Group #26050232 — subgroups

Subgroups — C₄₇×#10534 #26050232

Order 12032.

358 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₄₇×#3255	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1561	C₂	1	2	yes	maximal
4	C₄₇×#1493	C₂	1	2	yes	maximal
5	C₄₇×#1309	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#533	C₂²	1	2	yes	—
7	C₄₇×#475	—	2	2	no	—
8	C₄₇×#470	—	2	2	no	—
9	C₄₇×#462	—	2	2	no	—
10	C₂×#108×C₄₇	—	2	2	no	—
11	C₄₇×#423	C₂²	1	1	yes	characteristic
12	C₄₇×#423	C₂²	1	2	yes	—
13	C₄₇×#423	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#332	—	2	2	no	—
16	C₄₇×#332	—	2	2	no	—
17	C₄₇×Q₈⋊C₈	—	2	4	no	—
18	C₂×Q₁₆×C₄₇	—	2	2	no	—
19	C₂×M₁₆×C₄₇	—	2	1	no	—
20	C₂×M₁₆×C₄₇	—	2	2	no	—
21	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
22	C₂²×C₈×C₄₇	—	2	1	no	—
23	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
24	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
25	#125×C₄₇	—	2	2	no	—
26	C₂²⋊Q₈×C₄₇	—	2	2	no	—
27	C₂²⋊Q₈×C₄₇	—	2	2	no	—
28	C₄×Q₈×C₄₇	—	2	2	no	—
29	C₄×Q₈×C₄₇	—	2	2	no	—
30	#117×C₄₇	—	2	2	no	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	#108×C₄₇	—	4	4	no	—
33	#106×C₄₇	—	2	2	no	—
34	C₄⋊C₈×C₄₇	—	2	2	no	—
35	C₄⋊C₈×C₄₇	—	2	2	no	—
36	C₄⋊C₈×C₄₇	—	2	2	no	—
37	C₄⋊C₈×C₄₇	—	2	2	no	—
38	#103×C₄₇	—	2	4	no	—
39	C₄×C₈×C₄₇	—	2	1	no	—
40	C₄×C₈×C₄₇	D₄	1	2	yes	—
41	C₂×Q₈×C₄₇	—	2	2	no	—
42	C₂×Q₈×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	#41	1	2	yes	—
45	Q₁₆×C₄₇	—	8	2	no	—
46	M₁₆×C₄₇	—	4	2	no	—
47	M₁₆×C₄₇	—	4	2	no	—
48	M₁₆×C₄₇	—	8	2	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	—	4	1	no	—
54	C₂×C₈×C₄₇	—	4	2	no	—
55	C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₄⋊C₄×C₄₇	—	2	2	no	—
57	C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₄⋊C₄×C₄₇	—	4	2	no	—
60	C₄⋊C₄×C₄₇	—	4	2	no	—
61	C₂²⋊C₄×C₄₇	—	4	2	no	—
62	C₄²×C₄₇	—	4	2	no	—
63	C₄²×C₄₇	C₂×D₄	1	2	yes	—
64	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₄₇	#41	1	1	yes	characteristic
66	C₂³×C₄₇	#124	1	1	yes	characteristic
67	Q₈×C₄₇	—	4	2	no	—
68	Q₈×C₄₇	—	4	2	no	—
69	Q₈×C₄₇	—	4	2	no	—
70	Q₈×C₄₇	—	4	2	no	—
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	—	2	1	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	4	2	no	—
76	C₂×C₄×C₄₇	—	4	2	no	—
77	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₄₇	#121	1	2	yes	—
80	C₂×C₄×C₄₇	#123	1	2	yes	—
81	C₈×C₄₇	—	2	4	no	—
82	C₈×C₄₇	—	4	2	no	—
83	C₈×C₄₇	—	8	1	no	—
84	C₈×C₄₇	—	8	2	no	—
85	#10534	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₄₇	—	2	2	no	—
87	C₂²×C₄₇	—	2	1	no	—
88	C₂²×C₄₇	#394	1	1	yes	characteristic
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	2	1	no	—
91	C₄×C₄₇	—	8	2	no	—
92	C₄×C₄₇	—	8	2	no	—
93	C₄×C₄₇	#453	1	1	yes	characteristic
94	C₄×C₄₇	#456	1	1	yes	characteristic
95	C₄×C₄₇	#471	1	2	yes	—
96	C₄×C₄₇	#473	1	2	yes	—
97	#3255	C₂×C₄₇	1	1	yes	characteristic
98	#1848	C₂×C₄₇	1	1	yes	characteristic
99	#1561	C₂×C₄₇	1	2	yes	—
100	#1493	C₂×C₄₇	1	2	yes	—
101	#1309	C₂×C₄₇	1	1	yes	characteristic
102	C₂×C₄₇	—	4	1	no	—
103	C₂×C₄₇	#2039	1	1	yes	characteristic
104	C₂×C₄₇	#2052	1	2	yes	—
105	#533	C₂²×C₄₇	1	2	yes	—
106	#475	—	2	2	no	—
107	#470	—	2	2	no	—
108	#462	—	2	2	no	—
109	C₂×#108	—	2	2	no	—
110	#423	C₂²×C₄₇	1	1	yes	characteristic
111	#423	C₂²×C₄₇	1	2	yes	—
112	#423	C₂²×C₄₇	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
114	#332	—	2	2	no	—
115	#332	—	2	2	no	—
116	Q₈⋊C₈	—	2	4	no	—
117	C₄₇	#10534	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×M₁₆	—	2	1	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
122	C₂²×C₈	—	2	1	no	—
123	C₄⋊Q₈	D₄×C₄₇	1	2	yes	—
124	C₄⋊Q₈	D₄×C₄₇	1	2	yes	—
125	#125	—	2	2	no	—
126	C₂²⋊Q₈	—	2	2	no	—
127	C₂²⋊Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	C₄×Q₈	—	2	2	no	—
130	#117	—	2	2	no	—
131	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
132	#108	—	4	4	no	—
133	#106	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	#103	—	2	4	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	D₄×C₄₇	1	2	yes	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
144	C₂²×C₄	#41×C₄₇	1	2	yes	—
145	Q₁₆	—	8	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	8	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂×D₄×C₄₇	1	2	yes	—
164	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
165	C₄²	#41×C₄₇	1	1	yes	characteristic
166	C₂³	#124×C₄₇	1	1	yes	characteristic
167	Q₈	—	4	2	no	—
168	Q₈	—	4	2	no	—
169	Q₈	—	4	2	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
178	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
179	C₂×C₄	#121×C₄₇	1	2	yes	—
180	C₂×C₄	#123×C₄₇	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	4	2	no	—
183	C₈	—	8	1	no	—
184	C₈	—	8	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	2	no	—
187	C₂²	C₄₇×#394	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	—	8	2	no	—
192	C₄	C₄₇×#453	1	1	yes	characteristic
193	C₄	C₄₇×#456	1	1	yes	characteristic
194	C₄	C₄₇×#471	1	2	yes	—
195	C₄	C₄₇×#473	1	2	yes	—
196	C₂	—	4	1	no	—
197	C₂	C₄₇×#2039	1	1	yes	characteristic
198	C₂	C₄₇×#2052	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₇×#3255

C₂

yes

maximal, characteristic

C₄₇×#1848

C₂

yes

maximal, characteristic

C₄₇×#1561

C₂

yes

maximal

C₄₇×#1493

C₂

yes

maximal

C₄₇×#1309

C₂

yes

maximal, characteristic

C₄₇×#533

C₂²

yes

—

C₄₇×#475

—

C₄₇×#470

—

C₄₇×#462

—

C₂×#108×C₄₇

—

C₄₇×#423

C₂²

yes

characteristic

C₄₇×#423

C₂²

yes

—

C₄₇×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#332

—

C₄₇×#332

—

C₄₇×Q₈⋊C₈

—

C₂×Q₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

—

C₄⋊Q₈×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

D₄

yes

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

#106×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

#103×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

Q₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₂³×C₄₇

yes

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

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

#10534

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#394

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#473

yes

—

#3255

C₂×C₄₇

yes

characteristic

#1848

C₂×C₄₇

yes

characteristic

#1561

C₂×C₄₇

yes

—

100

#1493

C₂×C₄₇

yes

—

101

#1309

C₂×C₄₇

yes

characteristic

102

C₂×C₄₇

—

103

C₂×C₄₇

#2039

yes

characteristic

104

C₂×C₄₇

#2052

yes

—

105

#533

C₂²×C₄₇

yes

—

106

#475

—

107

#470

—

108

#462

—

109

C₂×#108

—

110

#423

C₂²×C₄₇

yes

characteristic

111

#423

C₂²×C₄₇

yes

—

112

#423

C₂²×C₄₇

yes

characteristic

113

C₂×C₄×C₈

C₂²×C₄₇

yes

characteristic

114

#332

—

115

#332

—

116

Q₈⋊C₈

—

117

C₄₇

#10534

yes

characteristic, direct product factor, Sylow

118

C₂×Q₁₆

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

D₄×C₄₇

yes

—

122

C₂²×C₈

—

123

C₄⋊Q₈

D₄×C₄₇

yes

—

124

C₄⋊Q₈

D₄×C₄₇

yes

—

125

#125

—

126

C₂²⋊Q₈

—

127

C₂²⋊Q₈

—

128

C₄×Q₈

—

129

C₄×Q₈

—

130

#117

—

131

C₂×C₄²

C₂³×C₄₇

yes

characteristic

132

#108

—

133

#106

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

139

C₄×C₈

—

140

C₄×C₈

D₄×C₄₇

yes

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

C₂×D₄×C₄₇

yes

characteristic

144

C₂²×C₄

#41×C₄₇

yes

—

145

Q₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

C₂×D₄×C₄₇

yes

—

164

C₄²

C₂×D₄×C₄₇

yes

characteristic

165

C₄²

#41×C₄₇

yes

characteristic

166

C₂³

#124×C₄₇

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

#120×C₄₇

yes

characteristic

178

C₂×C₄

#121×C₄₇

yes

characteristic

179

C₂×C₄

#121×C₄₇

yes

—

180

C₂×C₄

#123×C₄₇

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₄₇×#394

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

C₄₇×#453

yes

characteristic

193

C₄

C₄₇×#456

yes

characteristic

194

C₄

C₄₇×#471

yes

—

195

C₄

C₄₇×#473

yes

—

196

C₂

—

197

C₂

C₄₇×#2039

yes

characteristic

198

C₂

C₄₇×#2052

yes

—

	subgroup	G/H
center, socle	C₂²×C₄₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
prime radical	C₂³×C₄₇	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₄₇

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₄₇

prime radical

C₂³×C₄₇

#124

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
85. #10534	C₄₇	direct (N × H)
117. C₄₇	#10534	direct (N × H)

H ≅ G/N

type

85. #10534

C₄₇

direct (N × H)

117. C₄₇

#10534

direct (N × H)

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

subgroup	index
C₄₇×#3255	2
C₄₇×#1848	2
C₄₇×#1561	2
C₄₇×#1493	2
C₄₇×#1309	2
#10534	47

subgroup

index

C₄₇×#3255

C₄₇×#1848

C₄₇×#1561

C₄₇×#1493

C₄₇×#1309

#10534

Subgroups — C₄₇×#10534 #26050232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#10534 #26050232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#10534 #26050232