Group #26121072 — subgroups

Subgroups — #26121072

Order 12032.

310 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20244208	C₂	1	1	yes	maximal, characteristic
2	#20242061	C₂	1	2	yes	maximal
3	#20241248	C₂	1	2	yes	maximal
4	#20241233	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	#15689176	C₂²	1	2	yes	—
8	C₂×#13093949	—	2	1	no	—
9	C₂×#13093949	—	2	1	no	—
10	#15688989	C₂²	1	1	yes	characteristic
11	#15688870	—	2	4	no	—
12	C₂×#108×C₄₇	—	2	1	no	—
13	C₂×#108×C₄₇	—	2	1	no	—
14	C₄₇×#423	C₂²	1	2	yes	—
15	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₉₄	—	2	1	no	—
17	#13094024	—	2	2	no	—
18	#13094024	—	2	1	no	—
19	#13094024	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	1	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	#13093949	—	4	2	no	—
24	#13093949	—	4	2	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₄₇⋊#95	—	2	2	no	—
28	C₂×M₁₆×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	#108×C₄₇	—	4	2	no	—
33	#108×C₄₇	—	4	2	no	—
34	C₄⋊C₈×C₄₇	—	2	2	no	—
35	C₄⋊C₈×C₄₇	—	2	2	no	—
36	#97×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	#41	1	2	yes	—
39	C₂²×Dic₄₇	—	4	1	no	—
40	C₂²×Dic₄₇	—	4	1	no	—
41	C₂×Dic₉₄	—	2	2	no	—
42	C₂×Dic₉₄	—	4	1	no	—
43	C₂×Dic₉₄	—	4	1	no	—
44	C₂×Dic₉₄	—	4	1	no	—
45	C₂×Dic₉₄	—	4	1	no	—
46	C₂×Dic₉₄	—	4	1	no	—
47	M₁₆×C₄₇	—	4	2	no	—
48	M₁₆×C₄₇	—	4	2	no	—
49	C₂×C₈×C₄₇	—	2	1	no	—
50	C₂×C₈×C₄₇	—	2	1	no	—
51	C₂×C₈×C₄₇	—	2	1	no	—
52	C₂×C₈×C₄₇	—	2	1	no	—
53	C₂×C₈×C₄₇	—	4	2	no	—
54	C₂×C₈×C₄₇	D₈	1	2	yes	—
55	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
56	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₇	#41	1	1	yes	characteristic
58	C₄²×C₄₇	#41	1	2	yes	—
59	C₄⋊Dic₄₇	—	2	4	no	—
60	C₄⋊Dic₄₇	—	4	2	no	—
61	C₄⋊Dic₄₇	—	4	1	no	—
62	C₄⋊Dic₄₇	—	4	1	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂³×C₄₇	#124	1	1	yes	characteristic
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#123	1	2	yes	—
69	C₂×C₄×C₄₇	#126	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
74	C₂×Dic₄₇	—	4	2	no	—
75	C₂×Dic₄₇	—	8	1	no	—
76	C₂×Dic₄₇	—	8	1	no	—
77	C₂×Dic₄₇	—	8	2	no	—
78	C₂×Dic₄₇	—	8	1	no	—
79	Dic₉₄	—	4	2	no	—
80	Dic₉₄	—	4	2	no	—
81	Dic₉₄	—	8	1	no	—
82	Dic₉₄	—	8	1	no	—
83	C₈×C₄₇	—	2	2	no	—
84	C₈×C₄₇	—	2	2	no	—
85	C₈×C₄₇	—	8	2	no	—
86	C₄₇⋊C₈	—	8	2	no	—
87	#10632	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	—	2	2	no	—
89	C₂²×C₄₇	#397	1	1	yes	characteristic
90	C₂²×C₄₇	#486	1	1	yes	characteristic
91	C₂²×C₄₇	#489	1	1	yes	characteristic
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	2	2	no	—
94	C₄×C₄₇	#459	1	1	yes	characteristic
95	C₄×C₄₇	#461	1	1	yes	characteristic
96	C₄×C₄₇	#480	1	1	yes	characteristic
97	C₄×C₄₇	#481	1	1	yes	characteristic
98	Dic₄₇	—	8	2	no	—
99	Dic₄₇	—	16	1	no	—
100	#1960	—	47	1	no	—
101	#1849	D₄₇	1	1	yes	characteristic
102	#1546	—	47	2	no	—
103	#1286	—	47	2	no	—
104	#1281	—	47	1	no	—
105	C₂×C₄₇	—	2	1	no	—
106	C₂×C₄₇	—	2	1	no	—
107	C₂×C₄₇	#2071	1	1	yes	characteristic
108	C₂×C₄₇	#2082	1	2	yes	—
109	C₂×C₄⋊Q₈	—	47	1	no	—
110	#431	—	47	2	no	—
111	C₂×#108	—	2	1	no	—
112	C₂×#108	—	2	1	no	—
113	#423	C₂×D₄₇	1	2	yes	—
114	C₂×#103	—	94	1	no	—
115	C₂×#103	—	94	1	no	—
116	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
117	#382	—	47	1	no	—
118	#332	—	94	4	no	—
119	C₄₇	#10632	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	94	1	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₄⋊Q₈	—	47	2	no	—
125	C₄⋊Q₈	—	94	2	no	—
126	C₄⋊Q₈	—	94	1	no	—
127	C₂×C₄⋊C₄	—	94	2	no	—
128	C₂×C₄⋊C₄	—	94	1	no	—
129	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
130	#108	—	4	2	no	—
131	#108	—	4	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	94	2	no	—
135	#103	—	188	2	no	—
136	#103	—	188	2	no	—
137	C₂²⋊C₈	—	94	2	no	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	94	2	no	—
140	#95	—	94	2	no	—
141	C₂×Q₈	—	94	2	no	—
142	C₂×Q₈	—	188	1	no	—
143	C₂×Q₈	—	188	1	no	—
144	C₂×Q₈	—	188	1	no	—
145	C₂×Q₈	—	188	1	no	—
146	C₂×Q₈	—	188	1	no	—
147	C₂²×C₄	—	188	1	no	—
148	C₂²×C₄	—	188	1	no	—
149	C₂²×C₄	#10625354	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	188	2	no	—
159	C₂×C₈	#10625334	1	2	yes	—
160	C₂×C₈	#10625336	1	2	yes	—
161	C₄⋊C₄	—	94	4	no	—
162	C₄⋊C₄	—	188	1	no	—
163	C₄⋊C₄	—	188	1	no	—
164	C₄⋊C₄	—	188	2	no	—
165	C₄²	#10625350	1	2	yes	—
166	C₄²	D₄×D₄₇	1	1	yes	characteristic
167	C₄²	#10625352	1	1	yes	characteristic
168	C₂³	#13094013	1	1	yes	characteristic
169	Q₈	—	376	1	no	—
170	Q₈	—	376	1	no	—
171	Q₈	—	188	2	no	—
172	Q₈	—	188	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	376	1	no	—
175	C₂×C₄	—	376	1	no	—
176	C₂×C₄	—	376	2	no	—
177	C₂×C₄	—	376	1	no	—
178	C₂×C₄	—	188	2	no	—
179	C₂×C₄	#13093995	1	1	yes	characteristic
180	C₂×C₄	#13094007	1	1	yes	characteristic
181	C₂×C₄	#13094009	1	2	yes	—
182	C₂×C₄	#13094019	1	2	yes	—
183	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
184	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
185	C₂×C₄	#13094050	1	1	yes	characteristic
186	C₂×C₄	#13094055	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	376	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	#15689049	1	1	yes	characteristic
193	C₂²	#15689471	1	1	yes	characteristic
194	C₂²	#15689485	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	376	2	no	—
198	C₄	—	752	1	no	—
199	C₄	#15689315	1	1	yes	characteristic
200	C₄	#15689325	1	1	yes	characteristic
201	C₄	#15689436	1	1	yes	characteristic
202	C₄	#15689441	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#20244792	1	1	yes	characteristic
206	C₂	#20244853	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244208

C₂

yes

maximal, characteristic

#20242061

C₂

yes

maximal

#20241248

C₂

yes

maximal

#20241233

C₂

yes

maximal, characteristic

C₄₇×#1849

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

characteristic

#15689176

C₂²

yes

—

C₂×#13093949

—

C₂×#13093949

—

#15688989

C₂²

yes

characteristic

#15688870

—

C₂×#108×C₄₇

—

C₂×#108×C₄₇

—

C₄₇×#423

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂²×Dic₉₄

—

#13094024

—

#13094024

—

#13094024

D₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

#13093949

—

#13093949

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄₇⋊#95

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

#108×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

#97×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#10632

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

100

#1960

—

101

#1849

D₄₇

yes

characteristic

102

#1546

—

103

#1286

—

104

#1281

—

105

C₂×C₄₇

—

106

C₂×C₄₇

—

107

C₂×C₄₇

#2071

yes

characteristic

108

C₂×C₄₇

#2082

yes

—

109

C₂×C₄⋊Q₈

—

110

#431

—

111

C₂×#108

—

112

C₂×#108

—

113

#423

C₂×D₄₇

yes

—

114

C₂×#103

—

115

C₂×#103

—

116

C₂×#97

C₂×D₄₇

yes

characteristic

117

#382

—

118

#332

—

119

C₄₇

#10632

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

121

C₂×M₁₆

—

122

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

123

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄²

C₂²×D₄₇

yes

characteristic

130

#108

—

131

#108

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

188

—

136

#103

—

188

—

137

C₂²⋊C₈

—

138

#97

—

139

C₄×C₈

—

140

#95

—

141

C₂×Q₈

—

142

C₂×Q₈

—

188

—

143

C₂×Q₈

—

188

—

144

C₂×Q₈

—

188

—

145

C₂×Q₈

—

188

—

146

C₂×Q₈

—

188

—

147

C₂²×C₄

—

188

—

148

C₂²×C₄

—

188

—

149

C₂²×C₄

#10625354

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

188

—

159

C₂×C₈

#10625334

yes

—

160

C₂×C₈

#10625336

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

188

—

163

C₄⋊C₄

—

188

—

164

C₄⋊C₄

—

188

—

165

C₄²

#10625350

yes

—

166

C₄²

D₄×D₄₇

yes

characteristic

167

C₄²

#10625352

yes

characteristic

168

C₂³

#13094013

yes

characteristic

169

Q₈

—

376

—

170

Q₈

—

376

—

171

Q₈

—

188

—

172

Q₈

—

188

—

173

C₂×C₄

—

174

C₂×C₄

—

376

—

175

C₂×C₄

—

376

—

176

C₂×C₄

—

376

—

177

C₂×C₄

—

376

—

178

C₂×C₄

—

188

—

179

C₂×C₄

#13093995

yes

characteristic

180

C₂×C₄

#13094007

yes

characteristic

181

C₂×C₄

#13094009

yes

—

182

C₂×C₄

#13094019

yes

—

183

C₂×C₄

C₂×#10625334

yes

characteristic

184

C₂×C₄

C₂×#10625336

yes

characteristic

185

C₂×C₄

#13094050

yes

characteristic

186

C₂×C₄

#13094055

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

376

—

191

C₂²

—

192

C₂²

#15689049

yes

characteristic

193

C₂²

#15689471

yes

characteristic

194

C₂²

#15689485

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

376

—

198

C₄

—

752

—

199

C₄

#15689315

yes

characteristic

200

C₄

#15689325

yes

characteristic

201

C₄

#15689436

yes

characteristic

202

C₄

#15689441

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

#20244792

yes

characteristic

206

C₂

#20244853

yes

—

	subgroup	G/H
center	C₂²	#15689049
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1849	C₂
socle	C₂²×C₄₇	#397
prime radical	C₂³×C₄₇	#124
hypercenter	#1849	D₄₇
hypocenter	C₄₇	#10632
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂²

#15689049

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1849

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
119. C₄₇	#10632	semidirect

H ≅ G/N

type

119. C₄₇

#10632

semidirect

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

subgroup	index
#20244208	2
#20242061	2
#20241248	2
#20241233	2
C₄₇×#1849	2
#10632	47

subgroup

index

C₄₇×#1849

Subgroups — #26121072

Conjugacy classes of subgroups

Complements

Maximal subgroups