Group #26109947 — subgroups

Subgroups — #26109947

Order 12032.

362 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20243539	C₂	1	2	yes	maximal
2	#20243133	C₂	1	1	yes	maximal, characteristic
3	#20241223	C₂	1	1	yes	maximal, characteristic
4	#20241222	C₂	1	2	yes	maximal
5	C₄₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₄₇⋊C₈	—	2	2	no	—
8	C₂×Dic₄₇⋊C₈	—	2	2	no	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15688877	C₄	1	2	yes	—
12	#15688877	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
20	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	Dic₄₇⋊C₈	—	4	4	no	—
25	Dic₄₇⋊C₈	—	4	4	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	—	2	2	no	—
31	C₄⋊C₈×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
37	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₄₇	—	2	2	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	4	2	no	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	C₂×C₈×C₄₇	—	4	2	no	—
44	C₄²×C₄₇	#41	1	2	yes	—
45	C₄²×C₄₇	#41	1	2	yes	—
46	C₄×Dic₄₇	—	2	2	no	—
47	C₄×Dic₄₇	—	2	2	no	—
48	C₄×Dic₄₇	—	4	2	no	—
49	C₄×Dic₄₇	D₈	1	2	yes	—
50	C₄×Dic₄₇	SD₁₆	1	4	yes	—
51	C₄×Dic₄₇	Q₁₆	1	2	yes	—
52	C₂×C₄₇⋊C₈	—	2	1	no	—
53	C₂×C₄₇⋊C₈	—	2	1	no	—
54	C₂×C₄₇⋊C₈	—	2	2	no	—
55	C₂×C₄₇⋊C₈	—	2	2	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	—	4	2	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#124	1	2	yes	—
64	C₂×C₄×C₄₇	#125	1	2	yes	—
65	C₂×C₄×C₄₇	#131	1	2	yes	—
66	C₂×C₄×C₄₇	#131	1	2	yes	—
67	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₄₇	—	2	2	no	—
71	C₂×Dic₄₇	—	2	2	no	—
72	C₂×Dic₄₇	—	2	2	no	—
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	#106	1	4	yes	—
75	C₂×Dic₄₇	#107	1	4	yes	—
76	C₈×C₄₇	—	4	2	no	—
77	C₈×C₄₇	—	8	2	no	—
78	C₄₇⋊C₈	—	4	1	no	—
79	C₄₇⋊C₈	—	4	1	no	—
80	C₄₇⋊C₈	—	8	2	no	—
81	#8787	—	47	1	no	maximal, Sylow
82	C₂²×C₄₇	#382	1	1	yes	characteristic
83	C₂²×C₄₇	#424	1	2	yes	—
84	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
85	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
86	C₂²×C₄₇	#430	1	2	yes	—
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	#433	1	2	yes	—
89	C₄×C₄₇	#433	1	1	yes	characteristic
90	C₄×C₄₇	#433	1	1	yes	characteristic
91	C₄×C₄₇	#486	1	2	yes	—
92	C₄×C₄₇	#487	1	2	yes	—
93	Dic₄₇	—	2	4	no	—
94	Dic₄₇	—	2	4	no	—
95	#1842	—	47	2	no	—
96	#1744	—	47	1	no	—
97	#1277	—	47	2	no	—
98	#1277	—	47	1	no	—
99	#1277	D₄₇	1	1	yes	characteristic
100	C₂×C₄₇	#1570	1	2	yes	—
101	C₂×C₄₇	#1571	1	2	yes	—
102	C₂×C₄₇	#1832	1	1	yes	characteristic
103	C₂×C₄₇	#1834	1	1	yes	characteristic
104	C₂×C₄₇	#1838	1	1	yes	characteristic
105	C₂×C₄⋊C₈	—	47	2	no	—
106	C₂×C₄⋊C₈	—	94	2	no	—
107	C₂×C₄⋊C₈	—	94	2	no	—
108	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
109	C₂×C₄×C₈	—	47	1	no	—
110	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
111	C₄³	—	47	1	no	—
112	#336	—	47	2	no	—
113	#336	—	47	2	no	—
114	C₄₇	#8787	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	—	47	1	no	—
117	C₂²×C₈	—	47	1	no	—
118	C₂²×C₈	—	94	2	no	—
119	C₂²×C₈	C₄×D₄₇	1	2	yes	—
120	C₂×C₄²	—	47	1	no	—
121	C₂×C₄²	—	47	1	no	—
122	C₂×C₄²	—	47	2	no	—
123	C₂×C₄²	—	47	2	no	—
124	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	94	2	no	—
128	C₄⋊C₈	—	94	2	no	—
129	C₄⋊C₈	—	188	4	no	—
130	C₄⋊C₈	—	188	4	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	94	2	no	—
133	C₂²×C₄	—	47	2	no	—
134	C₂²×C₄	—	47	2	no	—
135	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
137	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	94	1	no	—
145	C₂×C₈	—	94	1	no	—
146	C₂×C₈	—	94	2	no	—
147	C₂×C₈	—	94	2	no	—
148	C₂×C₈	—	188	2	no	—
149	C₂×C₈	—	188	2	no	—
150	C₂×C₈	—	188	2	no	—
151	C₄²	—	47	2	no	—
152	C₄²	—	47	4	no	—
153	C₄²	—	47	2	no	—
154	C₄²	—	94	2	no	—
155	C₄²	—	94	2	no	—
156	C₄²	—	188	2	no	—
157	C₄²	#10625350	1	2	yes	—
158	C₄²	#10625352	1	2	yes	—
159	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	47	4	no	—
162	C₂×C₄	—	47	4	no	—
163	C₂×C₄	—	94	2	no	—
164	C₂×C₄	—	94	2	no	—
165	C₂×C₄	—	94	2	no	—
166	C₂×C₄	—	94	2	no	—
167	C₂×C₄	#13093980	1	1	yes	characteristic
168	C₂×C₄	#13093981	1	1	yes	characteristic
169	C₂×C₄	#13094010	1	2	yes	—
170	C₂×C₄	#13094014	1	2	yes	—
171	C₂×C₄	#13094030	1	2	yes	—
172	C₂×C₄	#13094032	1	2	yes	—
173	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
174	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
175	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
176	C₈	—	4	2	no	—
177	C₈	—	8	2	no	—
178	C₈	—	376	2	no	—
179	C₈	—	188	1	no	—
180	C₈	—	188	1	no	—
181	C₂²	#15688987	1	1	yes	characteristic
182	C₂²	#15689148	1	2	yes	—
183	C₂²	#106×D₄₇	1	1	yes	characteristic
184	C₂²	#107×D₄₇	1	1	yes	characteristic
185	C₂²	#15689170	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	94	4	no	—
188	C₄	—	94	4	no	—
189	C₄	#15689181	1	2	yes	—
190	C₄	#15689182	1	1	yes	characteristic
191	C₄	#15689185	1	1	yes	characteristic
192	C₄	#15689468	1	2	yes	—
193	C₄	#15689472	1	2	yes	—
194	C₂	#20242157	1	2	yes	—
195	C₂	#20242161	1	2	yes	—
196	C₂	#20243506	1	1	yes	characteristic
197	C₂	#20243514	1	1	yes	characteristic
198	C₂	#20243527	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243539

C₂

yes

maximal

#20243133

C₂

yes

maximal, characteristic

#20241223

C₂

yes

maximal, characteristic

#20241222

C₂

yes

maximal

C₄₇×#1277

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×Dic₄₇⋊C₈

—

C₂×Dic₄₇⋊C₈

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄²×Dic₄₇

C₂²

yes

characteristic

#15688877

C₄

yes

—

#15688877

C₄

yes

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄×Dic₄₇

Q₈

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

Dic₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×Dic₄₇

C₄⋊C₄

yes

—

C₂²×Dic₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

#41

yes

—

C₄²×C₄₇

#41

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₈

yes

—

C₄×Dic₄₇

SD₁₆

yes

—

C₄×Dic₄₇

Q₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#125

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#106

yes

—

C₂×Dic₄₇

#107

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

#433

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#487

yes

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic

100

C₂×C₄₇

#1570

yes

—

101

C₂×C₄₇

#1571

yes

—

102

C₂×C₄₇

#1832

yes

characteristic

103

C₂×C₄₇

#1834

yes

characteristic

104

C₂×C₄₇

#1838

yes

characteristic

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

111

C₄³

—

112

#336

—

113

#336

—

114

C₄₇

#8787

yes

characteristic, complemented, Sylow

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₄×D₄₇

yes

—

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

—

124

C₂×C₄²

C₂²×D₄₇

yes

characteristic

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

188

—

130

C₄⋊C₈

—

188

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

136

C₂²×C₄

D₄×D₄₇

yes

characteristic

137

C₂²×C₄

Q₈×D₄₇

yes

characteristic

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

188

—

149

C₂×C₈

—

188

—

150

C₂×C₈

—

188

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

188

—

157

C₄²

#10625350

yes

—

158

C₄²

#10625352

yes

—

159

C₂³

C₄⋊C₄×D₄₇

yes

characteristic

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

#13093980

yes

characteristic

168

C₂×C₄

#13093981

yes

characteristic

169

C₂×C₄

#13094010

yes

—

170

C₂×C₄

#13094014

yes

—

171

C₂×C₄

#13094030

yes

—

172

C₂×C₄

#13094032

yes

—

173

C₂×C₄

D₈×D₄₇

yes

characteristic

174

C₂×C₄

SD₁₆×D₄₇

yes

—

175

C₂×C₄

Q₁₆×D₄₇

yes

characteristic

176

C₈

—

177

C₈

—

178

C₈

—

376

—

179

C₈

—

188

—

180

C₈

—

188

—

181

C₂²

#15688987

yes

characteristic

182

C₂²

#15689148

yes

—

183

C₂²

#106×D₄₇

yes

characteristic

184

C₂²

#107×D₄₇

yes

characteristic

185

C₂²

#15689170

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#15689181

yes

—

190

C₄

#15689182

yes

characteristic

191

C₄

#15689185

yes

characteristic

192

C₄

#15689468

yes

—

193

C₄

#15689472

yes

—

194

C₂

#20242157

yes

—

195

C₂

#20242161

yes

—

196

C₂

#20243506

yes

characteristic

197

C₂

#20243514

yes

characteristic

198

C₂

#20243527

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1277	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1277	D₄₇
hypocenter	C₄₇	#8787
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂²×C₄

D₄×D₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1277

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
114. C₄₇	#8787	semidirect

H ≅ G/N

type

114. C₄₇

#8787

semidirect

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

subgroup	index
#20243539	2
#20243133	2
#20241223	2
#20241222	2
C₄₇×#1277	2
#8787	47

subgroup

index

C₄₇×#1277

Subgroups — #26109947

Conjugacy classes of subgroups

Complements

Maximal subgroups