Group #11777788 — subgroups

Subgroups — #11777788

Order 1152.

249 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 86 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	#10588522	C₂	1	1	yes	maximal, characteristic
2	#10588509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#3664	C₂	1	1	yes	maximal, characteristic
4	C₉×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588162	C₂	1	1	yes	maximal, characteristic
6	#10588160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#68523	—	3	1	no	maximal
9	C₂×#3664	C₄	1	1	yes	characteristic
10	C₂×#3663	—	2	1	no	—
11	C₄×D₃₆	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₈	C₂²	1	1	yes	characteristic
13	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
16	#63380	C₄	1	1	yes	characteristic
17	(C₂×D₉)⋊C₈	—	2	1	no	—
18	#63377	C₄	1	1	yes	characteristic
19	#63376	C₄	1	1	yes	characteristic
20	#63375	—	2	1	no	—
21	Dic₉⋊C₈	—	2	1	no	—
22	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
24	#4838	—	3	1	no	—
25	#4831	—	3	1	no	—
26	C₄×#204	—	3	1	no	—
27	C₃×#334	D₃	1	1	yes	characteristic
28	#4484	—	3	1	no	—
29	#4483	—	3	1	no	—
30	C₈×C₃⋊C₈	—	3	1	no	—
31	C₂×D₃₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₉	—	2	1	no	—
33	C₂×Dic₁₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₉	—	2	1	no	—
35	C₂×C₈×C₉	—	2	1	no	—
36	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₉	C₂³	1	1	yes	characteristic
39	(C₂×D₉)⋊C₄	—	2	1	no	—
40	C₄⋊Dic₉	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₉	C₂×C₄	1	1	yes	characteristic
42	Dic₉⋊C₄	—	2	1	no	—
43	C₄×Dic₉	—	2	1	no	—
44	C₂×C₉⋊C₈	—	2	1	no	—
45	C₂×C₉⋊C₈	D₄	1	1	yes	characteristic
46	C₂×C₉⋊C₈	D₄	1	1	yes	characteristic
47	#3664	—	2	2	no	—
48	#3663	—	4	2	no	—
49	#1590	—	9	1	no	Sylow
50	C₂×#204	—	3	1	no	—
51	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
52	C₄×D₁₂	—	3	1	no	—
53	C₄×Dic₆	—	3	1	no	—
54	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
55	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
57	#821	—	3	1	no	—
58	(C₂×D₃)⋊C₈	—	6	1	no	—
59	#817	—	3	1	no	—
60	#816	—	3	1	no	—
61	#815	—	6	1	no	—
62	Dic₃⋊C₈	—	6	1	no	—
63	C₄×C₃⋊C₈	—	3	1	no	—
64	C₄×C₃⋊C₈	—	3	1	no	—
65	C₂²×D₉	—	2	1	no	—
66	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₉	#41	1	1	yes	characteristic
69	C₂×Dic₉	—	2	1	no	—
70	C₂×Dic₉	—	2	1	no	—
71	C₂×Dic₉	—	2	1	no	—
72	D₃₆	—	2	1	no	—
73	D₃₆	—	2	1	no	—
74	C₄×D₉	—	4	1	no	—
75	C₄×D₉	—	4	1	no	—
76	Dic₁₈	—	2	1	no	—
77	Dic₁₈	—	2	1	no	—
78	C₈×C₉	—	2	1	no	—
79	C₈×C₉	—	4	1	no	—
80	C₈×C₉	—	4	1	no	—
81	C₈×C₉	M₁₆	1	2	yes	complemented
82	C₉⋊C₈	—	2	2	no	—
83	C₉⋊C₈	—	2	1	no	—
84	C₉⋊C₈	SD₁₆	1	2	yes	complemented
85	#446	—	9	1	no	—
86	C₄×SD₁₆	—	9	1	no	—
87	#436	—	9	1	no	—
88	#334	D₉	1	1	yes	characteristic, complemented
89	Q₈⋊C₈	—	9	1	no	—
90	D₄⋊C₈	—	9	1	no	—
91	C₈²	—	9	1	no	—
92	C₂×D₁₂	—	3	1	no	—
93	C₂×C₄×D₃	—	6	1	no	—
94	C₂×Dic₆	—	3	1	no	—
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	—	2	1	no	—
97	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
98	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
99	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
100	(C₂×D₃)⋊C₄	—	6	1	no	—
101	C₄⋊Dic₃	—	3	1	no	—
102	C₄⋊Dic₃	—	3	1	no	—
103	Dic₃⋊C₄	—	6	1	no	—
104	C₄×Dic₃	—	6	1	no	—
105	C₂×C₃⋊C₈	—	3	1	no	—
106	C₂×C₃⋊C₈	—	3	1	no	—
107	C₂×C₃⋊C₈	—	6	1	no	—
108	#204	—	6	2	no	—
109	ΓL(1,𝔽₂₅)	—	12	2	no	—
110	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
111	C₂×D₉	—	4	1	no	—
112	C₂×D₉	—	4	1	no	—
113	C₂×D₉	—	4	1	no	—
114	C₄×C₉	—	2	1	no	—
115	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
116	C₄×C₉	#131	1	1	yes	characteristic
117	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₄×C₉	#135	1	1	yes	characteristic
119	Dic₉	—	4	1	no	—
120	Dic₉	—	4	1	no	—
121	Dic₉	—	4	1	no	—
122	C₂×SD₁₆	—	9	1	no	—
123	C₂×M₁₆	—	18	1	no	—
124	C₄×Q₈	—	9	1	no	—
125	C₄×D₄	—	9	1	no	—
126	#106	—	9	1	no	—
127	C₄⋊C₈	—	18	1	no	—
128	C₄⋊C₈	C₂×D₉	1	1	yes	characteristic
129	C₄⋊C₈	C₂×D₉	1	1	yes	characteristic
130	#103	—	9	1	no	—
131	#102	—	9	1	no	—
132	C₂²⋊C₈	—	18	1	no	—
133	#97	—	18	1	no	—
134	C₄×C₈	—	9	1	no	—
135	C₄×C₈	—	9	1	no	—
136	C₄×C₈	C₂×D₉	1	1	yes	characteristic
137	C₂²×D₃	—	6	1	no	—
138	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	#239	1	1	yes	characteristic
141	C₂×Dic₃	—	6	1	no	—
142	C₂×Dic₃	—	6	1	no	—
143	C₂×Dic₃	—	6	1	no	—
144	D₁₂	—	6	1	no	—
145	D₁₂	—	6	1	no	—
146	C₄×D₃	—	12	1	no	—
147	C₄×D₃	—	12	1	no	—
148	Dic₆	—	6	1	no	—
149	Dic₆	—	6	1	no	—
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	—	4	1	no	—
152	C₃×C₈	—	4	1	no	—
153	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
154	C₃⋊C₈	—	3	2	no	—
155	C₃⋊C₈	—	6	2	no	—
156	C₃⋊C₈	—	6	1	no	—
157	C₂×C₉	#436	1	1	yes	characteristic
158	C₂×C₉	C₄×SD₁₆	1	1	yes	characteristic
159	C₂×C₉	#443	1	1	yes	characteristic
160	D₉	—	8	1	no	—
161	C₂×Q₈	—	9	1	no	—
162	C₂×D₄	—	9	1	no	—
163	C₂²×C₄	—	18	1	no	—
164	SD₁₆	—	18	2	no	—
165	M₁₆	—	36	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	9	1	no	—
169	C₂×C₈	—	9	1	no	—
170	C₂×C₈	—	18	1	no	—
171	C₂×C₈	C₄×D₉	1	1	yes	characteristic
172	C₂×C₈	C₄×D₉	1	1	yes	characteristic
173	C₄⋊C₄	—	9	1	no	—
174	C₄⋊C₄	—	9	1	no	—
175	C₄⋊C₄	—	18	1	no	—
176	C₂²⋊C₄	—	18	1	no	—
177	C₄²	—	18	1	no	—
178	C₄²	C₂²×D₉	1	1	yes	characteristic
179	C₂²×C₃	#893	1	1	yes	characteristic
180	C₂×D₃	—	12	1	no	—
181	C₂×D₃	—	12	1	no	—
182	C₂×D₃	—	12	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
185	C₃×C₄	#907	1	1	yes	characteristic
186	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
187	C₃×C₄	#916	1	1	yes	characteristic
188	Dic₃	—	12	1	no	—
189	Dic₃	—	12	1	no	—
190	Dic₃	—	12	1	no	—
191	C₉	#1590	1	1	yes	characteristic, complemented, Sylow
192	C₂³	—	18	1	no	—
193	Q₈	—	18	1	no	—
194	Q₈	—	18	1	no	—
195	D₄	—	18	1	no	—
196	D₄	—	18	1	no	—
197	C₂×C₄	—	18	1	no	—
198	C₂×C₄	—	18	1	no	—
199	C₂×C₄	—	18	1	no	—
200	C₂×C₄	—	36	1	no	—
201	C₂×C₄	—	36	1	no	—
202	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
203	C₂×C₄	D₄×D₉	1	1	yes	characteristic
204	C₂×C₄	#3701	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	4	1	no	—
207	C₈	—	4	1	no	—
208	C₈	—	9	2	no	—
209	C₈	—	18	2	no	—
210	C₈	—	18	1	no	—
211	C₈	#3663	1	2	yes	complemented
212	C₂×C₃	#4838	1	1	yes	characteristic
213	C₂×C₃	#4852	1	1	yes	characteristic
214	C₂×C₃	#4895	1	1	yes	characteristic
215	D₃	—	24	1	no	—
216	C₂²	—	36	1	no	—
217	C₂²	—	36	1	no	—
218	C₂²	—	36	1	no	—
219	C₂²	#63455	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	36	1	no	—
222	C₄	—	36	1	no	—
223	C₄	—	36	1	no	—
224	C₄	C₂×#3663	1	1	yes	characteristic
225	C₄	#63469	1	1	yes	characteristic
226	C₄	SD₁₆×D₉	1	1	yes	characteristic
227	C₄	#63478	1	1	yes	characteristic
228	C₃	#68523	1	1	yes	characteristic
229	C₂	—	72	1	no	—
230	C₂	#10588522	1	1	yes	characteristic
231	C₂	#10588532	1	1	yes	characteristic
232	C₂	#10588575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10588522

C₂

yes

maximal, characteristic

#10588509

C₂

yes

maximal, characteristic, complemented

C₄×#3664

C₂

yes

maximal, characteristic

C₉×#334

C₂

yes

maximal, characteristic, complemented

#10588162

C₂

yes

maximal, characteristic

#10588160

C₂

yes

maximal, characteristic, complemented

C₈×C₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#68523

—

maximal

C₂×#3664

C₄

yes

characteristic

C₂×#3663

—

C₄×D₃₆

C₂²

yes

characteristic

C₄×Dic₁₈

C₂²

yes

characteristic

C₉×C₄⋊C₈

C₂²

yes

characteristic

C₉×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₉

C₂²

yes

characteristic

#63380

C₄

yes

characteristic

(C₂×D₉)⋊C₈

—

#63377

C₄

yes

characteristic

#63376

C₄

yes

characteristic

#63375

—

Dic₉⋊C₈

—

C₄×C₉⋊C₈

C₂²

yes

characteristic

C₄×C₉⋊C₈

C₂²

yes

—

—

—

D₃

yes

characteristic

#4484

—

#4483

—

C₈×C₃⋊C₈

—

C₂×D₃₆

C₂×C₄

yes

characteristic

C₂×C₄×D₉

—

C₂×Dic₁₈

C₂×C₄

yes

characteristic

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₄²×C₉

C₂³

yes

characteristic

(C₂×D₉)⋊C₄

—

C₄⋊Dic₉

C₂×C₄

yes

characteristic

C₄⋊Dic₉

C₂×C₄

yes

characteristic

Dic₉⋊C₄

—

C₄×Dic₉

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

D₄

yes

characteristic

C₂×C₉⋊C₈

D₄

yes

—

—

—

—

C₂×ΓL(1,𝔽₂₅)

—

C₄×D₁₂

—

C₄×Dic₆

—

C₃×C₄⋊C₈

C₂×D₃

yes

characteristic

C₃×C₄⋊C₈

C₂×D₃

yes

characteristic

C₃×C₄×C₈

C₂×D₃

yes

characteristic

#821

—

(C₂×D₃)⋊C₈

—

#817

—

#816

—

#815

—

Dic₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂²×D₉

—

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

D₃₆

—

D₃₆

—

C₄×D₉

—

C₄×D₉

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

M₁₆

yes

complemented

C₉⋊C₈

—

C₉⋊C₈

—

C₉⋊C₈

SD₁₆

yes

complemented

#446

—

C₄×SD₁₆

—

#436

—

#334

D₉

yes

characteristic, complemented

Q₈⋊C₈

—

D₄⋊C₈

—

C₈²

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

characteristic

C₂×C₃×C₈

C₄×D₃

yes

characteristic

C₃×C₄²

C₂²×D₃

yes

characteristic

100

(C₂×D₃)⋊C₄

—

101

C₄⋊Dic₃

—

102

C₄⋊Dic₃

—

103

Dic₃⋊C₄

—

104

C₄×Dic₃

—

105

C₂×C₃⋊C₈

—

106

C₂×C₃⋊C₈

—

107

C₂×C₃⋊C₈

—

108

#204

—

109

ΓL(1,𝔽₂₅)

—

110

C₂²×C₉

C₄×D₄

yes

characteristic

111

C₂×D₉

—

112

C₂×D₉

—

113

C₂×D₉

—

114

C₄×C₉

—

115

C₄×C₉

C₂×M₁₆

yes

characteristic

116

C₄×C₉

#131

yes

characteristic

117

C₄×C₉

C₂×SD₁₆

yes

characteristic

118

C₄×C₉

#135

yes

characteristic

119

Dic₉

—

120

Dic₉

—

121

Dic₉

—

122

C₂×SD₁₆

—

123

C₂×M₁₆

—

124

C₄×Q₈

—

125

C₄×D₄

—

126

#106

—

127

C₄⋊C₈

—

128

C₄⋊C₈

C₂×D₉

yes

characteristic

129

C₄⋊C₈

C₂×D₉

yes

characteristic

130

#103

—

131

#102

—

132

C₂²⋊C₈

—

133

#97

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

C₂×D₉

yes

characteristic

137

C₂²×D₃

—

138

C₂×C₃×C₄

C₂×C₄×D₃

yes

characteristic

139

C₂×C₃×C₄

D₃×D₄

yes

characteristic

140

C₂×C₃×C₄

#239

yes

characteristic

141

C₂×Dic₃

—

142

C₂×Dic₃

—

143

C₂×Dic₃

—

144

D₁₂

—

145

D₁₂

—

146

C₄×D₃

—

147

C₄×D₃

—

148

Dic₆

—

149

Dic₆

—

150

C₃×C₈

—

151

C₃×C₈

—

152

C₃×C₈

—

153

C₃×C₈

ΓL(1,𝔽₂₅)

yes

—

154

C₃⋊C₈

—

155

C₃⋊C₈

—

156

C₃⋊C₈

—

157

C₂×C₉

#436

yes

characteristic

158

C₂×C₉

C₄×SD₁₆

yes

characteristic

159

C₂×C₉

#443

yes

characteristic

160

D₉

—

161

C₂×Q₈

—

162

C₂×D₄

—

163

C₂²×C₄

—

164

SD₁₆

—

165

M₁₆

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

C₄×D₉

yes

characteristic

172

C₂×C₈

C₄×D₉

yes

characteristic

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₂²⋊C₄

—

177

C₄²

—

178

C₄²

C₂²×D₉

yes

characteristic

179

C₂²×C₃

#893

yes

characteristic

180

C₂×D₃

—

181

C₂×D₃

—

182

C₂×D₃

—

183

C₃×C₄

—

184

C₃×C₄

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

185

C₃×C₄

#907

yes

characteristic

186

C₃×C₄

D₃×SD₁₆

yes

characteristic

187

C₃×C₄

#916

yes

characteristic

188

Dic₃

—

189

Dic₃

—

190

Dic₃

—

191

C₉

#1590

yes

characteristic, complemented, Sylow

192

C₂³

—

193

Q₈

—

194

Q₈

—

195

D₄

—

196

D₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

C₂×C₄×D₉

yes

characteristic

203

C₂×C₄

D₄×D₉

yes

characteristic

204

C₂×C₄

#3701

yes

characteristic

205

C₈

—

206

C₈

—

207

C₈

—

208

C₈

—

209

C₈

—

210

C₈

—

211

C₈

#3663

yes

complemented

212

C₂×C₃

#4838

yes

characteristic

213

C₂×C₃

#4852

yes

characteristic

214

C₂×C₃

#4895

yes

characteristic

215

D₃

—

216

C₂²

—

217

C₂²

—

218

C₂²

—

219

C₂²

#63455

yes

characteristic

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

C₂×#3663

yes

characteristic

225

C₄

#63469

yes

characteristic

226

C₄

SD₁₆×D₉

yes

characteristic

227

C₄

#63478

yes

characteristic

228

C₃

#68523

yes

characteristic

229

C₂

—

230

C₂

#10588522

yes

characteristic

231

C₂

#10588532

yes

characteristic

232

C₂

#10588575

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#334	C₂
socle	C₂²×C₃	#893
prime radical	C₂×D₃₆	C₂×C₄
cosocle	C₄²×C₉	C₂³
hypercenter	#334	D₉
hypocenter	C₉	#1590
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₉

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₃×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂×D₃₆

C₂×C₄

cosocle

C₄²×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #10588509	C₂	semidirect
4. C₉×#334	C₂	semidirect
6. #10588160	C₂	semidirect
7. C₈×C₉⋊C₈	C₂	semidirect
81. C₈×C₉	M₁₆	semidirect
84. C₉⋊C₈	SD₁₆	semidirect
88. #334	D₉	semidirect
191. C₉	#1590	semidirect
211. C₈	#3663	semidirect

H ≅ G/N

type

2. #10588509

C₂

semidirect

4. C₉×#334

C₂

semidirect

6. #10588160

C₂

semidirect

7. C₈×C₉⋊C₈

C₂

semidirect

81. C₈×C₉

M₁₆

semidirect

84. C₉⋊C₈

SD₁₆

semidirect

88. #334

D₉

semidirect

191. C₉

#1590

semidirect

211. C₈

#3663

semidirect

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

subgroup	index
#10588522	2
#10588509	2
C₄×#3664	2
C₉×#334	2
#10588162	2
#10588160	2
C₈×C₉⋊C₈	2
#68523	3

subgroup

index

C₈×C₉⋊C₈

Subgroups — #11777788

Conjugacy classes of subgroups

Complements

Maximal subgroups