Group #26053208 — subgroups

Subgroups — C₄₇×#13510 #26053208

Order 12032.

270 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 114 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×C₄₇×#365	C₂	1	2	yes	maximal
2	C₄₇×#1944	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1849	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1387	C₂	1	2	yes	maximal
5	C₄₇×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
7	C₂×#108×C₄₇	—	2	1	no	—
8	C₂×#108×C₄₇	C₂²	1	2	yes	—
9	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#423	—	2	1	no	—
12	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#389	—	2	1	no	—
14	C₄₇×#365	—	2	4	no	—
15	C₄₇×#340	—	2	1	no	—
16	C₂×M₁₆×C₄₇	—	2	2	no	—
17	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
20	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
22	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
23	M₃₂×C₄₇	—	2	2	no	—
24	M₃₂×C₄₇	Q₈	1	4	yes	—
25	C₂×C₁₆×C₄₇	—	2	2	no	—
26	#108×C₄₇	—	2	2	no	—
27	#108×C₄₇	—	2	2	no	—
28	#108×C₄₇	—	4	2	no	—
29	#107×C₄₇	—	2	2	no	—
30	#106×C₄₇	—	2	1	no	—
31	#106×C₄₇	D₄	1	2	yes	—
32	C₄⋊C₈×C₄₇	—	4	2	no	—
33	#97×C₄₇	—	2	2	no	—
34	#95×C₄₇	—	4	1	no	—
35	C₂²×C₄×C₄₇	—	4	1	no	—
36	C₂²×C₄×C₄₇	—	4	1	no	—
37	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₄₇	—	4	2	no	—
41	M₁₆×C₄₇	—	4	2	no	—
42	C₂×C₈×C₄₇	—	2	1	no	—
43	C₂×C₈×C₄₇	—	2	1	no	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₄₇	#41	1	2	yes	—
49	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₄⋊C₄×C₄₇	—	4	1	no	—
52	C₄⋊C₄×C₄₇	—	4	1	no	—
53	C₄⋊C₄×C₄₇	—	4	1	no	—
54	C₄⋊C₄×C₄₇	—	8	2	no	—
55	C₄²×C₄₇	—	2	1	no	—
56	C₄²×C₄₇	—	2	1	no	—
57	C₁₆×C₄₇	—	2	4	no	—
58	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	4	1	no	—
62	C₂×C₄×C₄₇	—	4	1	no	—
63	C₂×C₄×C₄₇	—	4	2	no	—
64	C₂×C₄×C₄₇	—	8	1	no	—
65	C₂×C₄×C₄₇	—	8	1	no	—
66	C₂×C₄×C₄₇	—	8	1	no	—
67	C₂×C₄×C₄₇	—	8	1	no	—
68	C₂×C₄×C₄₇	—	8	1	no	—
69	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
75	C₈×C₄₇	—	2	2	no	—
76	C₈×C₄₇	—	8	2	no	—
77	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₄₇	#125	1	2	yes	—
79	#13510	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₄₇	—	2	1	no	—
81	C₂²×C₄₇	—	2	1	no	—
82	C₂²×C₄₇	#461	1	1	yes	characteristic
83	C₂²×C₄₇	#481	1	1	yes	characteristic
84	C₂²×C₄₇	#498	1	1	yes	characteristic
85	C₄×C₄₇	—	4	1	no	—
86	C₄×C₄₇	—	4	1	no	—
87	C₄×C₄₇	—	8	2	no	—
88	C₄×C₄₇	—	16	1	no	—
89	C₄×C₄₇	#398	1	1	yes	characteristic
90	C₄×C₄₇	#460	1	1	yes	characteristic
91	C₄×C₄₇	#483	1	1	yes	characteristic
92	C₄×C₄₇	#501	1	1	yes	characteristic
93	C₂×#365	C₂×C₄₇	1	2	yes	—
94	#1944	C₂×C₄₇	1	1	yes	characteristic
95	#1849	C₂×C₄₇	1	1	yes	characteristic
96	#1387	C₂×C₄₇	1	2	yes	—
97	#1377	C₂×C₄₇	1	1	yes	characteristic
98	C₂×C₄₇	—	2	1	no	—
99	C₂×C₄₇	—	2	1	no	—
100	C₂×C₄₇	#2084	1	1	yes	characteristic
101	C₂×C₄₇	#2238	1	1	yes	characteristic
102	C₂×C₄₇	#2239	1	1	yes	characteristic
103	C₂×M₃₂	C₂²×C₄₇	1	2	yes	—
104	C₂×#108	—	2	1	no	—
105	C₂×#108	C₂²×C₄₇	1	2	yes	—
106	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
107	C₂×#106	C₂²×C₄₇	1	1	yes	characteristic
108	#423	—	2	1	no	—
109	C₂×#97	C₂²×C₄₇	1	1	yes	characteristic
110	#389	—	2	1	no	—
111	#365	—	2	4	no	—
112	#340	—	2	1	no	—
113	C₄₇	#13510	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×M₁₆	—	2	2	no	—
115	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	1	no	—
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₄⋊C₄	—	4	1	no	—
120	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
121	M₃₂	—	2	2	no	—
122	M₃₂	Q₈×C₄₇	1	4	yes	—
123	C₂×C₁₆	—	2	2	no	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	4	2	no	—
127	#107	—	2	2	no	—
128	#106	—	2	1	no	—
129	#106	D₄×C₄₇	1	2	yes	—
130	C₄⋊C₈	—	4	2	no	—
131	#97	—	2	2	no	—
132	#95	—	4	1	no	—
133	C₂²×C₄	—	4	1	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	SD₁₆×C₄₇	1	1	yes	characteristic
136	C₂²×C₄	SD₁₆×C₄₇	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
138	M₁₆	—	4	2	no	—
139	M₁₆	—	4	2	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	SD₁₆×C₄₇	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₄₇	1	2	yes	—
146	C₂×C₈	#41×C₄₇	1	2	yes	—
147	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	4	1	no	—
152	C₄⋊C₄	—	8	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₁₆	—	2	4	no	—
156	C₂³	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	1	no	—
166	C₂×C₄	—	8	1	no	—
167	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
168	C₂×C₄	#123×C₄₇	1	1	yes	characteristic
169	C₂×C₄	C₄⋊Q₈×C₄₇	1	1	yes	characteristic
170	C₂×C₄	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
171	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
172	C₂×C₄	#137×C₄₇	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	8	2	no	—
175	C₈	C₂²⋊Q₈×C₄₇	1	2	yes	—
176	C₈	#125×C₄₇	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	C₄₇×#461	1	1	yes	characteristic
180	C₂²	C₄₇×#481	1	1	yes	characteristic
181	C₂²	C₄₇×#498	1	1	yes	characteristic
182	C₄	—	4	1	no	—
183	C₄	—	4	1	no	—
184	C₄	—	8	2	no	—
185	C₄	—	16	1	no	—
186	C₄	C₄₇×#398	1	1	yes	characteristic
187	C₄	C₄₇×#460	1	1	yes	characteristic
188	C₄	C₄₇×#483	1	1	yes	characteristic
189	C₄	C₄₇×#501	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	C₄₇×#2084	1	1	yes	characteristic
193	C₂	C₄₇×#2238	1	1	yes	characteristic
194	C₂	C₄₇×#2239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄₇×#365

C₂

yes

maximal

C₄₇×#1944

C₂

yes

maximal, characteristic

C₄₇×#1849

C₂

yes

maximal, characteristic

C₄₇×#1387

C₂

yes

maximal

C₄₇×#1377

C₂

yes

maximal, characteristic

C₂×M₃₂×C₄₇

C₂²

yes

—

C₂×#108×C₄₇

—

C₂×#108×C₄₇

C₂²

yes

—

C₂×#107×C₄₇

C₂²

yes

characteristic

C₂×#106×C₄₇

C₂²

yes

characteristic

C₄₇×#423

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₄₇×#389

—

C₄₇×#365

—

C₄₇×#340

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

M₃₂×C₄₇

—

M₃₂×C₄₇

Q₈

yes

—

C₂×C₁₆×C₄₇

—

#108×C₄₇

—

#108×C₄₇

—

#108×C₄₇

—

#107×C₄₇

—

#106×C₄₇

—

#106×C₄₇

D₄

yes

—

C₄⋊C₈×C₄₇

—

#97×C₄₇

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

SD₁₆

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₁₆×C₄₇

—

C₂³×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₈×C₄₇

#125

yes

—

#13510

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₂×C₄₇

yes

—

#1944

C₂×C₄₇

yes

characteristic

#1849

C₂×C₄₇

yes

characteristic

#1387

C₂×C₄₇

yes

—

#1377

C₂×C₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

100

C₂×C₄₇

#2084

yes

characteristic

101

C₂×C₄₇

#2238

yes

characteristic

102

C₂×C₄₇

#2239

yes

characteristic

103

C₂×M₃₂

C₂²×C₄₇

yes

—

104

C₂×#108

—

105

C₂×#108

C₂²×C₄₇

yes

—

106

C₂×#107

C₂²×C₄₇

yes

characteristic

107

C₂×#106

C₂²×C₄₇

yes

characteristic

108

#423

—

109

C₂×#97

C₂²×C₄₇

yes

characteristic

110

#389

—

111

#365

—

112

#340

—

113

C₄₇

#13510

yes

characteristic, direct product factor, Sylow

114

C₂×M₁₆

—

115

C₂²×C₈

D₄×C₄₇

yes

characteristic

116

C₂²×C₈

C₂³×C₄₇

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

D₄×C₄₇

yes

characteristic

121

M₃₂

—

122

M₃₂

Q₈×C₄₇

yes

—

123

C₂×C₁₆

—

124

#108

—

125

#108

—

126

#108

—

127

#107

—

128

#106

—

129

#106

D₄×C₄₇

yes

—

130

C₄⋊C₈

—

131

#97

—

132

#95

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

SD₁₆×C₄₇

yes

characteristic

136

C₂²×C₄

SD₁₆×C₄₇

yes

characteristic

137

C₂²×C₄

C₂×D₄×C₄₇

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

SD₁₆×C₄₇

yes

—

144

C₂×C₈

C₂×D₄×C₄₇

yes

characteristic

145

C₂×C₈

C₂×Q₈×C₄₇

yes

—

146

C₂×C₈

#41×C₄₇

yes

—

147

C₂×C₈

#41×C₄₇

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₁₆

—

156

C₂³

C₂×SD₁₆×C₄₇

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

#121×C₄₇

yes

characteristic

168

C₂×C₄

#123×C₄₇

yes

characteristic

169

C₂×C₄

C₄⋊Q₈×C₄₇

yes

characteristic

170

C₂×C₄

C₂×SD₁₆×C₄₇

yes

characteristic

171

C₂×C₄

AGL(1,ℤ/8)×C₄₇

yes

characteristic

172

C₂×C₄

#137×C₄₇

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

C₂²⋊Q₈×C₄₇

yes

—

176

C₈

#125×C₄₇

yes

—

177

C₂²

—

178

C₂²

—

179

C₂²

C₄₇×#461

yes

characteristic

180

C₂²

C₄₇×#481

yes

characteristic

181

C₂²

C₄₇×#498

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

C₄₇×#398

yes

characteristic

187

C₄

C₄₇×#460

yes

characteristic

188

C₄

C₄₇×#483

yes

characteristic

189

C₄

C₄₇×#501

yes

characteristic

190

C₂

—

191

C₂

—

192

C₂

C₄₇×#2084

yes

characteristic

193

C₂

C₄₇×#2238

yes

characteristic

194

C₂

C₄₇×#2239

yes

characteristic

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

subgroup

G/H

center, socle

C₂²×C₄₇

#481

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₄₇

prime radical

C₂³×C₄₇

C₂×SD₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ C₂×#97×C₄₇ ◁ G

N	H ≅ G/N	type
79. #13510	C₄₇	direct (N × H)
113. C₄₇	#13510	direct (N × H)

H ≅ G/N

type

79. #13510

C₄₇

direct (N × H)

113. C₄₇

#13510

direct (N × H)

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

subgroup	index
C₂×C₄₇×#365	2
C₄₇×#1944	2
C₄₇×#1849	2
C₄₇×#1387	2
C₄₇×#1377	2
#13510	47

subgroup

index

C₂×C₄₇×#365

C₄₇×#1944

C₄₇×#1849

C₄₇×#1387

C₄₇×#1377

#13510

Subgroups — C₄₇×#13510 #26053208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#13510 #26053208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#13510 #26053208