Group #26111228 — subgroups

Subgroups — #26111228

Order 12032.

416 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20252459	C₂	1	1	yes	maximal, characteristic
2	#20242050	C₂	1	1	yes	maximal, characteristic
3	#20241861	C₂	1	2	yes	maximal, complemented
4	#20241287	C₂	1	2	yes	maximal, complemented
5	C₄₇×#3135	C₂	1	1	yes	maximal, characteristic
6	#15689195	—	2	4	no	—
7	#15689180	C₂²	1	2	yes	—
8	#15689176	C₂²	1	2	yes	—
9	#15689090	—	2	2	no	—
10	#15688872	—	2	1	no	—
11	#15688867	—	2	1	no	—
12	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
13	C₄₇×#533	C₂²	1	1	yes	characteristic
14	C₄₇×#532	C₂²	1	1	yes	characteristic
15	C₄₇×#491	—	2	1	no	—
16	C₄₇×#490	—	2	1	no	—
17	C₄₇×#489	C₄	1	4	yes	—
18	C₄₇×#416	—	2	2	no	—
19	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
27	#13093935	—	2	2	no	—
28	C₄×C₄₇⋊C₈	—	2	2	no	—
29	C₂×#41×C₄₇	D₄	1	2	yes	—
30	C₂²×C₈×C₄₇	—	2	1	no	—
31	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
33	#127×C₄₇	D₄	1	1	yes	characteristic
34	#125×C₄₇	—	2	1	no	—
35	#124×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²⋊Q₈×C₄₇	—	2	2	no	—
37	#121×C₄₇	—	2	2	no	—
38	C₄×Q₈×C₄₇	—	2	2	no	—
39	C₄×D₄×C₄₇	—	2	2	no	—
40	#117×C₄₇	—	2	1	no	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	#103×C₄₇	—	2	4	no	—
43	#102×C₄₇	—	2	4	no	—
44	#97×C₄₇	—	2	1	no	—
45	#97×C₄₇	C₂×C₄	1	2	yes	—
46	#41×C₄₇	—	4	2	no	—
47	#41×C₄₇	—	4	2	no	—
48	C₂×Q₈×C₄₇	—	2	2	no	—
49	C₂×Q₈×C₄₇	C₂²⋊C₄	1	2	yes	—
50	C₂×D₄×C₄₇	—	2	2	no	—
51	C₂×D₄×C₄₇	—	2	2	no	—
52	C₂×D₄×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	#41	1	2	yes	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	4	2	no	—
58	C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₄⋊C₄×C₄₇	—	4	2	no	—
62	C₂²⋊C₄×C₄₇	—	2	4	no	—
63	C₂²⋊C₄×C₄₇	—	4	1	no	—
64	C₄²×C₄₇	—	4	1	no	—
65	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₄₇	C₂×D₄	1	2	yes	—
68	#10625329	—	8	2	no	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂×C₄₇⋊C₈	—	4	2	no	—
71	C₂×C₄₇⋊C₈	—	8	2	no	—
72	C₂³×C₄₇	—	2	2	no	—
73	C₂³×C₄₇	#123	1	1	yes	characteristic
74	Q₈×C₄₇	—	4	2	no	—
75	Q₈×C₄₇	—	4	4	no	—
76	D₄×C₄₇	—	4	2	no	—
77	D₄×C₄₇	—	8	2	no	—
78	D₄×C₄₇	—	8	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	—	4	2	no	—
83	C₂×C₄×C₄₇	—	4	2	no	—
84	C₂×C₄×C₄₇	—	4	2	no	—
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#121	1	2	yes	—
89	C₈×C₄₇	—	4	2	no	—
90	C₄₇⋊C₈	—	4	4	no	—
91	C₄₇⋊C₈	—	8	2	no	—
92	#9005	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	—	2	1	no	—
95	C₂²×C₄₇	—	4	4	no	—
96	C₂²×C₄₇	—	4	2	no	—
97	C₂²×C₄₇	#386	1	1	yes	characteristic
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	—	2	2	no	—
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	4	2	no	—
102	C₄×C₄₇	—	8	2	no	—
103	C₄×C₄₇	#455	1	2	yes	—
104	#3135	D₄₇	1	1	yes	characteristic
105	#2980	—	47	1	no	—
106	#1543	—	47	1	no	—
107	#1497	—	47	2	no	—
108	#1302	—	47	2	no	—
109	C₂×C₄₇	—	4	1	no	—
110	C₂×C₄₇	—	8	2	no	—
111	C₂×C₄₇	#1881	1	2	yes	—
112	C₂×C₄₇	#1885	1	1	yes	characteristic
113	#533	C₂×D₄₇	1	1	yes	characteristic
114	#532	C₂×D₄₇	1	1	yes	characteristic
115	#491	—	2	1	no	—
116	#490	—	2	1	no	—
117	#489	Dic₄₇	1	4	yes	—
118	#435	—	94	4	no	—
119	#432	—	47	2	no	—
120	#431	—	47	2	no	—
121	#416	—	2	2	no	—
122	#408	—	94	2	no	—
123	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
124	#333	—	94	1	no	—
125	#331	—	94	1	no	—
126	Q₈⋊C₈	—	94	4	no	—
127	C₄₇	#9005	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	C₂²⋊D₄₇	1	2	yes	—
129	C₂×M₁₆	—	94	2	no	—
130	C₂²×C₈	—	2	1	no	—
131	C₂²×C₈	—	94	2	no	—
132	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
133	C₄⋊Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
134	#127	C₂²⋊D₄₇	1	1	yes	characteristic
135	#125	—	2	1	no	—
136	#124	C₂×Dic₄₇	1	2	yes	—
137	C₂²⋊Q₈	—	2	2	no	—
138	#121	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	C₄×D₄	—	2	2	no	—
141	#117	—	2	1	no	—
142	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
143	C₄⋊C₈	—	47	4	no	—
144	C₄⋊C₈	—	94	2	no	—
145	#103	—	2	4	no	—
146	#102	—	2	4	no	—
147	C₂²⋊C₈	—	94	4	no	—
148	C₂²⋊C₈	—	94	2	no	—
149	C₂²⋊C₈	—	94	2	no	—
150	#97	—	2	1	no	—
151	#97	—	94	2	no	—
152	#97	C₂×Dic₄₇	1	2	yes	—
153	C₄×C₈	—	94	2	no	—
154	#41	—	4	2	no	—
155	#41	—	4	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	C₂²⋊Dic₄₇	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₂²⋊Dic₄₇	1	2	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	#10625352	1	2	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
164	M₁₆	—	376	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	94	4	no	—
168	C₂×C₈	—	376	2	no	—
169	C₂×C₈	—	188	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	4	1	no	—
176	C₄²	—	4	1	no	—
177	C₄²	D₄×D₄₇	1	2	yes	—
178	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
179	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	#13094005	1	1	yes	characteristic
182	Q₈	—	4	2	no	—
183	Q₈	—	4	4	no	—
184	D₄	—	4	2	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
194	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
195	C₂×C₄	#13093992	1	1	yes	characteristic
196	C₂×C₄	#13093996	1	2	yes	—
197	C₈	—	4	2	no	—
198	C₈	—	376	2	no	—
199	C₈	—	188	4	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	4	4	no	—
203	C₂²	—	4	2	no	—
204	C₂²	#15689003	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	2	no	—
209	C₄	—	8	2	no	—
210	C₄	#15689300	1	2	yes	—
211	C₂	—	4	1	no	—
212	C₂	—	8	2	no	—
213	C₂	#20243752	1	2	yes	—
214	C₂	#20243777	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20252459

C₂

yes

maximal, characteristic

#20242050

C₂

yes

maximal, characteristic

#20241861

C₂

yes

maximal, complemented

#20241287

C₂

yes

maximal, complemented

C₄₇×#3135

C₂

yes

maximal, characteristic

#15689195

—

#15689180

C₂²

yes

—

#15689176

C₂²

yes

—

#15689090

—

#15688872

—

#15688867

—

Q₈⋊(C₄₇⋊C₈)

—

C₄₇×#533

C₂²

yes

characteristic

C₄₇×#532

C₂²

yes

characteristic

C₄₇×#491

—

C₄₇×#490

—

C₄₇×#489

C₄

yes

—

C₄₇×#416

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₂×#41×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

C₄⋊Q₈×C₄₇

D₄

yes

characteristic

#127×C₄₇

D₄

yes

characteristic

#125×C₄₇

—

#124×C₄₇

C₂×C₄

yes

—

C₂²⋊Q₈×C₄₇

—

#121×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×D₄×C₄₇

—

#117×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#103×C₄₇

—

#102×C₄₇

—

#97×C₄₇

—

#97×C₄₇

C₂×C₄

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂²⋊C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

#123

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9005

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#386

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

100

C₄×C₄₇

—

101

C₄×C₄₇

—

102

C₄×C₄₇

—

103

C₄×C₄₇

#455

yes

—

104

#3135

D₄₇

yes

characteristic

105

#2980

—

106

#1543

—

107

#1497

—

108

#1302

—

109

C₂×C₄₇

—

110

C₂×C₄₇

—

111

C₂×C₄₇

#1881

yes

—

112

C₂×C₄₇

#1885

yes

characteristic

113

#533

C₂×D₄₇

yes

characteristic

114

#532

C₂×D₄₇

yes

characteristic

115

#491

—

116

#490

—

117

#489

Dic₄₇

yes

—

118

#435

—

119

#432

—

120

#431

—

121

#416

—

122

#408

—

123

C₂×#97

C₂×D₄₇

yes

characteristic

124

#333

—

125

#331

—

126

Q₈⋊C₈

—

127

C₄₇

#9005

yes

characteristic, complemented, Sylow

128

C₂×#41

C₂²⋊D₄₇

yes

—

129

C₂×M₁₆

—

130

C₂²×C₈

—

131

C₂²×C₈

—

132

C₄⋊Q₈

C₂×Dic₄₇

yes

—

133

C₄⋊Q₈

C₂²⋊D₄₇

yes

characteristic

134

#127

C₂²⋊D₄₇

yes

characteristic

135

#125

—

136

#124

C₂×Dic₄₇

yes

—

137

C₂²⋊Q₈

—

138

#121

—

139

C₄×Q₈

—

140

C₄×D₄

—

141

#117

—

142

C₂×C₄²

C₂²×D₄₇

yes

characteristic

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

#103

—

146

#102

—

147

C₂²⋊C₈

—

148

C₂²⋊C₈

—

149

C₂²⋊C₈

—

150

#97

—

151

#97

—

152

#97

C₂×Dic₄₇

yes

—

153

C₄×C₈

—

154

#41

—

155

#41

—

156

C₂×Q₈

—

157

C₂×Q₈

C₂²⋊Dic₄₇

yes

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

C₂²⋊Dic₄₇

yes

—

161

C₂²×C₄

—

162

C₂²×C₄

#10625352

yes

—

163

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

164

M₁₆

—

376

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

376

—

169

C₂×C₈

—

188

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

D₄×D₄₇

yes

—

178

C₄²

C₂²×Dic₄₇

yes

characteristic

179

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

180

C₂³

—

181

C₂³

#13094005

yes

characteristic

182

Q₈

—

183

Q₈

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

194

C₂×C₄

D₄×Dic₄₇

yes

—

195

C₂×C₄

#13093992

yes

characteristic

196

C₂×C₄

#13093996

yes

—

197

C₈

—

198

C₈

—

376

—

199

C₈

—

188

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

#15689003

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

#15689300

yes

—

211

C₂

—

212

C₂

—

213

C₂

#20243752

yes

—

214

C₂

#20243777

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093992
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#3135	C₂
socle	C₂²×C₄₇	#386
prime radical	C₄₇×#532	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3135	D₄₇
hypocenter	C₄₇	#9005
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

subgroup

G/H

center

C₂×C₄

#13093992

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#3135

C₂

socle

C₂²×C₄₇

#386

prime radical

C₄₇×#532

C₂²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

N	H ≅ G/N	type
3. #20241861	C₂	semidirect
4. #20241287	C₂	semidirect
127. C₄₇	#9005	semidirect

H ≅ G/N

type

3. #20241861

C₂

semidirect

4. #20241287

C₂

semidirect

127. C₄₇

#9005

semidirect

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

subgroup	index
#20252459	2
#20242050	2
#20241861	2
#20241287	2
C₄₇×#3135	2
#9005	47

subgroup

index

C₄₇×#3135

Subgroups — #26111228

Conjugacy classes of subgroups

Complements

Maximal subgroups