Group #26216526 — subgroups

Subgroups — #26216526

Order 12032.

1062 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 299 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	C₄³×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20243531	C₂	1	1	yes	maximal, characteristic, complemented
3	#20243530	C₂	1	6	yes	maximal, complemented
4	#20243080	C₂	1	6	yes	maximal
5	C₄₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10625324	C₂²	1	4	yes	—
7	C₂×C₄²×D₄₇	C₄	1	3	yes	—
8	C₂×C₄²×D₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄²×D₄₇	C₂²	1	3	yes	—
10	C₄²×Dic₄₇	C₄	1	3	yes	—
11	C₄²×Dic₄₇	C₄	1	1	yes	characteristic
12	C₄²×Dic₄₇	C₂²	1	3	yes	—
13	#15688878	C₂²	1	6	yes	—
14	#15688877	C₂²	1	12	yes	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	C₂²	1	6	yes	—
17	C₂²×C₄×D₄₇	C₂×C₄	1	3	yes	—
18	C₂²×C₄×D₄₇	C₂×C₄	1	3	yes	—
19	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
20	C₂×#10625324	—	2	24	no	—
21	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
22	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
23	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
24	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
25	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
26	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
27	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
28	C₂×C₄×Dic₄₇	C₂³	1	3	yes	—
29	C₄²×D₄₇	—	2	12	no	—
30	C₄²×D₄₇	—	2	4	no	—
31	C₄²×D₄₇	—	2	12	no	—
32	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
33	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
36	C₂³×D₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
39	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
40	C₂²×Dic₄₇	C₂²×C₄	1	3	yes	—
41	C₂²×Dic₄₇	C₂²×C₄	1	3	yes	—
42	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×D₄₇	—	2	6	no	—
44	C₂×C₄×D₄₇	—	2	6	no	—
45	C₂×C₄×D₄₇	—	2	6	no	—
46	C₂×C₄×D₄₇	—	2	6	no	—
47	C₂×C₄×D₄₇	—	2	6	no	—
48	C₂×C₄×D₄₇	—	2	6	no	—
49	C₂×C₄×D₄₇	#41	1	12	yes	—
50	C₂×C₈×C₄₇	—	2	12	no	—
51	C₄²×C₄₇	—	2	6	no	—
52	C₄²×C₄₇	—	2	6	no	—
53	C₄²×C₄₇	M₁₆	1	4	yes	complemented
54	C₄×Dic₄₇	—	2	6	no	—
55	C₄×Dic₄₇	—	2	6	no	—
56	C₄×Dic₄₇	—	2	6	no	—
57	C₄×Dic₄₇	—	2	6	no	—
58	C₄×Dic₄₇	—	2	6	no	—
59	C₄×Dic₄₇	—	2	6	no	—
60	C₄×Dic₄₇	M₁₆	1	12	yes	complemented
61	C₂×C₄₇⋊C₈	—	2	12	no	—
62	#10625324	—	4	16	no	—
63	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
64	C₂²×D₄₇	—	2	3	no	—
65	C₂²×D₄₇	—	2	1	no	—
66	C₂²×D₄₇	#117	1	6	yes	—
67	C₂×C₄×C₄₇	—	2	3	no	—
68	C₂×C₄×C₄₇	—	2	3	no	—
69	C₂×C₄×C₄₇	—	2	3	no	—
70	C₂×C₄×C₄₇	—	2	3	no	—
71	C₂×C₄×C₄₇	C₂×M₁₆	1	6	yes	—
72	C₂×C₄×C₄₇	C₂×M₁₆	1	6	yes	—
73	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
74	C₂×Dic₄₇	—	2	3	no	—
75	C₂×Dic₄₇	—	2	3	no	—
76	C₂×Dic₄₇	—	2	3	no	—
77	C₂×Dic₄₇	—	2	3	no	—
78	C₂×Dic₄₇	#117	1	6	yes	—
79	C₂×Dic₄₇	C₂×M₁₆	1	6	yes	—
80	C₂×Dic₄₇	C₂×M₁₆	1	6	yes	—
81	C₄×D₄₇	—	2	24	no	—
82	C₄×D₄₇	—	2	24	no	—
83	C₄×D₄₇	#126	1	16	yes	—
84	C₈×C₄₇	—	4	4	no	—
85	C₄₇⋊C₈	—	4	4	no	—
86	#20772	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	C₂×#117	1	3	yes	—
88	C₂²×C₄₇	C₂²×M₁₆	1	3	yes	—
89	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
90	C₂×D₄₇	—	2	6	no	—
91	C₂×D₄₇	—	2	6	no	—
92	C₂×D₄₇	#383	1	4	yes	—
93	C₄×C₄₇	—	2	6	no	—
94	C₄×C₄₇	—	2	6	no	—
95	C₄×C₄₇	C₂×#126	1	4	yes	—
96	Dic₄₇	—	2	6	no	—
97	Dic₄₇	—	2	6	no	—
98	Dic₄₇	#383	1	4	yes	—
99	C₂×C₄³	—	47	1	no	—
100	#1839	—	47	1	no	—
101	#1839	—	47	6	no	—
102	#1839	D₄₇	1	1	yes	characteristic, complemented
103	#1728	—	47	6	no	—
104	C₂×C₄₇	C₂×#383	1	1	yes	characteristic
105	C₂×C₄₇	#2920	1	3	yes	—
106	C₂×C₄₇	#2920	1	3	yes	—
107	D₄₇	—	2	4	no	—
108	C₂²×M₁₆	—	47	4	no	—
109	C₂²×C₄²	—	47	3	no	—
110	C₂²×C₄²	—	47	1	no	—
111	C₂²×C₄²	—	47	3	no	—
112	C₄³	—	47	3	no	—
113	C₄³	—	47	1	no	—
114	C₄³	—	47	3	no	—
115	C₄³	C₂×D₄₇	1	1	yes	characteristic
116	#336	—	47	12	no	—
117	#336	—	47	6	no	—
118	#336	C₂×D₄₇	1	6	yes	—
119	C₄₇	#20772	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	—	47	1	no	—
121	C₂³×C₄	—	47	3	no	—
122	C₂³×C₄	—	47	3	no	—
123	C₂×M₁₆	—	94	24	no	—
124	C₂²×C₈	—	47	4	no	—
125	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
126	C₂×C₄²	—	47	3	no	—
127	C₂×C₄²	—	47	3	no	—
128	C₂×C₄²	—	47	3	no	—
129	C₂×C₄²	—	47	3	no	—
130	C₂×C₄²	—	47	3	no	—
131	C₂×C₄²	—	47	3	no	—
132	C₂×C₄²	—	47	3	no	—
133	C₂×C₄²	—	94	12	no	—
134	C₂×C₄²	—	94	12	no	—
135	C₂×C₄²	—	94	4	no	—
136	C₂×C₄²	C₄×D₄₇	1	3	yes	—
137	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
139	C₂⁴	—	47	1	no	—
140	C₂²×C₄	—	47	1	no	—
141	C₂²×C₄	—	47	12	no	—
142	C₂²×C₄	—	47	3	no	—
143	C₂²×C₄	—	47	3	no	—
144	C₂²×C₄	—	94	6	no	—
145	C₂²×C₄	—	94	6	no	—
146	C₂²×C₄	—	94	6	no	—
147	C₂²×C₄	—	94	6	no	—
148	C₂²×C₄	—	94	6	no	—
149	C₂²×C₄	—	94	6	no	—
150	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
151	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
152	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
153	M₁₆	—	188	16	no	—
154	C₂×C₈	—	2	12	no	—
155	C₂×C₈	—	94	12	no	—
156	C₄²	—	2	6	no	—
157	C₄²	—	2	6	no	—
158	C₄²	—	47	12	no	—
159	C₄²	—	94	6	no	—
160	C₄²	—	94	6	no	—
161	C₄²	—	94	6	no	—
162	C₄²	—	94	6	no	—
163	C₄²	—	94	6	no	—
164	C₄²	—	94	6	no	—
165	C₄²	#10625324	1	4	yes	complemented
166	C₂³	—	47	6	no	—
167	C₂³	—	94	3	no	—
168	C₂³	—	94	1	no	—
169	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	2	3	no	—
172	C₂×C₄	—	2	3	no	—
173	C₂×C₄	—	2	3	no	—
174	C₂×C₄	—	47	6	no	—
175	C₂×C₄	—	47	16	no	—
176	C₂×C₄	—	47	6	no	—
177	C₂×C₄	—	47	6	no	—
178	C₂×C₄	—	94	24	no	—
179	C₂×C₄	—	94	24	no	—
180	C₂×C₄	—	94	3	no	—
181	C₂×C₄	—	94	3	no	—
182	C₂×C₄	—	94	3	no	—
183	C₂×C₄	—	94	3	no	—
184	C₂×C₄	M₁₆×D₄₇	1	6	yes	—
185	C₂×C₄	C₂×#10625324	1	6	yes	—
186	C₂×C₄	#41×D₄₇	1	6	yes	—
187	C₈	—	4	4	no	—
188	C₈	—	188	4	no	—
189	C₂²	—	47	4	no	—
190	C₂²	—	94	6	no	—
191	C₂²	—	94	6	no	—
192	C₂²	#117×D₄₇	1	3	yes	—
193	C₂²	C₂×M₁₆×D₄₇	1	3	yes	—
194	C₂²	C₂²×#10625324	1	1	yes	characteristic
195	C₄	—	2	6	no	—
196	C₄	—	2	6	no	—
197	C₄	—	47	4	no	—
198	C₄	—	94	6	no	—
199	C₄	—	94	6	no	—
200	C₄	#126×D₄₇	1	4	yes	—
201	C₂	—	94	4	no	—
202	C₂	#383×D₄₇	1	1	yes	characteristic
203	C₂	#20252016	1	3	yes	—
204	C₂	#20252018	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄³×D₄₇

C₂

yes

maximal, characteristic

#20243531

C₂

yes

maximal, characteristic, complemented

#20243530

C₂

yes

maximal, complemented

#20243080

C₂

yes

maximal

C₄₇×#1839

C₂

yes

maximal, characteristic, complemented

C₂²×#10625324

C₂²

yes

—

C₂×C₄²×D₄₇

C₄

yes

—

C₂×C₄²×D₄₇

C₄

yes

characteristic

C₂×C₄²×D₄₇

C₂²

yes

—

C₄²×Dic₄₇

C₄

yes

—

C₄²×Dic₄₇

C₄

yes

characteristic

C₄²×Dic₄₇

C₂²

yes

—

#15688878

C₂²

yes

—

#15688877

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₂²

yes

—

C₂²×C₄×D₄₇

C₂×C₄

yes

—

C₂²×C₄×D₄₇

C₂×C₄

yes

—

C₂²×C₄×D₄₇

C₂³

yes

characteristic

C₂×#10625324

—

C₂²×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂³

yes

—

C₄²×D₄₇

—

C₄²×D₄₇

—

C₄²×D₄₇

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

—

C₂³×D₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

M₁₆

yes

complemented

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

M₁₆

yes

complemented

C₂×C₄₇⋊C₈

—

#10625324

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

#117

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#117

yes

—

C₂×Dic₄₇

C₂×M₁₆

yes

—

C₂×Dic₄₇

C₂×M₁₆

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

#126

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#20772

—

maximal, Sylow

C₂²×C₄₇

C₂×#117

yes

—

C₂²×C₄₇

C₂²×M₁₆

yes

—

C₂²×C₄₇

C₂²×M₁₆

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

#383

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#126

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

#383

yes

—

C₂×C₄³

—

100

#1839

—

101

#1839

—

102

#1839

D₄₇

yes

characteristic, complemented

103

#1728

—

104

C₂×C₄₇

C₂×#383

yes

characteristic

105

C₂×C₄₇

#2920

yes

—

106

C₂×C₄₇

#2920

yes

—

107

D₄₇

—

108

C₂²×M₁₆

—

109

C₂²×C₄²

—

110

C₂²×C₄²

—

111

C₂²×C₄²

—

112

C₄³

—

113

C₄³

—

114

C₄³

—

115

C₄³

C₂×D₄₇

yes

characteristic

116

#336

—

117

#336

—

118

#336

C₂×D₄₇

yes

—

119

C₄₇

#20772

yes

characteristic, complemented, Sylow

120

C₂³×C₄

—

121

C₂³×C₄

—

122

C₂³×C₄

—

123

C₂×M₁₆

—

124

C₂²×C₈

—

125

C₂²×C₈

C₂²×D₄₇

yes

—

126

C₂×C₄²

—

127

C₂×C₄²

—

128

C₂×C₄²

—

129

C₂×C₄²

—

130

C₂×C₄²

—

131

C₂×C₄²

—

132

C₂×C₄²

—

133

C₂×C₄²

—

134

C₂×C₄²

—

135

C₂×C₄²

—

136

C₂×C₄²

C₄×D₄₇

yes

—

137

C₂×C₄²

C₄×D₄₇

yes

characteristic

138

C₂×C₄²

C₂²×D₄₇

yes

—

139

C₂⁴

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×C₄×D₄₇

yes

—

151

C₂²×C₄

C₂×C₄×D₄₇

yes

—

152

C₂²×C₄

C₂³×D₄₇

yes

characteristic

153

M₁₆

—

188

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

#10625324

yes

complemented

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

C₂²×C₄×D₄₇

yes

characteristic

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

M₁₆×D₄₇

yes

—

185

C₂×C₄

C₂×#10625324

yes

—

186

C₂×C₄

#41×D₄₇

yes

—

187

C₈

—

188

C₈

—

188

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

#117×D₄₇

yes

—

193

C₂²

C₂×M₁₆×D₄₇

yes

—

194

C₂²

C₂²×#10625324

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

#126×D₄₇

yes

—

201

C₂

—

202

C₂

#383×D₄₇

yes

characteristic

203

C₂

#20252016

yes

—

204

C₂

#20252018

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1839	C₂
prime radical	C₂³×D₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1839	D₄₇
hypocenter	C₄₇	#20772
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂³×C₄

Fitting

C₄₇×#1839

C₂

prime radical

C₂³×D₄₇

C₂²×C₄

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ #1839

N	H ≅ G/N	type
2. #20243531	C₂	semidirect
3. #20243530	C₂	semidirect
5. C₄₇×#1839	C₂	semidirect
53. C₄²×C₄₇	M₁₆	semidirect
60. C₄×Dic₄₇	M₁₆	semidirect
102. #1839	D₄₇	semidirect
119. C₄₇	#20772	semidirect
165. C₄²	#10625324	semidirect

H ≅ G/N

type

2. #20243531

C₂

semidirect

3. #20243530

C₂

semidirect

5. C₄₇×#1839

C₂

semidirect

53. C₄²×C₄₇

M₁₆

semidirect

60. C₄×Dic₄₇

M₁₆

semidirect

102. #1839

D₄₇

semidirect

119. C₄₇

#20772

semidirect

165. C₄²

#10625324

semidirect

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

subgroup	index
C₄³×D₄₇	2
#20243531	2
#20243530	2
#20243080	2
C₄₇×#1839	2
#20772	47

subgroup

index

C₄³×D₄₇

C₄₇×#1839

Subgroups — #26216526

Conjugacy classes of subgroups

Complements

Maximal subgroups