Group #26050326 — subgroups

Subgroups — C₄₇×#10628 #26050326

Order 12032.

292 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 106 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×#1995	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#1843	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1542	C₂	1	2	yes	maximal
4	C₄₇×#1301	C₂	1	2	yes	maximal
5	C₄₇×#1274	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#533	C₂²	1	1	yes	characteristic
7	C₄₇×#431	C₂²	1	2	yes	—
8	C₄₇×#423	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
10	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
11	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
12	C₄₇×#376	C₂²	1	1	yes	characteristic
13	C₄₇×#339	—	2	1	no	—
14	C₄₇×#339	—	2	1	no	—
15	C₄₇×#333	—	2	2	no	—
16	C₄₇×#332	—	2	2	no	—
17	C₂×M₁₆×C₄₇	—	2	1	no	—
18	C₂²×C₈×C₄₇	—	2	2	no	—
19	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
20	#125×C₄₇	—	2	1	no	—
21	C₂²⋊Q₈×C₄₇	—	2	2	no	—
22	C₄×Q₈×C₄₇	—	2	2	no	—
23	#117×C₄₇	—	2	1	no	—
24	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	2	no	—
28	C₄⋊C₈×C₄₇	—	2	2	no	—
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	C₄⋊C₈×C₄₇	—	2	2	no	—
31	C₄⋊C₈×C₄₇	—	4	2	no	—
32	C₄⋊C₈×C₄₇	—	4	2	no	—
33	C₂²⋊C₈×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	—	2	2	no	—
35	#95×C₄₇	—	2	2	no	—
36	C₂×Q₈×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	1	no	—
38	C₂²×C₄×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	#41	1	2	yes	—
41	M₁₆×C₄₇	—	4	1	no	—
42	M₁₆×C₄₇	—	4	1	no	—
43	C₂×C₈×C₄₇	—	4	2	no	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	—	4	1	no	—
46	C₂×C₈×C₄₇	—	4	2	no	—
47	C₂×C₈×C₄₇	—	4	2	no	—
48	C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₄⋊C₄×C₄₇	—	4	2	no	—
51	C₂²⋊C₄×C₄₇	—	4	1	no	—
52	C₄²×C₄₇	—	2	1	no	—
53	C₄²×C₄₇	—	2	1	no	—
54	C₄²×C₄₇	—	4	1	no	—
55	C₄²×C₄₇	SD₁₆	1	2	yes	—
56	C₄²×C₄₇	Q₁₆	1	2	yes	—
57	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₇	#41	1	1	yes	characteristic
59	C₄²×C₄₇	#41	1	2	yes	—
60	C₂³×C₄₇	#123	1	1	yes	characteristic
61	Q₈×C₄₇	—	8	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	—	4	2	no	—
67	C₂×C₄×C₄₇	—	4	2	no	—
68	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#123	1	2	yes	—
70	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#126	1	2	yes	—
72	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
76	C₈×C₄₇	—	8	2	no	—
77	C₈×C₄₇	—	8	2	no	—
78	C₈×C₄₇	—	8	1	no	—
79	#10628	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₄₇	—	2	2	no	—
81	C₂²×C₄₇	#397	1	1	yes	characteristic
82	C₂²×C₄₇	#483	1	1	yes	characteristic
83	C₂²×C₄₇	#484	1	1	yes	characteristic
84	C₄×C₄₇	—	2	2	no	—
85	C₄×C₄₇	—	2	2	no	—
86	C₄×C₄₇	—	4	2	no	—
87	C₄×C₄₇	—	4	2	no	—
88	C₄×C₄₇	—	8	2	no	—
89	C₄×C₄₇	#460	1	1	yes	characteristic
90	C₄×C₄₇	#462	1	1	yes	characteristic
91	C₄×C₄₇	#487	1	1	yes	characteristic
92	C₄×C₄₇	#489	1	1	yes	characteristic
93	#1995	C₂×C₄₇	1	1	yes	characteristic
94	#1843	C₂×C₄₇	1	1	yes	characteristic
95	#1542	C₂×C₄₇	1	2	yes	—
96	#1301	C₂×C₄₇	1	2	yes	—
97	#1274	C₂×C₄₇	1	1	yes	characteristic
98	C₂×C₄₇	—	2	1	no	—
99	C₂×C₄₇	—	2	1	no	—
100	C₂×C₄₇	#2072	1	1	yes	characteristic
101	C₂×C₄₇	#2080	1	2	yes	—
102	#533	C₂²×C₄₇	1	1	yes	characteristic
103	#431	C₂²×C₄₇	1	2	yes	—
104	#423	C₂²×C₄₇	1	1	yes	characteristic
105	C₂×C₄⋊C₈	—	2	1	no	—
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
108	#376	C₂²×C₄₇	1	1	yes	characteristic
109	#339	—	2	1	no	—
110	#339	—	2	1	no	—
111	#333	—	2	2	no	—
112	#332	—	2	2	no	—
113	C₄₇	#10628	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×M₁₆	—	2	1	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₄⋊Q₈	D₄×C₄₇	1	2	yes	—
117	#125	—	2	1	no	—
118	C₂²⋊Q₈	—	2	2	no	—
119	C₄×Q₈	—	2	2	no	—
120	#117	—	2	1	no	—
121	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
122	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	4	2	no	—
129	C₄⋊C₈	—	4	2	no	—
130	C₂²⋊C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
137	C₂²×C₄	#41×C₄₇	1	2	yes	—
138	M₁₆	—	4	1	no	—
139	M₁₆	—	4	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₂²⋊C₄	—	4	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	4	1	no	—
152	C₄²	SD₁₆×C₄₇	1	2	yes	—
153	C₄²	Q₁₆×C₄₇	1	2	yes	—
154	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
155	C₄²	#41×C₄₇	1	1	yes	characteristic
156	C₄²	#41×C₄₇	1	2	yes	—
157	C₂³	#123×C₄₇	1	1	yes	characteristic
158	Q₈	—	8	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
166	C₂×C₄	#123×C₄₇	1	2	yes	—
167	C₂×C₄	#124×C₄₇	1	1	yes	characteristic
168	C₂×C₄	#126×C₄₇	1	2	yes	—
169	C₂×C₄	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
170	C₂×C₄	C₂×Q₁₆×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	—	8	2	no	—
174	C₈	—	8	2	no	—
175	C₈	—	8	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	C₄₇×#397	1	1	yes	characteristic
178	C₂²	C₄₇×#483	1	1	yes	characteristic
179	C₂²	C₄₇×#484	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	8	2	no	—
185	C₄	C₄₇×#460	1	1	yes	characteristic
186	C₄	C₄₇×#462	1	1	yes	characteristic
187	C₄	C₄₇×#487	1	1	yes	characteristic
188	C₄	C₄₇×#489	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	C₄₇×#2072	1	1	yes	characteristic
192	C₂	C₄₇×#2080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₇×#1995

C₂

yes

maximal, characteristic

C₄₇×#1843

C₂

yes

maximal, characteristic

C₄₇×#1542

C₂

yes

maximal

C₄₇×#1301

C₂

yes

maximal

C₄₇×#1274

C₂

yes

maximal, characteristic

C₄₇×#533

C₂²

yes

characteristic

C₄₇×#431

C₂²

yes

—

C₄₇×#423

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₄₇

—

C₂×C₄⋊C₈×C₄₇

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₄₇×#376

C₂²

yes

characteristic

C₄₇×#339

—

C₄₇×#339

—

C₄₇×#333

—

C₄₇×#332

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

—

C₄⋊Q₈×C₄₇

D₄

yes

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

M₁₆×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

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₂³×C₄₇

#123

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

#10628

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₂×C₄₇

yes

characteristic

#1843

C₂×C₄₇

yes

characteristic

#1542

C₂×C₄₇

yes

—

#1301

C₂×C₄₇

yes

—

#1274

C₂×C₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

100

C₂×C₄₇

#2072

yes

characteristic

101

C₂×C₄₇

#2080

yes

—

102

#533

C₂²×C₄₇

yes

characteristic

103

#431

C₂²×C₄₇

yes

—

104

#423

C₂²×C₄₇

yes

characteristic

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

C₂²×C₄₇

yes

—

108

#376

C₂²×C₄₇

yes

characteristic

109

#339

—

110

#339

—

111

#333

—

112

#332

—

113

C₄₇

#10628

yes

characteristic, direct product factor, Sylow

114

C₂×M₁₆

—

115

C₂²×C₈

—

116

C₄⋊Q₈

D₄×C₄₇

yes

—

117

#125

—

118

C₂²⋊Q₈

—

119

C₄×Q₈

—

120

#117

—

121

C₂×C₄²

D₄×C₄₇

yes

characteristic

122

C₂×C₄²

D₄×C₄₇

yes

characteristic

123

C₂×C₄²

C₂³×C₄₇

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₂²⋊C₈

—

131

C₄×C₈

—

132

#95

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×D₄×C₄₇

yes

characteristic

137

C₂²×C₄

#41×C₄₇

yes

—

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

SD₁₆×C₄₇

yes

—

153

C₄²

Q₁₆×C₄₇

yes

—

154

C₄²

C₂×D₄×C₄₇

yes

characteristic

155

C₄²

#41×C₄₇

yes

characteristic

156

C₄²

#41×C₄₇

yes

—

157

C₂³

#123×C₄₇

yes

characteristic

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#121×C₄₇

yes

characteristic

166

C₂×C₄

#123×C₄₇

yes

—

167

C₂×C₄

#124×C₄₇

yes

characteristic

168

C₂×C₄

#126×C₄₇

yes

—

169

C₂×C₄

C₂×SD₁₆×C₄₇

yes

characteristic

170

C₂×C₄

C₂×Q₁₆×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

—

176

C₂²

—

177

C₂²

C₄₇×#397

yes

characteristic

178

C₂²

C₄₇×#483

yes

characteristic

179

C₂²

C₄₇×#484

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

C₄₇×#460

yes

characteristic

186

C₄

C₄₇×#462

yes

characteristic

187

C₄

C₄₇×#487

yes

characteristic

188

C₄

C₄₇×#489

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₄₇×#2072

yes

characteristic

192

C₂

C₄₇×#2080

yes

—

	subgroup	G/H
center	C₂×C₄×C₄₇	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
socle	C₂²×C₄₇	#397
prime radical	C₂³×C₄₇	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₄₇

#124

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₄₇

C₂²×C₄₇

C₂³×C₄₇

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
79. #10628	C₄₇	direct (N × H)
113. C₄₇	#10628	direct (N × H)

H ≅ G/N

type

79. #10628

C₄₇

direct (N × H)

113. C₄₇

#10628

direct (N × H)

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

subgroup	index
C₄₇×#1995	2
C₄₇×#1843	2
C₄₇×#1542	2
C₄₇×#1301	2
C₄₇×#1274	2
#10628	47

subgroup

index

C₄₇×#1995

C₄₇×#1843

C₄₇×#1542

C₄₇×#1301

C₄₇×#1274

#10628

Subgroups — C₄₇×#10628 #26050326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#10628 #26050326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#10628 #26050326