Group #26136108 — subgroups

Subgroups — #26136108

Order 12032.

302 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 95 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×#15688939	C₂	1	2	yes	maximal
2	#20243563	C₂	1	1	yes	maximal, characteristic
3	#20241486	C₂	1	2	yes	maximal
4	#20241472	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1971	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	C₂×#13093960	—	2	1	no	—
8	C₂×#13093960	C₂²	1	2	yes	—
9	#15689147	—	2	1	no	—
10	#15688939	—	2	4	no	—
11	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#103×C₄₇	—	2	1	no	—
14	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#391	—	2	1	no	—
16	C₂×#10625329	—	2	2	no	—
17	#13093964	—	2	2	no	—
18	#13093964	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	—	2	2	no	—
20	#13093960	—	2	2	no	—
21	#13093960	—	2	2	no	—
22	#13093960	—	4	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
24	C₂²×Q₈×C₄₇	—	2	1	no	—
25	C₂×Q₁₆×C₄₇	—	2	2	no	—
26	C₂×Q₁₆×C₄₇	—	2	2	no	—
27	C₂×Q₁₆×C₄₇	—	2	1	no	—
28	C₂×Q₁₆×C₄₇	D₄	1	2	yes	—
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
33	#106×C₄₇	—	2	2	no	—
34	#103×C₄₇	—	4	2	no	—
35	#97×C₄₇	—	2	2	no	—
36	#95×C₄₇	—	4	2	no	—
37	C₂×Q₈×C₄₇	—	2	2	no	—
38	C₂×Q₈×C₄₇	—	4	1	no	—
39	C₂×Q₈×C₄₇	—	4	1	no	—
40	C₂×Q₈×C₄₇	—	4	1	no	—
41	C₂×Q₈×C₄₇	—	4	1	no	—
42	C₂×Q₈×C₄₇	—	4	1	no	—
43	C₂²×C₄×C₄₇	—	4	1	no	—
44	C₂²×C₄×C₄₇	—	4	1	no	—
45	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₄₇	—	2	4	no	—
48	Q₁₆×C₄₇	—	4	2	no	—
49	C₂×C₈×C₄₇	—	2	1	no	—
50	C₂×C₈×C₄₇	—	2	1	no	—
51	C₂×C₈×C₄₇	D₈	1	2	yes	—
52	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	#41	1	2	yes	—
54	C₂×C₈×C₄₇	#41	1	2	yes	—
55	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
56	C₄⋊C₄×C₄₇	—	4	1	no	—
57	C₄⋊C₄×C₄₇	—	4	1	no	—
58	C₄²×C₄₇	—	2	1	no	—
59	C₄²×C₄₇	—	2	1	no	—
60	#10625329	—	4	2	no	—
61	#10625329	—	4	2	no	—
62	C₂×C₄₇⋊C₈	—	4	2	no	—
63	C₄₇⋊C₁₆	—	4	2	no	—
64	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
65	Q₈×C₄₇	—	4	2	no	—
66	Q₈×C₄₇	—	4	2	no	—
67	Q₈×C₄₇	—	8	1	no	—
68	Q₈×C₄₇	—	8	1	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	4	2	no	—
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	—	8	1	no	—
73	C₂×C₄×C₄₇	—	8	1	no	—
74	C₂×C₄×C₄₇	—	8	2	no	—
75	C₂×C₄×C₄₇	—	8	1	no	—
76	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
82	C₈×C₄₇	—	2	2	no	—
83	C₈×C₄₇	#123	1	2	yes	—
84	C₈×C₄₇	#126	1	2	yes	—
85	C₄₇⋊C₈	—	8	2	no	—
86	#13488	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	2	2	no	—
88	C₂²×C₄₇	#461	1	1	yes	characteristic
89	C₂²×C₄₇	#481	1	1	yes	characteristic
90	C₂²×C₄₇	#486	1	1	yes	characteristic
91	C₄×C₄₇	—	4	2	no	—
92	C₄×C₄₇	—	8	2	no	—
93	C₄×C₄₇	—	16	1	no	—
94	C₄×C₄₇	#397	1	1	yes	characteristic
95	C₄×C₄₇	#459	1	1	yes	characteristic
96	C₄×C₄₇	#480	1	1	yes	characteristic
97	C₄×C₄₇	#489	1	1	yes	characteristic
98	C₂×#362	—	47	2	no	—
99	#1971	D₄₇	1	1	yes	characteristic
100	#1849	—	47	1	no	—
101	#1383	—	47	2	no	—
102	#1377	—	47	1	no	—
103	C₂×C₄₇	—	2	1	no	—
104	C₂×C₄₇	—	2	1	no	—
105	C₂×C₄₇	#2071	1	1	yes	characteristic
106	C₂×C₄₇	#2239	1	2	yes	—
107	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
108	C₂×M₃₂	—	47	2	no	—
109	C₂×#108	—	47	2	no	—
110	C₂×#108	—	94	1	no	—
111	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
112	#423	—	94	1	no	—
113	C₂×#103	—	2	1	no	—
114	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
115	#391	—	2	1	no	—
116	#362	—	94	4	no	—
117	C₄₇	#13488	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	2	1	no	—
119	C₂×Q₁₆	—	2	1	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	2	no	—
122	C₂×Q₁₆	D₁₈₈	1	2	yes	—
123	C₂×M₁₆	—	94	2	no	—
124	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
128	M₃₂	—	94	2	no	—
129	M₃₂	—	94	2	no	—
130	C₂×C₁₆	—	94	2	no	—
131	#108	—	94	2	no	—
132	#108	—	94	2	no	—
133	#108	—	188	2	no	—
134	#106	—	2	2	no	—
135	C₄⋊C₈	—	188	2	no	—
136	#103	—	4	2	no	—
137	#97	—	2	2	no	—
138	#95	—	4	2	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	4	1	no	—
141	C₂×Q₈	—	4	1	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	#10625335	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	Q₁₆	—	2	4	no	—
150	Q₁₆	—	4	2	no	—
151	M₁₆	—	188	2	no	—
152	M₁₆	—	188	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	188	2	no	—
156	C₂×C₈	#10625334	1	2	yes	—
157	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
158	C₂×C₈	#10625350	1	1	yes	characteristic
159	C₂×C₈	#10625352	1	2	yes	—
160	C₂×C₈	#10625352	1	2	yes	—
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	188	2	no	—
166	C₂³	C₂×#10625335	1	1	yes	characteristic
167	Q₈	—	4	2	no	—
168	Q₈	—	4	2	no	—
169	Q₈	—	8	1	no	—
170	Q₈	—	8	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	#13093998	1	1	yes	characteristic
179	C₂×C₄	#13094005	1	1	yes	characteristic
180	C₂×C₄	#13094011	1	1	yes	characteristic
181	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
182	C₂×C₄	#13094049	1	1	yes	characteristic
183	C₂×C₄	#13094054	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	376	2	no	—
186	C₈	#13094008	1	2	yes	—
187	C₈	#13094017	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#15689328	1	1	yes	characteristic
190	C₂²	#15689439	1	1	yes	characteristic
191	C₂²	#15689469	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	8	2	no	—
194	C₄	—	16	1	no	—
195	C₄	#15689048	1	1	yes	characteristic
196	C₄	#15689318	1	1	yes	characteristic
197	C₄	#15689434	1	1	yes	characteristic
198	C₄	#15689482	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#20244791	1	1	yes	characteristic
202	C₂	#20245605	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688939

C₂

yes

maximal

#20243563

C₂

yes

maximal, characteristic

#20241486

C₂

yes

maximal

#20241472

C₂

yes

maximal, characteristic

C₄₇×#1971

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

C₂×#13093960

—

C₂×#13093960

C₂²

yes

—

#15689147

—

#15688939

—

C₂²×Q₁₆×C₄₇

C₂²

yes

characteristic

C₂×#106×C₄₇

C₂²

yes

characteristic

C₂×#103×C₄₇

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₄₇×#391

—

C₂×#10625329

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093960

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×Q₈×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

#106×C₄₇

—

#103×C₄₇

—

#97×C₄₇

—

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×SD₁₆

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×D₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₈×C₄₇

#123

yes

—

C₈×C₄₇

#126

yes

—

C₄₇⋊C₈

—

#13488

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

D₄₇

yes

characteristic

100

#1849

—

101

#1383

—

102

#1377

—

103

C₂×C₄₇

—

104

C₂×C₄₇

—

105

C₂×C₄₇

#2071

yes

characteristic

106

C₂×C₄₇

#2239

yes

—

107

C₂²×Q₁₆

C₂×D₄₇

yes

characteristic

108

C₂×M₃₂

—

109

C₂×#108

—

110

C₂×#108

—

111

C₂×#106

C₂×D₄₇

yes

characteristic

112

#423

—

113

C₂×#103

—

114

C₂×#97

C₂×D₄₇

yes

characteristic

115

#391

—

116

#362

—

117

C₄₇

#13488

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

D₁₈₈

yes

—

123

C₂×M₁₆

—

124

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

125

C₂²×C₈

C₂²×D₄₇

yes

characteristic

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

133

#108

—

188

—

134

#106

—

135

C₄⋊C₈

—

188

—

136

#103

—

137

#97

—

138

#95

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

#10625335

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

149

Q₁₆

—

150

Q₁₆

—

151

M₁₆

—

188

—

152

M₁₆

—

188

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

188

—

156

C₂×C₈

#10625334

yes

—

157

C₂×C₈

C₂×D₁₈₈

yes

characteristic

158

C₂×C₈

#10625350

yes

characteristic

159

C₂×C₈

#10625352

yes

—

160

C₂×C₈

#10625352

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

188

—

166

C₂³

C₂×#10625335

yes

characteristic

167

Q₈

—

168

Q₈

—

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

#13093998

yes

characteristic

179

C₂×C₄

#13094005

yes

characteristic

180

C₂×C₄

#13094011

yes

characteristic

181

C₂×C₄

C₂×#10625334

yes

characteristic

182

C₂×C₄

#13094049

yes

characteristic

183

C₂×C₄

#13094054

yes

characteristic

184

C₈

—

185

C₈

—

376

—

186

C₈

#13094008

yes

—

187

C₈

#13094017

yes

—

188

C₂²

—

189

C₂²

#15689328

yes

characteristic

190

C₂²

#15689439

yes

characteristic

191

C₂²

#15689469

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

#15689048

yes

characteristic

196

C₄

#15689318

yes

characteristic

197

C₄

#15689434

yes

characteristic

198

C₄

#15689482

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#20244791

yes

characteristic

202

C₂

#20245605

yes

—

	subgroup	G/H
center	C₂²	#15689439
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1971	C₂
socle	C₂²×C₄₇	#481
prime radical	C₂³×C₄₇	C₂×SD₁₆
hypercenter	#1971	D₄₇
hypocenter	C₄₇	#13488
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

subgroup

G/H

center

C₂²

#15689439

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1971

C₂

socle

C₂²×C₄₇

#481

prime radical

C₂³×C₄₇

C₂×SD₁₆

D₄₇

C₄₇

G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

N	H ≅ G/N	type
117. C₄₇	#13488	semidirect

H ≅ G/N

type

117. C₄₇

#13488

semidirect

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

subgroup	index
C₂×#15688939	2
#20243563	2
#20241486	2
#20241472	2
C₄₇×#1971	2
#13488	47

subgroup

index

C₄₇×#1971

Subgroups — #26136108

Conjugacy classes of subgroups

Complements

Maximal subgroups