Group #11895141 — subgroups

Subgroups — #11895141

Order 1152.

555 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 182 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×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₃₆	C₂	1	1	yes	maximal, characteristic
3	#10588768	C₂	1	3	yes	maximal
4	#10588759	C₂	1	3	yes	maximal, complemented
5	C₄×#3675	C₂	1	3	yes	maximal
6	#10588702	C₂	1	3	yes	maximal, complemented
7	Q₈×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#83688	—	3	1	no	maximal
9	C₂×Q₈×D₉	—	2	1	no	—
10	C₉×C₄⋊Q₈	C₂²	1	3	yes	—
11	C₉×#125	—	2	3	no	—
12	C₄×Q₈×C₉	—	2	1	no	—
13	C₄×Q₈×C₉	C₂²	1	3	yes	—
14	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
15	C₂×#3675	C₂²	1	3	yes	—
16	#63459	—	2	3	no	—
17	C₄×D₃₆	C₂²	1	3	yes	—
18	#63431	C₂²	1	3	yes	—
19	#63395	C₂²	1	6	yes	—
20	#63395	C₂²	1	3	yes	—
21	#63369	C₂²	1	1	yes	characteristic
22	#63367	C₂²	1	3	yes	—
23	C₄⋊(C₉⋊C₈)	C₂²	1	3	yes	—
24	C₄×C₉⋊C₈	C₂²	1	3	yes	—
25	C₃×#557	D₃	1	1	yes	characteristic
26	Q₈×D₁₂	—	3	1	no	—
27	#5083	—	3	3	no	—
28	#5079	—	3	3	no	—
29	C₄×#215	—	3	3	no	—
30	Q₈×C₃⋊C₈	—	3	1	no	—
31	#5020	—	3	3	no	—
32	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
33	C₂×Q₈×C₉	C₂³	1	3	yes	—
34	Q₈×D₉	—	4	4	no	—
35	C₂×D₃₆	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₉	—	2	3	no	—
37	C₂×Dic₁₈	—	2	3	no	—
38	C₉×C₄⋊C₄	—	2	3	no	—
39	C₉×C₄⋊C₄	—	2	3	no	—
40	C₉×C₄⋊C₄	—	2	3	no	—
41	C₉×C₄⋊C₄	D₄	1	3	yes	—
42	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
43	C₄²×C₉	—	2	3	no	—
44	C₄²×C₉	C₂³	1	3	yes	—
45	#3675	—	2	6	no	—
46	(C₂×D₉)⋊C₄	—	2	3	no	—
47	C₄⋊Dic₉	—	2	3	no	—
48	C₄⋊Dic₉	C₂³	1	3	yes	—
49	C₂×C₉⋊C₈	C₂³	1	3	yes	—
50	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
51	#3359	—	9	1	no	Sylow
52	C₂×D₃×Q₈	—	6	1	no	—
53	C₃×C₄⋊Q₈	C₂×D₃	1	3	yes	—
54	C₃×#125	—	2	3	no	—
55	C₃×C₄×Q₈	—	2	1	no	—
56	C₃×C₄×Q₈	C₂×D₃	1	3	yes	—
57	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
58	C₂×#215	—	3	3	no	—
59	#897	—	6	3	no	—
60	C₄×D₁₂	—	3	3	no	—
61	#869	—	3	3	no	—
62	#835	—	3	6	no	—
63	#835	—	3	3	no	—
64	#809	—	3	1	no	—
65	#808	—	3	3	no	—
66	C₄⋊(C₃⋊C₈)	—	3	3	no	—
67	C₄×C₃⋊C₈	—	3	3	no	—
68	C₂²×D₉	—	2	1	no	—
69	Q₈×C₉	—	2	3	no	—
70	Q₈×C₉	—	2	3	no	—
71	Q₈×C₉	SD₁₆	1	4	yes	complemented
72	C₂×C₄×C₉	—	2	1	no	—
73	C₂×C₄×C₉	—	2	3	no	—
74	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
75	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
76	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₉	—	2	3	no	—
78	D₃₆	—	2	1	no	—
79	D₃₆	#41	1	2	yes	—
80	C₄×D₉	—	4	6	no	—
81	Dic₁₈	—	4	6	no	—
82	C₉⋊C₈	—	2	3	no	—
83	C₉⋊C₈	—	2	1	no	—
84	#557	D₉	1	1	yes	characteristic, complemented
85	D₄×Q₈	—	9	1	no	—
86	#487	—	9	3	no	—
87	#475	—	9	3	no	—
88	#461	—	9	3	no	—
89	C₈×Q₈	—	9	1	no	—
90	C₄×SD₁₆	—	9	3	no	—
91	C₂×C₃×Q₈	#67	1	1	yes	characteristic
92	C₂×C₃×Q₈	C₂²×D₃	1	3	yes	—
93	D₃×Q₈	—	12	4	no	—
94	C₂×D₁₂	—	3	1	no	—
95	C₂×C₄×D₃	—	6	3	no	—
96	C₂×Dic₆	—	6	3	no	—
97	C₃×C₄⋊C₄	—	2	3	no	—
98	C₃×C₄⋊C₄	—	2	3	no	—
99	C₃×C₄⋊C₄	—	2	3	no	—
100	C₃×C₄⋊C₄	#67	1	3	yes	—
101	C₃×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
102	C₃×C₄²	—	2	3	no	—
103	C₃×C₄²	C₂²×D₃	1	3	yes	—
104	#215	—	6	6	no	—
105	(C₂×D₃)⋊C₄	—	6	3	no	—
106	C₄⋊Dic₃	—	3	3	no	—
107	C₄⋊Dic₃	—	6	3	no	—
108	C₂×C₃⋊C₈	—	3	1	no	—
109	C₂×C₃⋊C₈	—	3	3	no	—
110	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×D₉	—	2	2	no	—
112	C₂×D₉	—	4	1	no	—
113	C₂×D₉	—	4	1	no	—
114	C₄×C₉	—	2	3	no	—
115	C₄×C₉	—	4	3	no	—
116	C₄×C₉	—	4	1	no	—
117	C₄×C₉	C₂×SD₁₆	1	6	yes	—
118	C₄×C₉	C₂×#41	1	1	yes	characteristic
119	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
120	Dic₉	—	4	3	no	—
121	C₂²×Q₈	—	18	1	no	—
122	C₂×SD₁₆	—	9	3	no	—
123	C₄⋊Q₈	—	9	3	no	—
124	C₄⋊Q₈	C₂×D₉	1	3	yes	—
125	#125	—	2	3	no	—
126	C₂²⋊Q₈	—	18	3	no	—
127	C₄×Q₈	—	2	1	no	—
128	C₄×Q₈	C₂×D₉	1	3	yes	—
129	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
130	C₄×D₄	—	9	3	no	—
131	#106	—	9	3	no	—
132	C₄⋊C₈	—	9	3	no	—
133	#103	—	9	3	no	—
134	#103	—	9	6	no	—
135	#102	—	9	1	no	—
136	C₄×C₈	—	9	3	no	—
137	C₂²×D₃	—	6	1	no	—
138	C₃×Q₈	—	2	3	no	—
139	C₃×Q₈	—	2	3	no	—
140	C₃×Q₈	#215	1	4	yes	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	3	no	—
143	C₂×C₃×C₄	C₂×#67	1	3	yes	—
144	C₂×C₃×C₄	C₂×#67	1	3	yes	—
145	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
146	C₂×Dic₃	—	6	3	no	—
147	D₁₂	—	3	2	no	—
148	D₁₂	—	6	1	no	—
149	C₄×D₃	—	12	6	no	—
150	Dic₆	—	12	6	no	—
151	C₃⋊C₈	—	6	3	no	—
152	C₃⋊C₈	—	6	1	no	—
153	C₂×C₉	#548	1	1	yes	characteristic
154	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
155	C₂×C₉	#578	1	1	yes	characteristic
156	D₉	—	4	2	no	—
157	C₂×Q₈	—	18	3	no	—
158	C₂×Q₈	—	36	4	no	—
159	C₂×Q₈	#611	1	1	yes	characteristic
160	C₂×Q₈	C₂²×D₉	1	3	yes	—
161	C₂×D₄	—	9	1	no	—
162	C₂²×C₄	—	18	3	no	—
163	SD₁₆	—	18	6	no	—
164	C₂×C₈	—	9	1	no	—
165	C₂×C₈	—	9	3	no	—
166	C₄⋊C₄	—	2	3	no	—
167	C₄⋊C₄	—	2	3	no	—
168	C₄⋊C₄	—	2	3	no	—
169	C₄⋊C₄	—	9	3	no	—
170	C₄⋊C₄	—	18	3	no	—
171	C₄⋊C₄	#611	1	3	yes	—
172	C₄⋊C₄	C₂²×D₉	1	1	yes	characteristic
173	C₂²⋊C₄	—	18	3	no	—
174	C₄²	—	2	3	no	—
175	C₄²	C₂²×D₉	1	3	yes	—
176	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
177	C₂×D₃	—	6	2	no	—
178	C₂×D₃	—	12	1	no	—
179	C₂×D₃	—	12	1	no	—
180	C₃×C₄	—	2	3	no	—
181	C₃×C₄	—	4	3	no	—
182	C₃×C₄	—	4	1	no	—
183	C₃×C₄	C₂×#215	1	6	yes	—
184	C₃×C₄	D₃×#41	1	1	yes	characteristic
185	C₃×C₄	#1010	1	1	yes	characteristic
186	Dic₃	—	12	3	no	—
187	C₉	#3359	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	18	1	no	—
189	Q₈	—	2	3	no	—
190	Q₈	—	2	3	no	—
191	Q₈	—	36	6	no	—
192	Q₈	#3675	1	4	yes	complemented
193	D₄	—	9	2	no	—
194	D₄	—	18	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	18	3	no	—
198	C₂×C₄	—	36	6	no	—
199	C₂×C₄	C₂×#611	1	3	yes	—
200	C₂×C₄	C₂×#611	1	3	yes	—
201	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
202	C₈	—	18	3	no	—
203	C₈	—	18	1	no	—
204	C₂×C₃	C₂²×#215	1	1	yes	characteristic
205	C₂×C₃	#5831	1	1	yes	characteristic
206	C₂×C₃	#5838	1	1	yes	characteristic
207	D₃	—	12	2	no	—
208	C₂²	—	18	2	no	—
209	C₂²	—	36	1	no	—
210	C₂²	—	36	1	no	—
211	C₂²	C₂²×#611	1	1	yes	characteristic
212	C₄	—	2	3	no	—
213	C₄	—	4	3	no	—
214	C₄	—	4	1	no	—
215	C₄	—	36	3	no	—
216	C₄	C₂×#3675	1	6	yes	—
217	C₄	#41×D₉	1	1	yes	characteristic
218	C₄	#63716	1	1	yes	characteristic
219	C₃	#83688	1	1	yes	characteristic
220	C₂	—	36	2	no	—
221	C₂	C₂²×#3675	1	1	yes	characteristic
222	C₂	#10589933	1	1	yes	characteristic
223	C₂	#10589940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#557

C₂

yes

maximal, characteristic, complemented

Q₈×D₃₆

C₂

yes

maximal, characteristic

#10588768

C₂

yes

maximal

#10588759

C₂

yes

maximal, complemented

C₄×#3675

C₂

yes

maximal

#10588702

C₂

yes

maximal, complemented

Q₈×C₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#83688

—

maximal

C₂×Q₈×D₉

—

C₉×C₄⋊Q₈

C₂²

yes

—

C₉×#125

—

C₄×Q₈×C₉

—

C₄×Q₈×C₉

C₂²

yes

—

C₄×Q₈×C₉

C₂²

yes

characteristic

C₂×#3675

C₂²

yes

—

#63459

—

C₄×D₃₆

C₂²

yes

—

#63431

C₂²

yes

—

#63395

C₂²

yes

—

#63395

C₂²

yes

—

#63369

C₂²

yes

characteristic

#63367

C₂²

yes

—

C₄⋊(C₉⋊C₈)

C₂²

yes

—

C₄×C₉⋊C₈

C₂²

yes

—

C₃×#557

D₃

yes

characteristic

Q₈×D₁₂

—

#5083

—

#5079

—

C₄×#215

—

Q₈×C₃⋊C₈

—

#5020

—

C₂×Q₈×C₉

D₄

yes

characteristic

C₂×Q₈×C₉

C₂³

yes

—

Q₈×D₉

—

C₂×D₃₆

C₂³

yes

characteristic

C₂×C₄×D₉

—

C₂×Dic₁₈

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

D₄

yes

—

C₉×C₄⋊C₄

C₂³

yes

characteristic

C₄²×C₉

—

C₄²×C₉

C₂³

yes

—

#3675

—

(C₂×D₉)⋊C₄

—

C₄⋊Dic₉

—

C₄⋊Dic₉

C₂³

yes

—

C₂×C₉⋊C₈

C₂³

yes

—

C₂×C₉⋊C₈

C₂³

yes

characteristic

#3359

—

Sylow

C₂×D₃×Q₈

—

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₃×#125

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

C₂×D₃

yes

—

C₃×C₄×Q₈

C₂×D₃

yes

characteristic

C₂×#215

—

#897

—

C₄×D₁₂

—

—

—

—

—

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

C₂²×D₉

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

SD₁₆

yes

complemented

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂⁴

yes

characteristic

C₂×Dic₉

—

D₃₆

—

D₃₆

#41

yes

—

C₄×D₉

—

Dic₁₈

—

C₉⋊C₈

—

C₉⋊C₈

—

#557

D₉

yes

characteristic, complemented

D₄×Q₈

—

#487

—

#475

—

#461

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂×C₃×Q₈

#67

yes

characteristic

C₂×C₃×Q₈

C₂²×D₃

yes

—

D₃×Q₈

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

100

C₃×C₄⋊C₄

#67

yes

—

101

C₃×C₄⋊C₄

C₂²×D₃

yes

characteristic

102

C₃×C₄²

—

103

C₃×C₄²

C₂²×D₃

yes

—

104

#215

—

105

(C₂×D₃)⋊C₄

—

106

C₄⋊Dic₃

—

107

C₄⋊Dic₃

—

108

C₂×C₃⋊C₈

—

109

C₂×C₃⋊C₈

—

110

C₂²×C₉

C₂²×D₄

yes

characteristic

111

C₂×D₉

—

112

C₂×D₉

—

113

C₂×D₉

—

114

C₄×C₉

—

115

C₄×C₉

—

116

C₄×C₉

—

117

C₄×C₉

C₂×SD₁₆

yes

—

118

C₄×C₉

C₂×#41

yes

characteristic

119

C₄×C₉

2₋¹⁺⁴

yes

characteristic

120

Dic₉

—

121

C₂²×Q₈

—

122

C₂×SD₁₆

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

C₂×D₉

yes

—

125

#125

—

126

C₂²⋊Q₈

—

127

C₄×Q₈

—

128

C₄×Q₈

C₂×D₉

yes

—

129

C₄×Q₈

C₂×D₉

yes

characteristic

130

C₄×D₄

—

131

#106

—

132

C₄⋊C₈

—

133

#103

—

134

#103

—

135

#102

—

136

C₄×C₈

—

137

C₂²×D₃

—

138

C₃×Q₈

—

139

C₃×Q₈

—

140

C₃×Q₈

#215

yes

—

141

C₂×C₃×C₄

—

142

C₂×C₃×C₄

—

143

C₂×C₃×C₄

C₂×#67

yes

—

144

C₂×C₃×C₄

C₂×#67

yes

—

145

C₂×C₃×C₄

C₂³×D₃

yes

characteristic

146

C₂×Dic₃

—

147

D₁₂

—

148

D₁₂

—

149

C₄×D₃

—

150

Dic₆

—

151

C₃⋊C₈

—

152

C₃⋊C₈

—

153

C₂×C₉

#548

yes

characteristic

154

C₂×C₉

C₂²×SD₁₆

yes

characteristic

155

C₂×C₉

#578

yes

characteristic

156

D₉

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

#611

yes

characteristic

160

C₂×Q₈

C₂²×D₉

yes

—

161

C₂×D₄

—

162

C₂²×C₄

—

163

SD₁₆

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

#611

yes

—

172

C₄⋊C₄

C₂²×D₉

yes

characteristic

173

C₂²⋊C₄

—

174

C₄²

—

175

C₄²

C₂²×D₉

yes

—

176

C₂²×C₃

C₂²×#67

yes

characteristic

177

C₂×D₃

—

178

C₂×D₃

—

179

C₂×D₃

—

180

C₃×C₄

—

181

C₃×C₄

—

182

C₃×C₄

—

183

C₃×C₄

C₂×#215

yes

—

184

C₃×C₄

D₃×#41

yes

characteristic

185

C₃×C₄

#1010

yes

characteristic

186

Dic₃

—

187

C₉

#3359

yes

characteristic, complemented, Sylow

188

C₂³

—

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

Q₈

#3675

yes

complemented

193

D₄

—

194

D₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

C₂×#611

yes

—

200

C₂×C₄

C₂×#611

yes

—

201

C₂×C₄

C₂³×D₉

yes

characteristic

202

C₈

—

203

C₈

—

204

C₂×C₃

C₂²×#215

yes

characteristic

205

C₂×C₃

#5831

yes

characteristic

206

C₂×C₃

#5838

yes

characteristic

207

D₃

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

C₂²×#611

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

C₂×#3675

yes

—

217

C₄

#41×D₉

yes

characteristic

218

C₄

#63716

yes

characteristic

219

C₃

#83688

yes

characteristic

220

C₂

—

221

C₂

C₂²×#3675

yes

characteristic

222

C₂

#10589933

yes

characteristic

223

C₂

#10589940

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#557	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂×D₃₆	C₂³
hypercenter	#557	D₉
hypocenter	C₉	#3359
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×#611

commutator, cosocle

C₂×C₄×C₉

C₂⁴

Frattini

C₂×C₃×C₄

C₂³×D₃

Fitting

C₉×#557

C₂

socle

C₂²×C₃

C₂²×#67

prime radical

C₂×D₃₆

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
1. C₉×#557	C₂	semidirect
4. #10588759	C₂	semidirect
6. #10588702	C₂	semidirect
7. Q₈×C₉⋊C₈	C₂	semidirect
71. Q₈×C₉	SD₁₆	semidirect
84. #557	D₉	semidirect
187. C₉	#3359	semidirect
192. Q₈	#3675	semidirect

H ≅ G/N

type

1. C₉×#557

C₂

semidirect

4. #10588759

C₂

semidirect

6. #10588702

C₂

semidirect

7. Q₈×C₉⋊C₈

C₂

semidirect

71. Q₈×C₉

SD₁₆

semidirect

84. #557

D₉

semidirect

187. C₉

#3359

semidirect

192. Q₈

#3675

semidirect

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

subgroup	index
C₉×#557	2
Q₈×D₃₆	2
#10588768	2
#10588759	2
C₄×#3675	2
#10588702	2
Q₈×C₉⋊C₈	2
#83688	3

subgroup

index

Q₈×D₃₆

Q₈×C₉⋊C₈

Subgroups — #11895141

Conjugacy classes of subgroups

Complements

Maximal subgroups