidirect

201. C₉

#1915

semidirect

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

subgroup	index
C₉×#536	2
#10589921	2
#10588915	2
C₂×#63394	2
#10588209	2
#71866	3

subgroup

index

Subgroups — #11788911

Conjugacy classes of subgroups

Complements

Maximal subgroups