Group #26132496 — subgroups

Subgroups — #26132496

Order 12032.

434 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 135 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×#15689078	C₂	1	1	yes	maximal, characteristic
2	#20244999	C₂	1	1	yes	maximal, characteristic
3	#20241471	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2112	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₄	1	1	yes	characteristic
6	C₂×#13093983	C₄	1	1	yes	characteristic
7	C₂×#13093964	C₂²	1	2	yes	—
8	#15689078	—	2	2	no	—
9	#15689078	—	2	2	no	—
10	C₂×#13093934	C₂²	1	2	yes	—
11	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
12	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
15	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
16	C₂×#10625329	—	2	2	no	—
17	C₂×#10625329	—	2	4	no	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
19	#13093983	—	2	2	no	—
20	#13093983	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	C₈	1	2	yes	—
22	C₂×C₂²⋊Dic₄₇	C₈	1	2	yes	—
23	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
24	#13093964	—	2	4	no	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	#13093934	—	2	2	no	—
27	#13093934	—	2	2	no	—
28	C₈×Dic₄₇	—	2	4	no	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	—	2	4	no	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	D₄	1	2	yes	—
33	C₂²×C₈×C₄₇	D₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	M₃₂×C₄₇	—	2	4	no	—
36	C₂×C₁₆×C₄₇	—	2	2	no	—
37	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₄₇	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₂×C₈×C₄₇	—	2	4	no	—
47	C₂×C₈×C₄₇	—	2	4	no	—
48	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	#41	1	4	yes	—
52	C₂²⋊Dic₄₇	—	2	4	no	—
53	C₄⋊Dic₄₇	—	2	4	no	—
54	C₄×Dic₄₇	—	2	2	no	—
55	C₄×Dic₄₇	—	2	2	no	—
56	#10625329	—	2	8	no	—
57	C₂×C₄₇⋊C₈	—	2	2	no	—
58	C₂×C₄₇⋊C₈	—	2	2	no	—
59	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
60	C₁₆×C₄₇	—	4	2	no	—
61	C₄₇⋊C₁₆	—	4	2	no	—
62	C₂³×C₄₇	—	2	2	no	—
63	C₂³×C₄₇	—	4	2	no	—
64	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
65	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#117	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₄₇	—	2	4	no	—
76	C₂×Dic₄₇	—	4	4	no	—
77	C₈×C₄₇	—	2	2	no	—
78	C₈×C₄₇	#123	1	2	yes	—
79	C₈×C₄₇	#123	1	2	yes	—
80	C₄₇⋊C₈	—	2	4	no	—
81	#12847	—	47	1	no	maximal, Sylow
82	C₂²×C₄₇	—	2	2	no	—
83	C₂²×C₄₇	—	2	2	no	—
84	C₂²×C₄₇	—	2	4	no	—
85	C₂²×C₄₇	—	4	4	no	—
86	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
87	C₂²×C₄₇	#431	1	2	yes	—
88	C₄×C₄₇	—	2	2	no	—
89	C₄×C₄₇	#381	1	1	yes	characteristic
90	C₄×C₄₇	#407	1	1	yes	characteristic
91	C₄×C₄₇	#432	1	2	yes	—
92	Dic₄₇	—	4	4	no	—
93	C₂×#405	—	47	1	no	—
94	#2112	—	47	1	no	—
95	#2112	D₄₇	1	1	yes	characteristic
96	#1377	—	47	4	no	—
97	C₂×C₄₇	—	2	2	no	—
98	C₂×C₄₇	—	4	2	no	—
99	C₂×C₄₇	#1817	1	1	yes	characteristic
100	C₂×C₄₇	#2115	1	2	yes	—
101	C₂²×M₁₆	—	47	1	no	—
102	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
103	C₂×#117	—	47	1	no	—
104	C₂×M₃₂	—	47	2	no	—
105	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
106	#405	—	94	2	no	—
107	#405	—	94	2	no	—
108	C₂×#97	—	47	2	no	—
109	C₂×C₄×C₈	—	47	2	no	—
110	C₂²⋊C₁₆	—	2	2	no	—
111	C₂²⋊C₁₆	—	94	2	no	—
112	C₄₇	#12847	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
114	C₂×M₁₆	—	94	4	no	—
115	C₂×M₁₆	—	94	2	no	—
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	—	47	2	no	—
118	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
119	C₂²×C₈	D₁₈₈	1	2	yes	—
120	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
121	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
122	#117	—	94	2	no	—
123	#117	—	94	2	no	—
124	C₂×C₄⋊C₄	—	47	2	no	—
125	C₂×C₂²⋊C₄	—	47	2	no	—
126	C₂×C₄²	—	47	2	no	—
127	M₃₂	—	2	4	no	—
128	M₃₂	—	94	4	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	94	2	no	—
131	#97	—	94	2	no	—
132	#97	—	94	2	no	—
133	C₄×C₈	—	94	4	no	—
134	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	47	4	no	—
138	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
139	C₂²×C₄	#10625324	1	2	yes	—
140	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
141	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
142	M₁₆	—	94	8	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	47	4	no	—
147	C₂×C₈	—	94	2	no	—
148	C₂×C₈	—	94	2	no	—
149	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
150	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
151	C₂×C₈	#10625352	1	4	yes	—
152	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
153	C₄⋊C₄	—	94	4	no	—
154	C₂²⋊C₄	—	94	4	no	—
155	C₄²	—	94	2	no	—
156	C₄²	—	94	2	no	—
157	C₁₆	—	4	2	no	—
158	C₁₆	—	188	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	4	2	no	—
161	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
162	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	94	4	no	—
168	C₂×C₄	—	188	4	no	—
169	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
170	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
171	C₂×C₄	#13093980	1	2	yes	—
172	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
173	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	94	4	no	—
176	C₈	#13094005	1	2	yes	—
177	C₈	#13094008	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
183	C₂²	#15689173	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	188	4	no	—
186	C₄	#15688984	1	1	yes	characteristic
187	C₄	#15689086	1	1	yes	characteristic
188	C₄	#15689177	1	2	yes	—
189	C₂	—	2	2	no	—
190	C₂	—	4	2	no	—
191	C₂	#20243454	1	1	yes	characteristic
192	C₂	#20245008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689078

C₂

yes

maximal, characteristic

#20244999

C₂

yes

maximal, characteristic

#20241471

C₂

yes

maximal, complemented

C₄₇×#2112

C₂

yes

maximal, characteristic

C₂²×#10625329

C₄

yes

characteristic

C₂×#13093983

C₄

yes

characteristic

C₂×#13093964

C₂²

yes

—

#15689078

—

#15689078

—

C₂×#13093934

C₂²

yes

—

C₂×C₈×Dic₄₇

C₄

yes

—

C₂²⋊(C₄₇⋊C₁₆)

—

C₂³×C₈×C₄₇

C₂²

yes

characteristic

C₂×M₃₂×C₄₇

C₂²

yes

—

C₄₇×C₂²⋊C₁₆

—

C₂×#10625329

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

#13093983

—

#13093983

—

C₂×C₄⋊Dic₄₇

C₈

yes

—

C₂×C₂²⋊Dic₄₇

C₈

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

#13093934

—

#13093934

—

C₈×Dic₄₇

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂⁴×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₈

yes

—

C₂³×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#123

yes

—

C₈×C₄₇

#123

yes

—

C₄₇⋊C₈

—

#12847

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₄₇

#431

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

C₂×#405

—

#2112

—

#2112

D₄₇

yes

characteristic

#1377

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1817

yes

characteristic

100

C₂×C₄₇

#2115

yes

—

101

C₂²×M₁₆

—

102

C₂³×C₈

C₂×D₄₇

yes

characteristic

103

C₂×#117

—

104

C₂×M₃₂

—

105

C₂×M₃₂

C₂×D₄₇

yes

—

106

#405

—

107

#405

—

108

C₂×#97

—

109

C₂×C₄×C₈

—

110

C₂²⋊C₁₆

—

111

C₂²⋊C₁₆

—

112

C₄₇

#12847

yes

characteristic, complemented, Sylow

113

C₂³×C₄

C₄×D₄₇

yes

characteristic

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

C₄×D₄₇

yes

characteristic

119

C₂²×C₈

D₁₈₈

yes

—

120

C₂²×C₈

C₂²⋊D₄₇

yes

—

121

C₂²×C₈

C₂²×D₄₇

yes

characteristic

122

#117

—

123

#117

—

124

C₂×C₄⋊C₄

—

125

C₂×C₂²⋊C₄

—

126

C₂×C₄²

—

127

M₃₂

—

128

M₃₂

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

#97

—

132

#97

—

133

C₄×C₈

—

134

C₂⁴

C₈×D₄₇

yes

characteristic

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₈×D₄₇

yes

characteristic

139

C₂²×C₄

#10625324

yes

—

140

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

141

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

142

M₁₆

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

150

C₂×C₈

C₂×D₁₈₈

yes

characteristic

151

C₂×C₈

#10625352

yes

—

152

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

153

C₄⋊C₄

—

154

C₂²⋊C₄

—

155

C₄²

—

156

C₄²

—

157

C₁₆

—

158

C₁₆

—

188

—

159

C₂³

—

160

C₂³

—

161

C₂³

(C₂×D₄₇)⋊C₈

yes

—

162

C₂³

C₂×C₈×D₄₇

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

188

—

169

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

—

170

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

171

C₂×C₄

#13093980

yes

—

172

C₂×C₄

M₁₆×D₄₇

yes

—

173

C₂×C₄

C₂×#10625324

yes

characteristic

174

C₈

—

175

C₈

—

176

C₈

#13094005

yes

—

177

C₈

#13094008

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

C₂×(C₂×D₄₇)⋊C₈

yes

characteristic

183

C₂²

#15689173

yes

—

184

C₄

—

185

C₄

—

188

—

186

C₄

#15688984

yes

characteristic

187

C₄

#15689086

yes

characteristic

188

C₄

#15689177

yes

—

189

C₂

—

190

C₂

—

191

C₂

#20243454

yes

characteristic

192

C₂

#20245008

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂³×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2112	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2112	D₄₇
hypocenter	C₄₇	#12847
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂³×C₄₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#2112

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₄₇

C₂×C₈

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
3. #20241471	C₂	semidirect
112. C₄₇	#12847	semidirect

H ≅ G/N

type

3. #20241471

C₂

semidirect

112. C₄₇

#12847

semidirect

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

subgroup	index
C₂×#15689078	2
#20244999	2
#20241471	2
C₄₇×#2112	2
#12847	47

subgroup

index

C₄₇×#2112

Subgroups — #26132496

Conjugacy classes of subgroups

Complements

Maximal subgroups