Group #26131401 — subgroups

Subgroups — #26131401

Order 12032.

382 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 151 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×#15689540	C₂	1	1	yes	maximal, characteristic
2	#20244992	C₂	1	2	yes	maximal, complemented
3	#20243188	C₂	1	1	yes	maximal, characteristic
4	#20241478	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2115	C₂	1	1	yes	maximal, characteristic
6	#15689540	—	2	4	no	—
7	C₂×#13093964	C₄	1	2	yes	complemented
8	C₂×#13093964	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	complemented
10	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	#15689107	C₄	1	1	yes	characteristic
12	#15689097	C₄	1	1	yes	characteristic
13	#15689078	C₂²	1	2	yes	—
14	#15688900	—	2	4	no	—
15	#15688894	C₄	1	2	yes	—
16	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#349	C₄	1	4	yes	complemented
19	C₂×#10625329	C₂×C₄	1	2	yes	—
20	#13093983	C₂×C₄	1	2	yes	—
21	#13093964	—	2	4	no	—
22	#13093964	D₄	1	4	yes	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	C₂×C₄₇⋊C₁₆	—	2	4	no	—
25	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
28	#13093943	—	2	1	no	—
29	#13093941	—	2	1	no	—
30	#13093939	—	2	2	no	—
31	#13093934	—	2	2	no	—
32	C₈×Dic₄₇	—	2	2	no	—
33	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
34	#131×C₄₇	—	2	2	no	—
35	#131×C₄₇	—	2	2	no	—
36	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
40	M₃₂×C₄₇	—	2	2	no	—
41	M₃₂×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₁₆×C₄₇	—	2	2	no	—
43	#41×C₄₇	—	2	2	no	—
44	#41×C₄₇	—	2	2	no	—
45	C₂×Q₈×C₄₇	M₁₆	1	1	yes	characteristic
46	C₂×D₄×C₄₇	M₁₆	1	2	yes	—
47	C₂×D₄×C₄₇	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₄₇	—	2	2	no	—
51	M₁₆×C₄₇	—	2	4	no	—
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	—	2	2	no	—
54	C₂×C₈×C₄₇	C₄²	1	2	yes	—
55	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₄₇	#41	1	2	yes	—
59	C₂²⋊Dic₄₇	—	2	2	no	—
60	C₄⋊Dic₄₇	—	2	2	no	—
61	C₄×Dic₄₇	—	2	2	no	—
62	#10625329	—	2	2	no	—
63	#10625329	—	2	2	no	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₁₆×C₄₇	—	2	4	no	—
66	C₄₇⋊C₁₆	—	2	1	no	—
67	C₄₇⋊C₁₆	—	2	1	no	—
68	C₄₇⋊C₁₆	C₂²⋊C₄	1	4	yes	—
69	C₂³×C₄₇	#97	1	2	yes	—
70	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₄₇	—	2	2	no	—
72	D₄×C₄₇	—	2	4	no	—
73	D₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	#97	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#131	1	2	yes	—
81	C₂×Dic₄₇	—	4	2	no	—
82	C₈×C₄₇	—	2	2	no	—
83	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₄₇	#117	1	1	yes	characteristic
85	C₈×C₄₇	C₄×D₄	1	2	yes	—
86	C₄₇⋊C₈	—	4	2	no	—
87	#12617	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	—	2	2	no	—
89	C₂²×C₄₇	—	4	1	no	—
90	C₂²×C₄₇	—	4	2	no	—
91	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
92	C₂²×C₄₇	#435	1	2	yes	—
93	C₄×C₄₇	—	2	2	no	—
94	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₄₇	#405	1	1	yes	characteristic
96	C₄×C₄₇	#435	1	2	yes	—
97	Dic₄₇	—	8	2	no	—
98	C₂×#504	—	47	1	no	—
99	#2115	D₄₇	1	1	yes	characteristic
100	#2110	—	47	2	no	—
101	#1757	—	47	1	no	—
102	#1379	—	47	2	no	—
103	C₂×C₄₇	—	2	2	no	—
104	C₂×C₄₇	—	2	1	no	—
105	C₂×C₄₇	—	4	2	no	—
106	C₂×C₄₇	#1752	1	1	yes	characteristic
107	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
108	#504	—	94	4	no	—
109	C₂×M₃₂	—	47	1	no	—
110	C₂×M₃₂	—	47	2	no	—
111	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
112	C₂²×C₁₆	—	47	2	no	—
113	C₂²×C₁₆	—	47	1	no	—
114	#413	—	47	1	no	—
115	#410	—	47	1	no	—
116	#405	—	47	2	no	—
117	#349	Dic₄₇	1	4	yes	complemented
118	#347	—	94	4	no	—
119	#344	—	47	2	no	—
120	C₄₇	#12617	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	C₄×D₄₇	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	—	47	2	no	—
125	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
126	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
127	C₂²×C₈	C₄×D₄₇	1	2	yes	—
128	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
129	#117	—	47	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	47	4	no	—
132	M₃₂	—	94	4	no	—
133	M₃₂	C₂×Dic₄₇	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	47	2	no	—
136	C₂×C₁₆	—	47	1	no	—
137	C₂×C₁₆	—	47	1	no	—
138	C₂×C₁₆	—	94	4	no	—
139	C₂×C₁₆	—	94	2	no	—
140	#101	—	94	1	no	—
141	#100	—	94	1	no	—
142	#99	—	94	2	no	—
143	#97	—	94	2	no	—
144	C₄×C₈	—	94	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	#10625324	1	1	yes	characteristic
148	C₂×D₄	#10625324	1	2	yes	—
149	C₂×D₄	#10625324	1	1	yes	characteristic
150	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	94	2	no	—
155	M₁₆	—	94	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	94	2	no	—
159	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
161	C₂×C₈	D₄×D₄₇	1	2	yes	—
162	C₂×C₈	#10625352	1	2	yes	—
163	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
164	C₄⋊C₄	—	94	2	no	—
165	C₂²⋊C₄	—	94	2	no	—
166	C₄²	—	94	2	no	—
167	C₁₆	—	2	4	no	—
168	C₁₆	—	47	4	no	—
169	C₁₆	—	94	1	no	—
170	C₁₆	—	94	1	no	—
171	C₂³	#13093934	1	2	yes	—
172	C₂³	C₂×#10625324	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	188	2	no	—
179	C₂×C₄	#13093934	1	2	yes	—
180	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
181	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
182	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
183	C₂×C₄	#13094032	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	188	2	no	—
186	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
187	C₈	#13093980	1	1	yes	characteristic
188	C₈	D₄×Dic₄₇	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#13093934	1	1	yes	characteristic
193	C₂²	#15689191	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	376	2	no	—
196	C₄	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
197	C₄	#15689081	1	1	yes	characteristic
198	C₄	#15689191	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	#20243167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689540

C₂

yes

maximal, characteristic

#20244992

C₂

yes

maximal, complemented

#20243188

C₂

yes

maximal, characteristic

#20241478

C₂

yes

maximal, complemented

C₄₇×#2115

C₂

yes

maximal, characteristic

#15689540

—

C₂×#13093964

C₄

yes

complemented

C₂×#13093964

C₂²

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₄

yes

complemented

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15689107

C₄

yes

characteristic

#15689097

C₄

yes

characteristic

#15689078

C₂²

yes

—

#15688900

—

#15688894

C₄

yes

—

C₂×#131×C₄₇

C₂²

yes

characteristic

C₂×M₃₂×C₄₇

C₂²

yes

—

C₄₇×#349

C₄

yes

complemented

C₂×#10625329

C₂×C₄

yes

—

#13093983

C₂×C₄

yes

—

#13093964

—

#13093964

D₄

yes

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₄₇

—

C₂×#41×C₄₇

C₂×C₄

yes

characteristic

#131×C₄₇

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

M₃₂×C₄₇

—

M₃₂×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

M₁₆

yes

characteristic

C₂×D₄×C₄₇

M₁₆

yes

—

C₂×D₄×C₄₇

M₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₄²

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

C₂²⋊C₄

yes

—

C₂³×C₄₇

#97

yes

—

C₂³×C₄₇

C₂×M₁₆

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₄×D₄

yes

—

C₄₇⋊C₈

—

#12617

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#97

yes

characteristic

C₂²×C₄₇

#435

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₄×C₂²⋊C₄

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

C₂×#504

—

#2115

D₄₇

yes

characteristic

100

#2110

—

101

#1757

—

102

#1379

—

103

C₂×C₄₇

—

104

C₂×C₄₇

—

105

C₂×C₄₇

—

106

C₂×C₄₇

#1752

yes

characteristic

107

C₂×#131

C₂×D₄₇

yes

characteristic

108

#504

—

109

C₂×M₃₂

—

110

C₂×M₃₂

—

111

C₂×M₃₂

C₂×D₄₇

yes

—

112

C₂²×C₁₆

—

113

C₂²×C₁₆

—

114

#413

—

115

#410

—

116

#405

—

117

#349

Dic₄₇

yes

complemented

118

#347

—

119

#344

—

120

C₄₇

#12617

yes

characteristic, complemented, Sylow

121

C₂×#41

C₄×D₄₇

yes

characteristic

122

#131

—

123

#131

—

124

C₂×M₁₆

—

125

C₂×M₁₆

C₄×D₄₇

yes

characteristic

126

C₂×M₁₆

C₂×Dic₄₇

yes

—

127

C₂²×C₈

C₄×D₄₇

yes

—

128

C₂²×C₈

C₂²×D₄₇

yes

characteristic

129

#117

—

130

M₃₂

—

131

M₃₂

—

132

M₃₂

—

133

M₃₂

C₂×Dic₄₇

yes

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

#101

—

141

#100

—

142

#99

—

143

#97

—

144

C₄×C₈

—

145

#41

—

146

#41

—

147

C₂×Q₈

#10625324

yes

characteristic

148

C₂×D₄

#10625324

yes

—

149

C₂×D₄

#10625324

yes

characteristic

150

C₂²×C₄

C₄×Dic₄₇

yes

—

151

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₄×Dic₄₇

yes

—

160

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

161

C₂×C₈

D₄×D₄₇

yes

—

162

C₂×C₈

#10625352

yes

—

163

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

164

C₄⋊C₄

—

165

C₂²⋊C₄

—

166

C₄²

—

167

C₁₆

—

168

C₁₆

—

169

C₁₆

—

170

C₁₆

—

171

C₂³

#13093934

yes

—

172

C₂³

C₂×#10625324

yes

characteristic

173

Q₈

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

188

—

179

C₂×C₄

#13093934

yes

—

180

C₂×C₄

C₂×C₄×Dic₄₇

yes

characteristic

181

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

182

C₂×C₄

C₂×#10625324

yes

characteristic

183

C₂×C₄

#13094032

yes

—

184

C₈

—

185

C₈

—

188

—

186

C₈

C₂²⋊C₄×D₄₇

yes

characteristic

187

C₈

#13093980

yes

characteristic

188

C₈

D₄×Dic₄₇

yes

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

C₂×#13093934

yes

characteristic

193

C₂²

#15689191

yes

—

194

C₄

—

195

C₄

—

376

—

196

C₄

C₂²⋊C₄×Dic₄₇

yes

characteristic

197

C₄

#15689081

yes

characteristic

198

C₄

#15689191

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

202

C₂

#20243167

yes

characteristic

	subgroup	G/H
center	C₈	#13093980
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2115	C₂
socle	C₂×C₄₇	#1752
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2115	D₄₇
hypocenter	C₄₇	#12617
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2115

subgroup

G/H

center

C₈

#13093980

commutator

C₂×C₄×C₄₇

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#2115

C₂

socle

C₂×C₄₇

#1752

prime radical

C₂×#41×C₄₇

C₂×C₄

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2115

N	H ≅ G/N	type
2. #20244992	C₂	semidirect
4. #20241478	C₂	semidirect
7. C₂×#13093964	C₄	semidirect
9. C₂²×C₄₇⋊C₁₆	C₄	semidirect
18. C₄₇×#349	C₄	semidirect
117. #349	Dic₄₇	semidirect
120. C₄₇	#12617	semidirect

H ≅ G/N

type

2. #20244992

C₂

semidirect

4. #20241478

C₂

semidirect

7. C₂×#13093964

C₄

semidirect

9. C₂²×C₄₇⋊C₁₆

C₄

semidirect

18. C₄₇×#349

C₄

semidirect

117. #349

Dic₄₇

semidirect

120. C₄₇

#12617

semidirect

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

subgroup	index
C₂×#15689540	2
#20244992	2
#20243188	2
#20241478	2
C₄₇×#2115	2
#12617	47

subgroup

index

C₄₇×#2115

Subgroups — #26131401

Conjugacy classes of subgroups

Complements

Maximal subgroups