Group #26121260 — subgroups

Subgroups — #26121260

Order 12032.

372 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20244186	C₂	1	1	yes	maximal, characteristic
2	#20243563	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242159	C₂	1	2	yes	maximal, complemented
4	#20241814	C₂	1	2	yes	maximal
5	C₄₇×#1275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×#13093960	—	2	2	no	—
8	#15689147	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₄₇)⋊C₈	—	2	2	no	—
11	#15688874	—	2	4	no	—
12	#15688852	—	2	4	no	—
13	C₄₇×#431	C₂²	1	2	yes	—
14	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₈₈	Q₈	1	1	yes	characteristic
16	C₂²×C₄×D₄₇	—	2	1	no	—
17	C₂×#10625329	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	1	no	—
19	C₄×D₁₈₈	—	2	2	no	—
20	C₄×D₁₈₈	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	Q₈	1	1	yes	characteristic
22	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
23	#13093960	—	4	4	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	(C₂×D₄₇)⋊C₈	—	4	4	no	—
28	C₂²×C₈×C₄₇	D₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	—	2	2	no	—
31	C₂²⋊C₈×C₄₇	—	2	2	no	—
32	#97×C₄₇	—	2	2	no	—
33	C₄×C₈×C₄₇	D₄	1	4	yes	—
34	C₂³×D₄₇	—	2	1	no	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	2	1	no	—
39	C₂×D₁₈₈	—	2	2	no	—
40	C₂×D₁₈₈	—	4	2	no	—
41	C₂×C₄×D₄₇	—	4	1	no	—
42	C₂×C₄×D₄₇	—	4	1	no	—
43	C₂×C₄×D₄₇	—	4	2	no	—
44	C₂×C₄×D₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₂×C₈×C₄₇	—	2	2	no	—
47	C₂×C₈×C₄₇	—	2	4	no	—
48	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
49	C₄²×C₄₇	C₂×D₄	1	2	yes	—
50	C₄²×C₄₇	#41	1	2	yes	—
51	(C₂×D₄₇)⋊C₄	—	4	2	no	—
52	C₄⋊Dic₄₇	—	2	2	no	—
53	#10625329	—	4	4	no	—
54	C₂×C₄₇⋊C₈	—	4	1	no	—
55	C₂×C₄₇⋊C₈	—	4	2	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂²×D₄₇	—	2	2	no	—
59	C₂²×D₄₇	—	4	1	no	—
60	C₂²×D₄₇	—	4	2	no	—
61	C₂²×D₄₇	—	4	1	no	—
62	C₂²×D₄₇	—	4	2	no	—
63	C₂×C₄×C₄₇	—	4	1	no	—
64	C₂×C₄×C₄₇	—	4	1	no	—
65	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
66	C₂×C₄×C₄₇	#123	1	2	yes	—
67	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
70	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₄₇	—	4	1	no	—
72	C₂×Dic₄₇	—	4	2	no	—
73	D₁₈₈	—	2	4	no	—
74	D₁₈₈	—	4	2	no	—
75	C₄×D₄₇	—	8	2	no	—
76	C₄×D₄₇	—	8	1	no	—
77	C₄×D₄₇	—	8	1	no	—
78	C₈×C₄₇	—	2	4	no	—
79	C₈×C₄₇	—	4	4	no	—
80	C₄₇⋊C₈	—	8	1	no	—
81	C₄₇⋊C₈	—	8	2	no	—
82	#10661	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	—	2	1	no	—
84	C₂²×C₄₇	—	2	1	no	—
85	C₂²×C₄₇	#399	1	1	yes	characteristic
86	C₂²×C₄₇	#476	1	2	yes	—
87	C₂×D₄₇	—	4	4	no	—
88	C₂×D₄₇	—	4	2	no	—
89	C₂×D₄₇	—	8	2	no	—
90	C₂×D₄₇	—	8	1	no	—
91	C₂×D₄₇	—	8	1	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	2	2	no	—
94	C₄×C₄₇	#481	1	2	yes	—
95	C₄×C₄₇	#492	1	1	yes	characteristic
96	C₄×C₄₇	#496	1	1	yes	characteristic
97	Dic₄₇	—	8	1	no	—
98	#1955	—	47	1	no	—
99	#1849	—	47	1	no	—
100	#1570	—	47	2	no	—
101	#1486	—	47	2	no	—
102	#1275	D₄₇	1	1	yes	characteristic, complemented
103	C₂×C₄₇	—	2	2	no	—
104	C₂×C₄₇	#2091	1	1	yes	characteristic
105	C₂×C₄₇	#2094	1	1	yes	characteristic
106	C₂×C₄₇	#2098	1	1	yes	characteristic
107	D₄₇	—	8	2	no	—
108	C₂×C₄×D₄	—	47	1	no	—
109	#431	C₂×D₄₇	1	2	yes	—
110	C₂×#108	—	94	2	no	—
111	#423	—	47	2	no	—
112	C₂×C₄⋊C₈	—	47	1	no	—
113	C₂×C₂²⋊C₈	—	94	2	no	—
114	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
115	#334	—	94	4	no	—
116	D₄⋊C₈	—	94	4	no	—
117	C₄₇	#10661	1	1	yes	characteristic, complemented, Sylow
118	C₂²×D₄	—	47	1	no	—
119	C₂³×C₄	—	94	1	no	—
120	C₂×M₁₆	—	94	2	no	—
121	C₂²×C₈	—	94	1	no	—
122	C₂²×C₈	D₁₈₈	1	2	yes	—
123	C₄×D₄	—	94	2	no	—
124	C₄×D₄	—	94	2	no	—
125	C₂×C₄⋊C₄	—	47	1	no	—
126	C₂×C₂²⋊C₄	—	94	1	no	—
127	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
128	#108	—	188	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	94	2	no	—
131	C₄⋊C₈	—	94	2	no	—
132	C₄⋊C₈	—	94	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	188	4	no	—
135	#97	—	2	2	no	—
136	C₄×C₈	D₁₈₈	1	4	yes	—
137	C₂⁴	—	94	1	no	—
138	C₂×D₄	—	94	2	no	—
139	C₂×D₄	—	188	2	no	—
140	C₂²×C₄	—	94	1	no	—
141	C₂²×C₄	—	188	1	no	—
142	C₂²×C₄	—	188	1	no	—
143	C₂²×C₄	—	188	2	no	—
144	C₂²×C₄	—	188	2	no	—
145	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
146	C₂²×C₄	#10625352	1	1	yes	characteristic
147	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
148	M₁₆	—	188	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	188	1	no	—
153	C₂×C₈	—	188	2	no	—
154	C₂×C₈	—	188	2	no	—
155	C₂×C₈	#10625325	1	4	yes	—
156	C₄⋊C₄	—	94	2	no	—
157	C₂²⋊C₄	—	188	2	no	—
158	C₄²	C₂×D₁₈₈	1	2	yes	—
159	C₄²	#10625352	1	2	yes	—
160	C₂³	—	94	2	no	—
161	C₂³	—	188	1	no	—
162	C₂³	—	188	2	no	—
163	C₂³	—	188	1	no	—
164	C₂³	—	188	2	no	—
165	C₂³	#13094004	1	1	yes	characteristic
166	D₄	—	94	4	no	—
167	D₄	—	188	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	376	1	no	—
171	C₂×C₄	—	376	1	no	—
172	C₂×C₄	—	376	2	no	—
173	C₂×C₄	—	188	1	no	—
174	C₂×C₄	—	188	2	no	—
175	C₂×C₄	#13094004	1	2	yes	—
176	C₂×C₄	#13094008	1	2	yes	—
177	C₂×C₄	#13094008	1	1	yes	characteristic
178	C₂×C₄	#13094021	1	1	yes	characteristic
179	C₂×C₄	C₂×#10625325	1	2	yes	—
180	C₂×C₄	#13094052	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	4	4	no	—
183	C₈	—	376	1	no	—
184	C₈	—	376	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	376	2	no	—
188	C₂²	—	376	1	no	—
189	C₂²	—	376	1	no	—
190	C₂²	—	188	4	no	—
191	C₂²	—	188	2	no	—
192	C₂²	#15689060	1	1	yes	characteristic
193	C₂²	#15689414	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	376	1	no	—
197	C₄	#15689442	1	2	yes	—
198	C₄	#15689496	1	1	yes	characteristic
199	C₄	#15689512	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	—	376	2	no	—
202	C₂	#20244905	1	1	yes	characteristic
203	C₂	#20244920	1	1	yes	characteristic
204	C₂	#20244942	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20244186

C₂

yes

maximal, characteristic

#20243563

C₂

yes

maximal, characteristic, complemented

#20242159

C₂

yes

maximal, complemented

#20241814

C₂

yes

maximal

C₄₇×#1275

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₁₈₈

C₂²

yes

characteristic

C₂×#13093960

—

#15689147

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₂×(C₂×D₄₇)⋊C₈

—

#15688874

—

#15688852

—

C₄₇×#431

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂²×D₁₈₈

Q₈

yes

characteristic

C₂²×C₄×D₄₇

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

—

C₄×D₁₈₈

—

C₄×D₁₈₈

—

C₂×C₄⋊Dic₄₇

Q₈

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

(C₂×D₄₇)⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

characteristic

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10661

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#399

yes

characteristic

C₂²×C₄₇

#476

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#481

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1955

—

#1849

—

100

#1570

—

101

#1486

—

102

#1275

D₄₇

yes

characteristic, complemented

103

C₂×C₄₇

—

104

C₂×C₄₇

#2091

yes

characteristic

105

C₂×C₄₇

#2094

yes

characteristic

106

C₂×C₄₇

#2098

yes

characteristic

107

D₄₇

—

108

C₂×C₄×D₄

—

109

#431

C₂×D₄₇

yes

—

110

C₂×#108

—

111

#423

—

112

C₂×C₄⋊C₈

—

113

C₂×C₂²⋊C₈

—

114

C₂×#97

C₂×D₄₇

yes

characteristic

115

#334

—

116

D₄⋊C₈

—

117

C₄₇

#10661

yes

characteristic, complemented, Sylow

118

C₂²×D₄

—

119

C₂³×C₄

—

120

C₂×M₁₆

—

121

C₂²×C₈

—

122

C₂²×C₈

D₁₈₈

yes

—

123

C₄×D₄

—

124

C₄×D₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

C₂²×D₄₇

yes

characteristic

128

#108

—

188

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

188

—

135

#97

—

136

C₄×C₈

D₁₈₈

yes

—

137

C₂⁴

—

138

C₂×D₄

—

139

C₂×D₄

—

188

—

140

C₂²×C₄

—

141

C₂²×C₄

—

188

—

142

C₂²×C₄

—

188

—

143

C₂²×C₄

—

188

—

144

C₂²×C₄

—

188

—

145

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

146

C₂²×C₄

#10625352

yes

characteristic

147

C₂²×C₄

Q₈×D₄₇

yes

characteristic

148

M₁₆

—

188

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

188

—

153

C₂×C₈

—

188

—

154

C₂×C₈

—

188

—

155

C₂×C₈

#10625325

yes

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

188

—

158

C₄²

C₂×D₁₈₈

yes

—

159

C₄²

#10625352

yes

—

160

C₂³

—

161

C₂³

—

188

—

162

C₂³

—

188

—

163

C₂³

—

188

—

164

C₂³

—

188

—

165

C₂³

#13094004

yes

characteristic

166

D₄

—

167

D₄

—

188

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

376

—

171

C₂×C₄

—

376

—

172

C₂×C₄

—

376

—

173

C₂×C₄

—

188

—

174

C₂×C₄

—

188

—

175

C₂×C₄

#13094004

yes

—

176

C₂×C₄

#13094008

yes

—

177

C₂×C₄

#13094008

yes

characteristic

178

C₂×C₄

#13094021

yes

characteristic

179

C₂×C₄

C₂×#10625325

yes

—

180

C₂×C₄

#13094052

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

376

—

184

C₈

—

376

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

376

—

188

C₂²

—

376

—

189

C₂²

—

376

—

190

C₂²

—

188

—

191

C₂²

—

188

—

192

C₂²

#15689060

yes

characteristic

193

C₂²

#15689414

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

376

—

197

C₄

#15689442

yes

—

198

C₄

#15689496

yes

characteristic

199

C₄

#15689512

yes

characteristic

200

C₂

—

201

C₂

—

376

—

202

C₂

#20244905

yes

characteristic

203

C₂

#20244920

yes

characteristic

204

C₂

#20244942

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13094021
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1275	C₂
socle	C₂²×C₄₇	#399
prime radical	C₂²×D₁₈₈	Q₈
hypercenter	#1275	D₄₇
hypocenter	C₄₇	#10661
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#13094021

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1275

C₂

socle

C₂²×C₄₇

#399

prime radical

C₂²×D₁₈₈

Q₈

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
2. #20243563	C₂	semidirect
3. #20242159	C₂	semidirect
5. C₄₇×#1275	C₂	semidirect
102. #1275	D₄₇	semidirect
117. C₄₇	#10661	semidirect

H ≅ G/N

type

2. #20243563

C₂

semidirect

3. #20242159

C₂

semidirect

5. C₄₇×#1275

C₂

semidirect

102. #1275

D₄₇

semidirect

117. C₄₇

#10661

semidirect

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

subgroup	index
#20244186	2
#20243563	2
#20242159	2
#20241814	2
C₄₇×#1275	2
#10661	47

subgroup

index

C₄₇×#1275

Subgroups — #26121260

Conjugacy classes of subgroups

Complements

Maximal subgroups