Group #26136136 — subgroups

Subgroups — #26136136

Order 12032.

342 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 105 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×#15688949	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₂×D₄₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20243414	C₂	1	2	yes	maximal
4	#20241479	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1380	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#13093958	C₂²	1	2	yes	—
9	#15688980	—	2	2	no	—
10	#15688949	—	2	2	no	—
11	(C₂×D₄₇)⋊C₁₆	—	2	2	no	—
12	#15688885	—	2	2	no	—
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₈×D₄₇	—	2	1	no	—
18	C₂×C₈×D₄₇	—	2	2	no	—
19	C₂×C₈×D₄₇	—	2	2	no	—
20	C₂×C₈×D₄₇	—	2	1	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×(C₂×D₄₇)⋊C₄	—	4	2	no	—
23	C₂×C₄₇⋊C₁₆	—	2	1	no	—
24	C₂×C₄₇⋊C₁₆	—	2	2	no	—
25	#13093958	—	2	2	no	—
26	#13093958	—	2	2	no	—
27	C₄₇⋊#95	—	4	2	no	—
28	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₁₆×C₄₇	—	2	2	no	—
31	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
32	#107×C₄₇	—	2	2	no	—
33	#107×C₄₇	—	2	2	no	—
34	C₂³×D₄₇	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	—	4	2	no	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₇	—	4	2	no	—
38	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
39	C₂×C₄×D₄₇	—	2	1	no	—
40	C₂×C₄×D₄₇	—	2	2	no	—
41	C₂×C₄×D₄₇	—	2	1	no	—
42	C₂×C₄×D₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	#41	1	2	yes	—
45	C₂×C₈×C₄₇	#41	1	2	yes	—
46	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₄₇	—	4	2	no	—
48	C₄⋊C₄×C₄₇	—	4	2	no	—
49	(C₂×D₄₇)⋊C₄	—	8	4	no	—
50	C₄⋊Dic₄₇	—	4	2	no	—
51	C₄⋊Dic₄₇	—	4	2	no	—
52	C₂×C₄₇⋊C₈	—	2	1	no	—
53	C₂×C₄₇⋊C₈	—	2	2	no	—
54	C₈×D₄₇	—	4	1	no	—
55	C₈×D₄₇	—	4	1	no	—
56	C₈×D₄₇	—	4	2	no	—
57	C₁₆×C₄₇	—	2	2	no	—
58	C₄₇⋊C₁₆	—	4	1	no	—
59	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
60	C₂²×D₄₇	—	2	1	no	—
61	C₂²×D₄₇	—	2	2	no	—
62	C₂²×D₄₇	—	2	1	no	—
63	C₂²×D₄₇	—	2	2	no	—
64	C₂²×D₄₇	D₁₆	1	2	yes	—
65	C₂×C₄×C₄₇	—	8	2	no	—
66	C₂×C₄×C₄₇	—	8	2	no	—
67	C₂×C₄×C₄₇	—	8	2	no	—
68	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#123	1	2	yes	—
70	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#137	1	2	yes	—
72	C₂×Dic₄₇	—	2	1	no	—
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	—	8	2	no	—
75	C₂×Dic₄₇	—	8	2	no	—
76	C₂×Dic₄₇	—	8	2	no	—
77	C₄×D₄₇	—	4	1	no	—
78	C₄×D₄₇	—	4	1	no	—
79	C₄×D₄₇	—	4	2	no	—
80	C₈×C₄₇	#123	1	1	yes	characteristic
81	C₈×C₄₇	#124	1	1	yes	characteristic
82	C₈×C₄₇	#126	1	2	yes	—
83	C₄₇⋊C₈	—	4	1	no	—
84	#13492	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	#466	1	1	yes	characteristic
86	C₂²×C₄₇	#480	1	2	yes	—
87	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
88	C₂²×C₄₇	#508	1	1	yes	characteristic
89	C₂²×C₄₇	#510	1	2	yes	—
90	C₂×D₄₇	—	2	2	no	—
91	C₂×D₄₇	—	2	4	no	—
92	C₂×D₄₇	—	4	1	no	—
93	C₂×D₄₇	—	4	1	no	—
94	C₂×D₄₇	—	4	2	no	—
95	C₄×C₄₇	—	16	2	no	—
96	C₄×C₄₇	#397	1	1	yes	characteristic
97	C₄×C₄₇	#470	1	1	yes	characteristic
98	C₄×C₄₇	#485	1	2	yes	—
99	Dic₄₇	—	4	1	no	—
100	Dic₄₇	—	16	2	no	—
101	C₂×#366	—	47	1	no	—
102	C₂×C₂²⋊C₁₆	—	47	1	no	—
103	#1808	—	47	2	no	—
104	#1380	—	47	2	no	—
105	#1380	D₄₇	1	1	yes	characteristic, complemented
106	C₂×C₄₇	#2075	1	1	yes	characteristic
107	C₂×C₄₇	#2215	1	1	yes	characteristic
108	C₂×C₄₇	#2219	1	1	yes	characteristic
109	C₂×C₄₇	#2232	1	2	yes	—
110	C₂×C₄₇	#2237	1	2	yes	—
111	D₄₇	—	4	2	no	—
112	C₂³×C₈	—	47	1	no	—
113	C₂²×C₁₆	—	47	1	no	—
114	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
115	C₂×#107	—	47	2	no	—
116	C₂×#107	C₂×D₄₇	1	2	yes	—
117	#380	—	94	2	no	—
118	#366	—	94	2	no	—
119	C₂²⋊C₁₆	—	94	2	no	—
120	#340	—	94	2	no	—
121	C₄₇	#13492	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	—	47	1	no	—
123	C₂²×C₈	—	47	1	no	—
124	C₂²×C₈	—	94	1	no	—
125	C₂²×C₈	—	94	2	no	—
126	C₂²×C₈	—	94	2	no	—
127	C₂²×C₈	—	94	1	no	—
128	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	94	2	no	—
131	C₂×C₂²⋊C₄	—	188	2	no	—
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	94	1	no	—
134	C₂×C₁₆	—	94	2	no	—
135	C₂×C₁₆	D₁₈₈	1	2	yes	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	—	94	2	no	—
139	#107	—	94	2	no	—
140	#95	—	188	2	no	—
141	C₂⁴	—	47	1	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	47	1	no	—
144	C₂²×C₄	—	94	1	no	—
145	C₂²×C₄	—	94	2	no	—
146	C₂²×C₄	—	94	1	no	—
147	C₂²×C₄	—	94	2	no	—
148	C₂²×C₄	—	188	2	no	—
149	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
150	C₂×C₈	—	94	1	no	—
151	C₂×C₈	—	94	2	no	—
152	C₂×C₈	—	188	1	no	—
153	C₂×C₈	—	188	1	no	—
154	C₂×C₈	—	188	2	no	—
155	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
156	C₂×C₈	#10625350	1	2	yes	—
157	C₂×C₈	#10625352	1	2	yes	—
158	C₂×C₈	#10625354	1	1	yes	characteristic
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	188	2	no	—
162	C₄⋊C₄	—	188	2	no	—
163	C₂²⋊C₄	—	376	4	no	—
164	C₁₆	—	2	2	no	—
165	C₁₆	—	188	1	no	—
166	C₂³	—	47	2	no	—
167	C₂³	—	94	1	no	—
168	C₂³	—	94	2	no	—
169	C₂³	—	94	1	no	—
170	C₂³	—	94	2	no	—
171	C₂³	D₈×D₄₇	1	1	yes	characteristic
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	94	1	no	—
176	C₂×C₄	—	94	2	no	—
177	C₂×C₄	—	376	2	no	—
178	C₂×C₄	—	376	2	no	—
179	C₂×C₄	—	376	2	no	—
180	C₂×C₄	—	188	1	no	—
181	C₂×C₄	—	188	1	no	—
182	C₂×C₄	—	188	2	no	—
183	C₂×C₄	#13093999	1	1	yes	characteristic
184	C₂×C₄	#13094006	1	2	yes	—
185	C₂×C₄	#13094047	1	1	yes	characteristic
186	C₂×C₄	#13094053	1	2	yes	—
187	C₈	—	188	1	no	—
188	C₈	#13094008	1	1	yes	characteristic
189	C₈	#13094012	1	1	yes	characteristic
190	C₈	#13094019	1	2	yes	—
191	C₂²	—	94	4	no	—
192	C₂²	—	94	2	no	—
193	C₂²	—	188	1	no	—
194	C₂²	—	188	2	no	—
195	C₂²	—	188	1	no	—
196	C₂²	#15689360	1	1	yes	characteristic
197	C₂²	#15689435	1	2	yes	—
198	C₂²	D₁₆×D₄₇	1	1	yes	characteristic
199	C₂²	#15689556	1	1	yes	characteristic
200	C₂²	#15689562	1	2	yes	—
201	C₄	—	16	2	no	—
202	C₄	—	188	1	no	—
203	C₄	—	752	2	no	—
204	C₄	#15689050	1	1	yes	characteristic
205	C₄	#15689382	1	1	yes	characteristic
206	C₄	#15689463	1	2	yes	—
207	C₂	—	188	2	no	—
208	C₂	#20244813	1	1	yes	characteristic
209	C₂	#20245475	1	1	yes	characteristic
210	C₂	#20245497	1	1	yes	characteristic
211	C₂	#20245567	1	2	yes	—
212	C₂	#20245595	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688949

C₂

yes

maximal, characteristic, complemented

C₂×(C₂×D₄₇)⋊C₁₆

C₂

yes

maximal, characteristic

#20243414

C₂

yes

maximal

#20241479

C₂

yes

maximal, complemented

C₄₇×#1380

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×#13093958

C₂²

yes

—

#15688980

—

#15688949

—

(C₂×D₄₇)⋊C₁₆

—

#15688885

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#107×C₄₇

C₂²

yes

—

C₂²×C₄×D₄₇

D₄

yes

characteristic

C₂²×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₈×D₄₇

—

C₂×C₈×D₄₇

—

C₂×C₈×D₄₇

—

C₂×C₈×D₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×(C₂×D₄₇)⋊C₄

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

#13093958

—

#13093958

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

#107×C₄₇

—

#107×C₄₇

—

C₂³×D₄₇

D₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₈

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₈×D₄₇

—

C₈×D₄₇

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

D₁₆

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×D₁₆

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

100

Dic₄₇

—

101

C₂×#366

—

102

C₂×C₂²⋊C₁₆

—

103

#1808

—

104

#1380

—

105

#1380

D₄₇

yes

characteristic, complemented

106

C₂×C₄₇

#2075

yes

characteristic

107

C₂×C₄₇

#2215

yes

characteristic

108

C₂×C₄₇

#2219

yes

characteristic

109

C₂×C₄₇

#2232

yes

—

110

C₂×C₄₇

#2237

yes

—

111

D₄₇

—

112

C₂³×C₈

—

113

C₂²×C₁₆

—

114

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

115

C₂×#107

—

116

C₂×#107

C₂×D₄₇

yes

—

117

#380

—

118

#366

—

119

C₂²⋊C₁₆

—

120

#340

—

121

C₄₇

#13492

yes

characteristic, complemented, Sylow

122

C₂³×C₄

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²×D₄₇

yes

characteristic

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₂²⋊C₄

—

188

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₂×C₁₆

D₁₈₈

yes

—

136

#107

—

137

#107

—

138

#107

—

139

#107

—

140

#95

—

188

—

141

C₂⁴

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

188

—

149

C₂²×C₄

D₄×D₄₇

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

188

—

153

C₂×C₈

—

188

—

154

C₂×C₈

—

188

—

155

C₂×C₈

C₂×D₁₈₈

yes

characteristic

156

C₂×C₈

#10625350

yes

—

157

C₂×C₈

#10625352

yes

—

158

C₂×C₈

#10625354

yes

characteristic

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

188

—

162

C₄⋊C₄

—

188

—

163

C₂²⋊C₄

—

376

—

164

C₁₆

—

165

C₁₆

—

188

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

D₈×D₄₇

yes

characteristic

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

376

—

178

C₂×C₄

—

376

—

179

C₂×C₄

—

376

—

180

C₂×C₄

—

188

—

181

C₂×C₄

—

188

—

182

C₂×C₄

—

188

—

183

C₂×C₄

#13093999

yes

characteristic

184

C₂×C₄

#13094006

yes

—

185

C₂×C₄

#13094047

yes

characteristic

186

C₂×C₄

#13094053

yes

—

187

C₈

—

188

—

188

C₈

#13094008

yes

characteristic

189

C₈

#13094012

yes

characteristic

190

C₈

#13094019

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

188

—

194

C₂²

—

188

—

195

C₂²

—

188

—

196

C₂²

#15689360

yes

characteristic

197

C₂²

#15689435

yes

—

198

C₂²

D₁₆×D₄₇

yes

characteristic

199

C₂²

#15689556

yes

characteristic

200

C₂²

#15689562

yes

—

201

C₄

—

202

C₄

—

188

—

203

C₄

—

752

—

204

C₄

#15689050

yes

characteristic

205

C₄

#15689382

yes

characteristic

206

C₄

#15689463

yes

—

207

C₂

—

188

—

208

C₂

#20244813

yes

characteristic

209

C₂

#20245475

yes

characteristic

210

C₂

#20245497

yes

characteristic

211

C₂

#20245567

yes

—

212

C₂

#20245595

yes

—

	subgroup	G/H
center	C₂³	D₈×D₄₇
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1380	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂³×D₄₇	D₈
hypercenter	#1380	D₄₇
hypocenter	C₄₇	#13492
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₄₇

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1380

C₂

socle

C₂³×C₄₇

C₂×D₈

prime radical

C₂³×D₄₇

D₈

D₄₇

C₄₇

G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
1. C₂×#15688949	C₂	semidirect
4. #20241479	C₂	semidirect
5. C₄₇×#1380	C₂	semidirect
105. #1380	D₄₇	semidirect
121. C₄₇	#13492	semidirect

H ≅ G/N

type

1. C₂×#15688949

C₂

semidirect

4. #20241479

C₂

semidirect

5. C₄₇×#1380

C₂

semidirect

105. #1380

D₄₇

semidirect

121. C₄₇

#13492

semidirect

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

subgroup	index
C₂×#15688949	2
C₂×(C₂×D₄₇)⋊C₁₆	2
#20243414	2
#20241479	2
C₄₇×#1380	2
#13492	47

subgroup

index

C₂×(C₂×D₄₇)⋊C₁₆

C₄₇×#1380

Subgroups — #26136136

Conjugacy classes of subgroups

Complements

Maximal subgroups