Group #26121028 — subgroups

Subgroups — #26121028

Order 12032.

350 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20244324	C₂	1	1	yes	maximal, characteristic
2	#20243390	C₂	1	2	yes	maximal
3	#20241271	C₂	1	2	yes	maximal
4	#20241223	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#13093947	—	2	2	no	—
10	#15688885	—	2	2	no	—
11	#15688877	—	2	2	no	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#384	C₂²	1	2	yes	—
14	C₄₇×#340	—	2	2	no	—
15	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	2	1	no	—
18	C₄×Dic₉₄	—	2	2	no	—
19	C₄×Dic₉₄	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
21	C₂×Dic₄₇⋊C₄	—	2	1	no	—
22	C₂×C₄×Dic₄₇	—	2	1	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	#13093947	—	4	4	no	—
27	C₂²×C₈×C₄₇	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	—	4	2	no	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
37	C₂²×Dic₄₇	—	2	1	no	—
38	C₂²×Dic₄₇	—	2	1	no	—
39	C₂×Dic₉₄	—	2	2	no	—
40	C₂×Dic₉₄	—	2	2	no	—
41	C₂×Dic₉₄	—	2	1	no	—
42	C₂×Dic₉₄	Q₁₆	1	2	yes	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₄⋊C₄×C₄₇	—	4	2	no	—
47	C₄⋊C₄×C₄₇	—	4	2	no	—
48	C₄⋊C₄×C₄₇	—	4	2	no	—
49	C₄⋊C₄×C₄₇	—	4	2	no	—
50	C₄⋊C₄×C₄₇	—	8	2	no	—
51	C₄²×C₄₇	#41	1	2	yes	—
52	C₄²×C₄₇	#41	1	2	yes	—
53	C₄⋊Dic₄₇	—	2	2	no	—
54	Dic₄₇⋊C₄	—	4	2	no	—
55	C₄×Dic₄₇	—	4	1	no	—
56	C₄×Dic₄₇	—	4	1	no	—
57	C₂×C₄₇⋊C₈	—	4	2	no	—
58	C₂×C₄₇⋊C₈	—	4	1	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂×C₄₇⋊C₈	—	4	2	no	—
61	C₂×C₄₇⋊C₈	—	4	2	no	—
62	C₂³×C₄₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	8	2	no	—
65	C₂×C₄×C₄₇	—	8	2	no	—
66	C₂×C₄×C₄₇	—	8	2	no	—
67	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#123	1	2	yes	—
69	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#126	1	2	yes	—
71	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#135	1	2	yes	—
73	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
74	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₄₇	#137	1	2	yes	—
76	C₂×Dic₄₇	—	2	2	no	—
77	C₂×Dic₄₇	—	4	1	no	—
78	C₂×Dic₄₇	—	4	1	no	—
79	C₂×Dic₄₇	—	4	1	no	—
80	C₂×Dic₄₇	—	4	2	no	—
81	Dic₉₄	—	2	4	no	—
82	Dic₉₄	—	4	2	no	—
83	C₈×C₄₇	—	4	2	no	—
84	C₄₇⋊C₈	—	8	2	no	—
85	C₄₇⋊C₈	—	8	1	no	—
86	#10624	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	#397	1	1	yes	characteristic
88	C₂²×C₄₇	#464	1	2	yes	—
89	C₂²×C₄₇	#467	1	1	yes	characteristic
90	C₂²×C₄₇	#469	1	1	yes	characteristic
91	C₂²×C₄₇	#471	1	2	yes	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	16	2	no	—
94	C₄×C₄₇	#482	1	2	yes	—
95	C₄×C₄₇	#484	1	2	yes	—
96	C₄×C₄₇	#485	1	2	yes	—
97	C₄×C₄₇	#488	1	1	yes	characteristic
98	C₄×C₄₇	#491	1	1	yes	characteristic
99	Dic₄₇	—	4	2	no	—
100	Dic₄₇	—	8	1	no	—
101	#1984	—	47	1	no	—
102	#1834	D₄₇	1	1	yes	characteristic
103	#1803	—	47	2	no	—
104	#1296	—	47	2	no	—
105	#1277	—	47	1	no	—
106	C₂×C₄₇	#1684	1	2	yes	—
107	C₂×C₄₇	#1702	1	2	yes	—
108	C₂×C₄₇	#2074	1	1	yes	characteristic
109	C₂×C₄₇	#2076	1	1	yes	characteristic
110	C₂×C₄₇	#2079	1	1	yes	characteristic
111	C₂×C₄×Q₈	—	47	1	no	—
112	C₂×C₄⋊C₈	—	47	2	no	—
113	C₂×C₄⋊C₈	—	47	1	no	—
114	C₂×#103	—	94	2	no	—
115	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
116	#384	C₂×D₄₇	1	2	yes	—
117	#340	—	2	2	no	—
118	#340	—	94	2	no	—
119	#336	—	94	2	no	—
120	C₄₇	#10624	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	47	1	no	—
122	C₂²×C₈	—	94	2	no	—
123	C₂²×C₈	—	94	1	no	—
124	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
125	C₄×Q₈	—	94	2	no	—
126	C₄×Q₈	—	94	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	47	1	no	—
131	C₂×C₄⋊C₄	—	94	1	no	—
132	C₂×C₄²	—	94	1	no	—
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	C₄⋊C₈	—	94	2	no	—
135	C₄⋊C₈	—	94	2	no	—
136	C₄⋊C₈	—	94	2	no	—
137	#103	—	188	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	47	2	no	—
140	C₂×Q₈	—	94	1	no	—
141	C₂×Q₈	—	94	2	no	—
142	C₂×Q₈	—	94	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	94	1	no	—
145	C₂²×C₄	—	94	1	no	—
146	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
147	C₂²×C₄	#10625352	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	188	2	no	—
153	C₂×C₈	—	188	1	no	—
154	C₂×C₈	—	188	2	no	—
155	C₂×C₈	—	188	2	no	—
156	C₂×C₈	—	188	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	8	2	no	—
162	C₄⋊C₄	—	94	2	no	—
163	C₄⋊C₄	—	188	2	no	—
164	C₄²	—	188	1	no	—
165	C₄²	—	188	1	no	—
166	C₄²	#10625350	1	2	yes	—
167	C₄²	#10625354	1	2	yes	—
168	C₂³	#13093995	1	1	yes	characteristic
169	Q₈	—	94	4	no	—
170	Q₈	—	188	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	94	2	no	—
176	C₂×C₄	—	188	1	no	—
177	C₂×C₄	—	188	2	no	—
178	C₂×C₄	—	188	1	no	—
179	C₂×C₄	—	188	1	no	—
180	C₂×C₄	#13094007	1	1	yes	characteristic
181	C₂×C₄	#13094009	1	2	yes	—
182	C₂×C₄	#13094013	1	1	yes	characteristic
183	C₂×C₄	#13094019	1	2	yes	—
184	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
185	C₂×C₄	#13094044	1	2	yes	—
186	C₂×C₄	#13094047	1	1	yes	characteristic
187	C₂×C₄	#13094048	1	2	yes	—
188	C₂×C₄	#13094054	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	376	1	no	—
191	C₈	—	376	2	no	—
192	C₂²	#15689049	1	1	yes	characteristic
193	C₂²	#15689342	1	2	yes	—
194	C₂²	#15689361	1	1	yes	characteristic
195	C₂²	#15689373	1	1	yes	characteristic
196	C₂²	#15689383	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	16	2	no	—
199	C₄	—	376	1	no	—
200	C₄	—	188	2	no	—
201	C₄	#15689450	1	2	yes	—
202	C₄	#15689460	1	2	yes	—
203	C₄	#15689464	1	2	yes	—
204	C₄	#15689480	1	1	yes	characteristic
205	C₄	#15689493	1	1	yes	characteristic
206	C₂	#20242843	1	2	yes	—
207	C₂	#20242953	1	2	yes	—
208	C₂	#20244807	1	1	yes	characteristic
209	C₂	#20244817	1	1	yes	characteristic
210	C₂	#20244834	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20244324

C₂

yes

maximal, characteristic

#20243390

C₂

yes

maximal

#20241271

C₂

yes

maximal

#20241223

C₂

yes

maximal, characteristic

C₄₇×#1834

C₂

yes

maximal, characteristic

C₂×C₄×Dic₉₄

C₂²

yes

characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

—

C₄₇×#340

—

C₂²×Dic₉₄

D₄

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

C₂×C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

#13093947

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

Q₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

#41

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#123

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#482

yes

—

C₄×C₄₇

#484

yes

—

C₄×C₄₇

#485

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

100

Dic₄₇

—

101

#1984

—

102

#1834

D₄₇

yes

characteristic

103

#1803

—

104

#1296

—

105

#1277

—

106

C₂×C₄₇

#1684

yes

—

107

C₂×C₄₇

#1702

yes

—

108

C₂×C₄₇

#2074

yes

characteristic

109

C₂×C₄₇

#2076

yes

characteristic

110

C₂×C₄₇

#2079

yes

characteristic

111

C₂×C₄×Q₈

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

—

114

C₂×#103

—

115

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

116

#384

C₂×D₄₇

yes

—

117

#340

—

118

#340

—

119

#336

—

120

C₄₇

#10624

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

—

124

C₂²×C₈

C₂²⋊D₄₇

yes

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄²

—

133

C₂×C₄²

C₂²×D₄₇

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#103

—

188

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

D₄×D₄₇

yes

characteristic

147

C₂²×C₄

#10625352

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

188

—

153

C₂×C₈

—

188

—

154

C₂×C₈

—

188

—

155

C₂×C₈

—

188

—

156

C₂×C₈

—

188

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

188

—

164

C₄²

—

188

—

165

C₄²

—

188

—

166

C₄²

#10625350

yes

—

167

C₄²

#10625354

yes

—

168

C₂³

#13093995

yes

characteristic

169

Q₈

—

170

Q₈

—

188

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

188

—

177

C₂×C₄

—

188

—

178

C₂×C₄

—

188

—

179

C₂×C₄

—

188

—

180

C₂×C₄

#13094007

yes

characteristic

181

C₂×C₄

#13094009

yes

—

182

C₂×C₄

#13094013

yes

characteristic

183

C₂×C₄

#13094019

yes

—

184

C₂×C₄

Q₁₆×D₄₇

yes

characteristic

185

C₂×C₄

#13094044

yes

—

186

C₂×C₄

#13094047

yes

characteristic

187

C₂×C₄

#13094048

yes

—

188

C₂×C₄

#13094054

yes

—

189

C₈

—

190

C₈

—

376

—

191

C₈

—

376

—

192

C₂²

#15689049

yes

characteristic

193

C₂²

#15689342

yes

—

194

C₂²

#15689361

yes

characteristic

195

C₂²

#15689373

yes

characteristic

196

C₂²

#15689383

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

376

—

200

C₄

—

188

—

201

C₄

#15689450

yes

—

202

C₄

#15689460

yes

—

203

C₄

#15689464

yes

—

204

C₄

#15689480

yes

characteristic

205

C₄

#15689493

yes

characteristic

206

C₂

#20242843

yes

—

207

C₂

#20242953

yes

—

208

C₂

#20244807

yes

characteristic

209

C₂

#20244817

yes

characteristic

210

C₂

#20244834

yes

characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1834	C₂
socle, prime radical	C₂³×C₄₇	#121
hypercenter	#1834	D₄₇
hypocenter	C₄₇	#10624
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#13093995

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1834

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
120. C₄₇	#10624	semidirect

H ≅ G/N

type

120. C₄₇

#10624

semidirect

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

subgroup	index
#20244324	2
#20243390	2
#20241271	2
#20241223	2
C₄₇×#1834	2
#10624	47

subgroup

index

C₄₇×#1834

Subgroups — #26121028

Conjugacy classes of subgroups

Complements

Maximal subgroups