Group #25001348 — subgroups

Subgroups — #25001348

Order 11008.

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	#19135531	C₂	1	1	yes	maximal, characteristic
2	#19134723	C₂	1	1	yes	maximal, characteristic
3	#19134601	C₂	1	2	yes	maximal
4	#19132479	C₂	1	2	yes	maximal
5	C₄₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#13063250	—	2	2	no	—
9	#14582769	C₂²	1	2	yes	—
10	#14582662	—	2	2	no	—
11	#14582662	—	2	2	no	—
12	#14582653	—	2	2	no	—
13	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₈	—	2	1	no	—
17	C₄×Dic₈₆	—	2	2	no	—
18	C₄×Dic₈₆	—	2	2	no	—
19	C₂×C₄⋊Dic₄₃	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	C₂×C₄⋊Dic₄₃	—	2	2	no	—
22	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
23	C₂×Dic₄₃⋊C₄	—	2	1	no	—
24	C₂×C₄×Dic₄₃	—	2	1	no	—
25	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
26	#13063250	—	4	4	no	—
27	C₂²×C₈×C₄₃	—	2	2	no	—
28	C₂²×C₈×C₄₃	D₄	1	2	yes	—
29	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₃	—	2	2	no	—
31	C₄⋊C₈×C₄₃	—	2	2	no	—
32	C₄×C₈×C₄₃	—	2	2	no	—
33	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
36	C₂²×Dic₄₃	—	2	1	no	—
37	C₂²×Dic₄₃	—	2	1	no	—
38	C₂²×Dic₄₃	—	4	2	no	—
39	C₂×Dic₈₆	—	2	2	no	—
40	C₂×Dic₈₆	—	2	2	no	—
41	C₂×Dic₈₆	—	2	1	no	—
42	C₂×Dic₈₆	Q₁₆	1	2	yes	—
43	C₂×C₈×C₄₃	—	2	2	no	—
44	C₂×C₈×C₄₃	—	2	2	no	—
45	C₂×C₈×C₄₃	—	2	2	no	—
46	C₂×C₈×C₄₃	—	4	2	no	—
47	C₂×C₈×C₄₃	—	4	2	no	—
48	C₂×C₈×C₄₃	—	4	2	no	—
49	C₄²×C₄₃	#41	1	2	yes	—
50	C₄²×C₄₃	#41	1	2	yes	—
51	C₄⋊Dic₄₃	—	2	2	no	—
52	C₄⋊Dic₄₃	—	4	2	no	—
53	C₄⋊Dic₄₃	—	4	2	no	—
54	C₄⋊Dic₄₃	—	4	2	no	—
55	C₄⋊Dic₄₃	—	4	2	no	—
56	C₄⋊Dic₄₃	—	8	2	no	—
57	Dic₄₃⋊C₄	—	4	2	no	—
58	C₄×Dic₄₃	—	4	1	no	—
59	C₄×Dic₄₃	—	4	1	no	—
60	C₂×C₄₃⋊C₈	—	4	1	no	—
61	C₂×C₄₃⋊C₈	—	4	2	no	—
62	C₂³×C₄₃	#121	1	1	yes	characteristic
63	C₂×C₄×C₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	#123	1	2	yes	—
65	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
67	C₂×C₄×C₄₃	#126	1	2	yes	—
68	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
70	C₂×C₄×C₄₃	#135	1	2	yes	—
71	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
72	C₂×C₄×C₄₃	#137	1	2	yes	—
73	C₂×Dic₄₃	—	2	2	no	—
74	C₂×Dic₄₃	—	4	1	no	—
75	C₂×Dic₄₃	—	4	1	no	—
76	C₂×Dic₄₃	—	4	1	no	—
77	C₂×Dic₄₃	—	4	2	no	—
78	C₂×Dic₄₃	—	8	2	no	—
79	C₂×Dic₄₃	—	8	2	no	—
80	C₂×Dic₄₃	—	8	2	no	—
81	Dic₈₆	—	2	4	no	—
82	Dic₈₆	—	4	2	no	—
83	C₈×C₄₃	—	4	2	no	—
84	C₈×C₄₃	—	8	2	no	—
85	C₄₃⋊C₈	—	8	1	no	—
86	#10624	—	43	1	no	maximal, Sylow
87	C₂²×C₄₃	#397	1	1	yes	characteristic
88	C₂²×C₄₃	#464	1	2	yes	—
89	C₂²×C₄₃	#467	1	1	yes	characteristic
90	C₂²×C₄₃	#469	1	1	yes	characteristic
91	C₂²×C₄₃	#471	1	2	yes	—
92	C₄×C₄₃	—	2	2	no	—
93	C₄×C₄₃	#482	1	2	yes	—
94	C₄×C₄₃	#484	1	2	yes	—
95	C₄×C₄₃	#485	1	2	yes	—
96	C₄×C₄₃	#488	1	1	yes	characteristic
97	C₄×C₄₃	#491	1	1	yes	characteristic
98	Dic₄₃	—	4	2	no	—
99	Dic₄₃	—	8	1	no	—
100	Dic₄₃	—	16	2	no	—
101	#1984	—	43	1	no	—
102	#1834	—	43	1	no	—
103	#1803	—	43	2	no	—
104	#1296	—	43	2	no	—
105	#1277	D₄₃	1	1	yes	characteristic
106	C₂×C₄₃	#1684	1	2	yes	—
107	C₂×C₄₃	#1702	1	2	yes	—
108	C₂×C₄₃	#2074	1	1	yes	characteristic
109	C₂×C₄₃	#2076	1	1	yes	characteristic
110	C₂×C₄₃	#2079	1	1	yes	characteristic
111	C₂×C₄×Q₈	—	43	1	no	—
112	C₂×C₄⋊C₈	—	43	1	no	—
113	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
114	C₂×#103	—	86	2	no	—
115	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
116	#384	—	43	2	no	—
117	#340	—	86	2	no	—
118	#340	—	86	2	no	—
119	#336	—	86	2	no	—
120	C₄₃	#10624	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	43	1	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	86	1	no	—
124	C₂²×C₈	D₁₇₂	1	2	yes	—
125	C₄×Q₈	—	86	2	no	—
126	C₄×Q₈	—	86	2	no	—
127	C₂×C₄⋊C₄	—	43	1	no	—
128	C₂×C₄⋊C₄	—	86	2	no	—
129	C₂×C₄⋊C₄	—	86	1	no	—
130	C₂×C₄⋊C₄	—	86	2	no	—
131	C₂×C₄⋊C₄	—	86	2	no	—
132	C₂×C₄²	—	86	1	no	—
133	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	86	2	no	—
137	#103	—	172	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	43	2	no	—
140	C₂×Q₈	—	86	1	no	—
141	C₂×Q₈	—	86	2	no	—
142	C₂×Q₈	—	86	2	no	—
143	C₂²×C₄	—	86	1	no	—
144	C₂²×C₄	—	86	1	no	—
145	C₂²×C₄	—	172	2	no	—
146	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
147	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
148	C₂²×C₄	#10621902	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	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	172	1	no	—
156	C₂×C₈	—	172	2	no	—
157	C₄⋊C₄	—	86	2	no	—
158	C₄⋊C₄	—	344	2	no	—
159	C₄⋊C₄	—	172	2	no	—
160	C₄⋊C₄	—	172	2	no	—
161	C₄⋊C₄	—	172	2	no	—
162	C₄⋊C₄	—	172	2	no	—
163	C₄⋊C₄	—	172	2	no	—
164	C₄²	—	172	1	no	—
165	C₄²	—	172	1	no	—
166	C₄²	#10621898	1	2	yes	—
167	C₄²	#10621900	1	2	yes	—
168	C₂³	#13063300	1	1	yes	characteristic
169	Q₈	—	86	4	no	—
170	Q₈	—	172	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	86	2	no	—
173	C₂×C₄	—	344	2	no	—
174	C₂×C₄	—	344	2	no	—
175	C₂×C₄	—	344	2	no	—
176	C₂×C₄	—	172	1	no	—
177	C₂×C₄	—	172	2	no	—
178	C₂×C₄	—	172	1	no	—
179	C₂×C₄	—	172	1	no	—
180	C₂×C₄	#13063307	1	2	yes	—
181	C₂×C₄	#13063309	1	1	yes	characteristic
182	C₂×C₄	#13063313	1	1	yes	characteristic
183	C₂×C₄	#13063320	1	2	yes	—
184	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
185	C₂×C₄	#13063344	1	1	yes	characteristic
186	C₂×C₄	#13063347	1	2	yes	—
187	C₂×C₄	#13063352	1	2	yes	—
188	C₂×C₄	#13063358	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	2	no	—
191	C₈	—	344	1	no	—
192	C₂²	#14582826	1	1	yes	characteristic
193	C₂²	#14583122	1	2	yes	—
194	C₂²	#14583141	1	1	yes	characteristic
195	C₂²	#14583153	1	1	yes	characteristic
196	C₂²	#14583163	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	344	1	no	—
199	C₄	—	172	2	no	—
200	C₄	—	688	2	no	—
201	C₄	#14583221	1	2	yes	—
202	C₄	#14583233	1	2	yes	—
203	C₄	#14583241	1	2	yes	—
204	C₄	#14583254	1	1	yes	characteristic
205	C₄	#14583268	1	1	yes	characteristic
206	C₂	#19134054	1	2	yes	—
207	C₂	#19134164	1	2	yes	—
208	C₂	#19136015	1	1	yes	characteristic
209	C₂	#19136025	1	1	yes	characteristic
210	C₂	#19136042	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19135531

C₂

yes

maximal, characteristic

#19134723

C₂

yes

maximal, characteristic

#19134601

C₂

yes

maximal

#19132479

C₂

yes

maximal

C₄₃×#1277

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₆

C₂²

yes

characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₂×#13063250

—

#14582769

C₂²

yes

—

#14582662

—

#14582662

—

#14582653

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

D₄

yes

characteristic

C₂²×C₄₃⋊C₈

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

D₄

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

—

#13063250

—

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

Q₁₆

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

#41

yes

—

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

#482

yes

—

C₄×C₄₃

#484

yes

—

C₄×C₄₃

#485

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

100

Dic₄₃

—

101

#1984

—

102

#1834

—

103

#1803

—

104

#1296

—

105

#1277

D₄₃

yes

characteristic

106

C₂×C₄₃

#1684

yes

—

107

C₂×C₄₃

#1702

yes

—

108

C₂×C₄₃

#2074

yes

characteristic

109

C₂×C₄₃

#2076

yes

characteristic

110

C₂×C₄₃

#2079

yes

characteristic

111

C₂×C₄×Q₈

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

C₂×D₄₃

yes

—

114

C₂×#103

—

115

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

116

#384

—

117

#340

—

118

#340

—

119

#336

—

120

C₄₃

#10624

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

—

124

C₂²×C₈

D₁₇₂

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×D₄₃

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#103

—

172

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

172

—

146

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

147

C₂²×C₄

D₄×D₄₃

yes

characteristic

148

C₂²×C₄

#10621902

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

172

—

156

C₂×C₈

—

172

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

344

—

159

C₄⋊C₄

—

172

—

160

C₄⋊C₄

—

172

—

161

C₄⋊C₄

—

172

—

162

C₄⋊C₄

—

172

—

163

C₄⋊C₄

—

172

—

164

C₄²

—

172

—

165

C₄²

—

172

—

166

C₄²

#10621898

yes

—

167

C₄²

#10621900

yes

—

168

C₂³

#13063300

yes

characteristic

169

Q₈

—

170

Q₈

—

172

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

344

—

174

C₂×C₄

—

344

—

175

C₂×C₄

—

344

—

176

C₂×C₄

—

172

—

177

C₂×C₄

—

172

—

178

C₂×C₄

—

172

—

179

C₂×C₄

—

172

—

180

C₂×C₄

#13063307

yes

—

181

C₂×C₄

#13063309

yes

characteristic

182

C₂×C₄

#13063313

yes

characteristic

183

C₂×C₄

#13063320

yes

—

184

C₂×C₄

Q₁₆×D₄₃

yes

characteristic

185

C₂×C₄

#13063344

yes

characteristic

186

C₂×C₄

#13063347

yes

—

187

C₂×C₄

#13063352

yes

—

188

C₂×C₄

#13063358

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

344

—

192

C₂²

#14582826

yes

characteristic

193

C₂²

#14583122

yes

—

194

C₂²

#14583141

yes

characteristic

195

C₂²

#14583153

yes

characteristic

196

C₂²

#14583163

yes

—

197

C₄

—

198

C₄

—

344

—

199

C₄

—

172

—

200

C₄

—

688

—

201

C₄

#14583221

yes

—

202

C₄

#14583233

yes

—

203

C₄

#14583241

yes

—

204

C₄

#14583254

yes

characteristic

205

C₄

#14583268

yes

characteristic

206

C₂

#19134054

yes

—

207

C₂

#19134164

yes

—

208

C₂

#19136015

yes

characteristic

209

C₂

#19136025

yes

characteristic

210

C₂

#19136042

yes

characteristic

	subgroup	G/H
center	C₂³	#13063300
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1277	C₂
socle, prime radical	C₂³×C₄₃	#121
hypercenter	#1277	D₄₃
hypocenter	C₄₃	#10624
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂³

#13063300

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1277

C₂

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
120. C₄₃	#10624	semidirect

H ≅ G/N

type

120. C₄₃

#10624

semidirect

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

subgroup	index
#19135531	2
#19134723	2
#19134601	2
#19132479	2
C₄₃×#1277	2
#10624	43

subgroup

index

C₄₃×#1277

Subgroups — #25001348

Conjugacy classes of subgroups

Complements

Maximal subgroups