Group #26134765 — subgroups

Subgroups — #26134765

Order 12032.

566 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₂×#15689500	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688937	C₂	1	4	yes	maximal, complemented
3	C₄×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₇₆	—	2	1	no	—
6	C₂²×D₃₇₆	C₂²	1	2	yes	—
7	C₂×#13094021	—	2	1	no	—
8	#15689500	—	2	2	no	—
9	#15689500	C₄	1	4	yes	—
10	C₂×#13093960	C₂²	1	2	yes	—
11	C₄×#10625329	—	2	1	no	—
12	#15688937	—	2	8	no	—
13	#15688888	—	2	1	no	—
14	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×C₄⋊C₁₆	—	2	2	no	—
17	C₂²×D₁₈₈	—	2	2	no	—
18	C₂²×D₁₈₈	—	4	2	no	—
19	C₂×D₃₇₆	—	2	4	no	—
20	C₂×D₃₇₆	—	2	4	no	—
21	C₂×D₃₇₆	—	2	4	no	—
22	C₂×D₃₇₆	—	2	2	no	—
23	C₂×D₃₇₆	—	4	4	no	—
24	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
25	C₂×#10625329	—	2	2	no	—
26	#13094021	—	2	4	no	—
27	#13094021	—	4	1	no	—
28	#13094021	—	4	1	no	—
29	#13093960	—	2	2	no	—
30	#13093960	D₄	1	4	yes	complemented
31	#13093935	—	4	1	no	—
32	C₄×C₄₇⋊C₈	—	4	1	no	—
33	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	M₃₂×C₄₇	—	2	4	no	—
37	C₂×C₁₆×C₄₇	—	2	2	no	—
38	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₇	D₄	1	2	yes	—
40	C₂³×D₄₇	—	4	2	no	—
41	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×D₁₈₈	—	2	4	no	—
44	C₂×D₁₈₈	—	4	4	no	—
45	C₂×D₁₈₈	—	4	4	no	—
46	C₂×D₁₈₈	—	4	4	no	—
47	C₂×D₁₈₈	—	4	4	no	—
48	C₂×D₁₈₈	—	4	2	no	—
49	C₂×D₁₈₈	—	8	8	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	#41	1	2	yes	—
56	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₄₇	D₈	1	1	yes	characteristic
58	C₄²×C₄₇	D₈	1	1	yes	characteristic
59	#10625329	—	2	4	no	—
60	#10625329	—	4	2	no	—
61	C₂×C₄₇⋊C₈	—	4	2	no	—
62	D₃₇₆	—	2	8	no	—
63	D₃₇₆	—	4	8	no	—
64	D₃₇₆	—	4	2	no	—
65	D₃₇₆	—	4	2	no	—
66	C₁₆×C₄₇	—	4	2	no	—
67	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₂²×D₄₇	—	4	4	no	—
69	C₂²×D₄₇	—	8	8	no	—
70	C₂²×D₄₇	—	8	4	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	#102	1	2	yes	—
73	C₂×C₄×C₄₇	#102	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#121	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
78	D₁₈₈	—	4	4	no	—
79	D₁₈₈	—	4	4	no	—
80	D₁₈₈	—	8	8	no	—
81	D₁₈₈	—	8	8	no	—
82	D₁₈₈	—	8	4	no	—
83	C₈×C₄₇	—	2	2	no	—
84	C₈×C₄₇	C₄×D₄	1	2	yes	—
85	C₈×C₄₇	#124	1	1	yes	characteristic
86	C₈×C₄₇	#127	1	1	yes	characteristic
87	C₄₇⋊C₈	—	4	4	no	—
88	#13257	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
91	C₂²×C₄₇	#460	1	2	yes	—
92	C₂×D₄₇	—	8	8	no	—
93	C₂×D₄₇	—	8	4	no	—
94	C₂×D₄₇	—	16	8	no	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	#361	1	4	yes	complemented
97	C₄×C₄₇	#390	1	1	yes	characteristic
98	C₄×C₄₇	#418	1	1	yes	characteristic
99	C₄×C₄₇	#459	1	2	yes	—
100	C₂×#493	—	47	1	no	—
101	#2150	D₄₇	1	1	yes	characteristic, complemented
102	C₂×#361	—	47	4	no	—
103	C₄×#108	—	47	1	no	—
104	C₂×C₄₇	—	2	2	no	—
105	C₂×C₄₇	#1982	1	1	yes	characteristic
106	C₂×C₄₇	C₂×#361	1	2	yes	—
107	D₄₇	—	16	4	no	—
108	C₂²×D₈	—	47	2	no	—
109	C₂²×D₈	—	94	1	no	—
110	C₂×#127	—	94	1	no	—
111	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
112	#493	—	47	4	no	—
113	#493	—	94	2	no	—
114	C₂×#108	—	47	2	no	—
115	C₄×M₁₆	—	94	1	no	—
116	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
117	C₄⋊C₁₆	—	2	2	no	—
118	#361	—	94	8	no	—
119	#341	—	94	1	no	—
120	C₄₇	#13257	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₄	—	94	2	no	—
122	C₂²×D₄	—	188	2	no	—
123	C₂×D₈	—	47	4	no	—
124	C₂×D₈	—	94	4	no	—
125	C₂×D₈	—	94	4	no	—
126	C₂×D₈	—	94	4	no	—
127	C₂×D₈	—	94	2	no	—
128	C₂×D₈	—	188	4	no	—
129	C₂×M₁₆	—	94	2	no	—
130	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
132	#127	—	94	4	no	—
133	#127	—	188	1	no	—
134	#127	—	188	1	no	—
135	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
136	M₃₂	—	2	4	no	—
137	C₂×C₁₆	—	2	2	no	—
138	#108	—	47	4	no	—
139	#108	—	94	2	no	—
140	#97	—	188	1	no	—
141	C₄×C₈	—	188	1	no	—
142	C₄×C₈	C₄×D₄₇	1	2	yes	—
143	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
144	C₂⁴	—	188	2	no	—
145	C₂×D₄	—	94	4	no	—
146	C₂×D₄	—	376	8	no	—
147	C₂×D₄	—	188	4	no	—
148	C₂×D₄	—	188	4	no	—
149	C₂×D₄	—	188	4	no	—
150	C₂×D₄	—	188	2	no	—
151	C₂×D₄	—	188	4	no	—
152	C₂²×C₄	#10625325	1	2	yes	—
153	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
154	D₈	—	94	8	no	—
155	D₈	—	188	8	no	—
156	D₈	—	188	2	no	—
157	D₈	—	188	2	no	—
158	M₁₆	—	94	4	no	—
159	M₁₆	—	188	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	188	2	no	—
162	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
164	C₂×C₈	D₄×D₄₇	1	2	yes	—
165	C₂×C₈	#10625354	1	2	yes	—
166	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
167	C₄²	D₃₇₆	1	1	yes	characteristic
168	C₄²	D₃₇₆	1	1	yes	characteristic
169	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
170	C₁₆	—	4	2	no	—
171	C₂³	—	376	8	no	—
172	C₂³	—	376	4	no	—
173	C₂³	—	188	4	no	—
174	C₂³	C₂×#10625325	1	1	yes	characteristic
175	D₄	—	376	8	no	—
176	D₄	—	376	4	no	—
177	D₄	—	376	8	no	—
178	D₄	—	188	4	no	—
179	D₄	—	188	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	#13093946	1	2	yes	—
182	C₂×C₄	#13093946	1	2	yes	—
183	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#13093999	1	2	yes	—
185	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
186	C₂×C₄	#13094052	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	188	4	no	—
189	C₈	#13093986	1	2	yes	—
190	C₈	#13094013	1	1	yes	characteristic
191	C₈	#13094020	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	376	4	no	—
194	C₂²	—	376	8	no	—
195	C₂²	—	752	8	no	—
196	C₂²	C₂×#13093946	1	1	yes	characteristic
197	C₂²	#15689324	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	#15688937	1	4	yes	complemented
200	C₄	#15689024	1	1	yes	characteristic
201	C₄	#15689129	1	1	yes	characteristic
202	C₄	#15689319	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	752	4	no	—
205	C₂	#20244314	1	1	yes	characteristic
206	C₂	C₂×#15688937	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689500

C₂

yes

maximal, characteristic

C₂×#15688937

C₂

yes

maximal, complemented

C₄×#13093960

C₂

yes

maximal, characteristic, complemented

C₄₇×#2150

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₇₆

—

C₂²×D₃₇₆

C₂²

yes

—

C₂×#13094021

—

#15689500

—

#15689500

C₄

yes

—

C₂×#13093960

C₂²

yes

—

C₄×#10625329

—

#15688937

—

#15688888

—

C₂×M₃₂×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

complemented

#13093935

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

C₁₆×C₄₇

—

C₂³×C₄₇

C₂×SD₁₆

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

C₄×D₄

yes

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₄₇⋊C₈

—

#13257

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#102

yes

characteristic

C₂²×C₄₇

#460

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

100

C₂×#493

—

101

#2150

D₄₇

yes

characteristic, complemented

102

C₂×#361

—

103

C₄×#108

—

104

C₂×C₄₇

—

105

C₂×C₄₇

#1982

yes

characteristic

106

C₂×C₄₇

C₂×#361

yes

—

107

D₄₇

—

108

C₂²×D₈

—

109

C₂²×D₈

—

110

C₂×#127

—

111

C₂×M₃₂

C₂×D₄₇

yes

—

112

#493

—

113

#493

—

114

C₂×#108

—

115

C₄×M₁₆

—

116

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

117

C₄⋊C₁₆

—

118

#361

—

119

#341

—

120

C₄₇

#13257

yes

characteristic, complemented, Sylow

121

C₂²×D₄

—

122

C₂²×D₄

—

188

—

123

C₂×D₈

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×D₈

—

127

C₂×D₈

—

128

C₂×D₈

—

188

—

129

C₂×M₁₆

—

130

C₂²×C₈

D₁₈₈

yes

characteristic

131

C₂²×C₈

C₂²×D₄₇

yes

characteristic

132

#127

—

133

#127

—

188

—

134

#127

—

188

—

135

C₂×C₄²

D₁₈₈

yes

characteristic

136

M₃₂

—

137

C₂×C₁₆

—

138

#108

—

139

#108

—

140

#97

—

188

—

141

C₄×C₈

—

188

—

142

C₄×C₈

C₄×D₄₇

yes

—

143

C₄×C₈

C₂²⋊D₄₇

yes

—

144

C₂⁴

—

188

—

145

C₂×D₄

—

146

C₂×D₄

—

376

—

147

C₂×D₄

—

188

—

148

C₂×D₄

—

188

—

149

C₂×D₄

—

188

—

150

C₂×D₄

—

188

—

151

C₂×D₄

—

188

—

152

C₂²×C₄

#10625325

yes

—

153

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

154

D₈

—

155

D₈

—

188

—

156

D₈

—

188

—

157

D₈

—

188

—

158

M₁₆

—

159

M₁₆

—

188

—

160

C₂×C₈

—

161

C₂×C₈

—

188

—

162

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

163

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

164

C₂×C₈

D₄×D₄₇

yes

—

165

C₂×C₈

#10625354

yes

—

166

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

167

C₄²

D₃₇₆

yes

characteristic

168

C₄²

D₃₇₆

yes

characteristic

169

C₄²

(C₂×D₄₇)⋊C₄

yes

—

170

C₁₆

—

171

C₂³

—

376

—

172

C₂³

—

376

—

173

C₂³

—

188

—

174

C₂³

C₂×#10625325

yes

characteristic

175

D₄

—

376

—

176

D₄

—

376

—

177

D₄

—

376

—

178

D₄

—

188

—

179

D₄

—

188

—

180

C₂×C₄

—

181

C₂×C₄

#13093946

yes

—

182

C₂×C₄

#13093946

yes

—

183

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

184

C₂×C₄

#13093999

yes

—

185

C₂×C₄

C₂×D₃₇₆

yes

characteristic

186

C₂×C₄

#13094052

yes

—

187

C₈

—

188

C₈

—

188

—

189

C₈

#13093986

yes

—

190

C₈

#13094013

yes

characteristic

191

C₈

#13094020

yes

characteristic

192

C₂²

—

193

C₂²

—

376

—

194

C₂²

—

376

—

195

C₂²

—

752

—

196

C₂²

C₂×#13093946

yes

characteristic

197

C₂²

#15689324

yes

—

198

C₄

—

199

C₄

#15688937

yes

complemented

200

C₄

#15689024

yes

characteristic

201

C₄

#15689129

yes

characteristic

202

C₄

#15689319

yes

—

203

C₂

—

204

C₂

—

752

—

205

C₂

#20244314

yes

characteristic

206

C₂

C₂×#15688937

yes

—

	subgroup	G/H
center	C₂²	C₂×#13093946
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2150	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂×#15689500	C₂
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2150	D₄₇
hypocenter	C₄₇	#13257
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

C₂×#13093946

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#2150

C₂

C₂²×C₄₇

C₂

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
2. C₂×#15688937	C₂	semidirect
3. C₄×#13093960	C₂	semidirect
4. C₄₇×#2150	C₂	semidirect
30. #13093960	D₄	semidirect
96. C₄×C₄₇	#361	semidirect
101. #2150	D₄₇	semidirect
120. C₄₇	#13257	semidirect
199. C₄	#15688937	semidirect

H ≅ G/N

type

2. C₂×#15688937

C₂

semidirect

3. C₄×#13093960

C₂

semidirect

4. C₄₇×#2150

C₂

semidirect

30. #13093960

D₄

semidirect

96. C₄×C₄₇

#361

semidirect

101. #2150

D₄₇

semidirect

120. C₄₇

#13257

semidirect

199. C₄

#15688937

semidirect

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

subgroup	index
C₂×#15689500	2
C₂×#15688937	2
C₄×#13093960	2
C₄₇×#2150	2
#13257	47

subgroup

index

C₄₇×#2150

Subgroups — #26134765

Conjugacy classes of subgroups

Complements

Maximal subgroups