Group #25001366 — subgroups

Subgroups — #25001366

Order 11008.

330 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19135427	C₂	1	1	yes	maximal, characteristic
2	#19133245	C₂	1	2	yes	maximal
3	#19132492	C₂	1	2	yes	maximal, complemented
4	#19132424	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583449	C₂²	1	1	yes	characteristic
7	#14582952	C₂²	1	2	yes	—
8	#14582923	C₂²	1	1	yes	characteristic
9	C₂×#13063253	—	2	1	no	—
10	C₂×#13063253	—	2	1	no	—
11	#14582646	—	2	2	no	—
12	#14582642	—	2	2	no	—
13	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
14	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
15	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
16	C₄₃×#376	C₂²	1	1	yes	characteristic
17	C₂×#10621898	—	2	1	no	—
18	C₂×#10621877	—	2	1	no	—
19	#13063325	D₄	1	1	yes	characteristic
20	#13063322	D₄	1	1	yes	characteristic
21	#13063313	—	2	1	no	—
22	#13063299	—	2	2	no	—
23	C₄×D₁₇₂	—	2	2	no	—
24	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
26	#13063253	—	4	2	no	—
27	#13063253	—	4	2	no	—
28	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
29	C₄×C₄₃⋊C₈	—	2	2	no	—
30	C₂²×C₈×C₄₃	—	2	2	no	—
31	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₄₃	—	2	2	no	—
35	C₄⋊C₈×C₄₃	—	2	2	no	—
36	C₄⋊C₈×C₄₃	—	4	2	no	—
37	C₄⋊C₈×C₄₃	—	4	2	no	—
38	#95×C₄₃	—	2	2	no	—
39	C₂²×C₄×C₄₃	—	2	1	no	—
40	C₂²×C₄×C₄₃	—	2	1	no	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	#41	1	2	yes	—
43	C₂×C₂²⋊D₄₃	—	4	1	no	—
44	#10621898	—	4	1	no	—
45	#10621898	—	4	1	no	—
46	#10621898	—	4	1	no	—
47	#10621898	—	4	1	no	—
48	C₂×D₁₇₂	—	2	1	no	—
49	C₂×D₁₇₂	—	2	2	no	—
50	C₂×C₄×D₄₃	—	4	1	no	—
51	C₂×Dic₈₆	—	2	1	no	—
52	C₂×C₈×C₄₃	—	4	2	no	—
53	C₂×C₈×C₄₃	—	4	2	no	—
54	C₂×C₈×C₄₃	—	4	2	no	—
55	C₄²×C₄₃	—	2	1	no	—
56	C₄²×C₄₃	—	2	1	no	—
57	C₄²×C₄₃	D₈	1	2	yes	—
58	C₄²×C₄₃	SD₁₆	1	2	yes	—
59	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₃	#41	1	1	yes	characteristic
61	C₄²×C₄₃	#41	1	2	yes	—
62	(C₂×D₄₃)⋊C₄	—	4	2	no	—
63	C₄⋊Dic₄₃	—	2	2	no	—
64	#10621877	—	4	1	no	—
65	#10621877	—	4	1	no	—
66	C₂×C₄₃⋊C₈	—	4	2	no	—
67	C₂×C₄₃⋊C₈	—	4	1	no	—
68	C₂³×C₄₃	#123	1	1	yes	characteristic
69	C₂²×D₄₃	—	4	1	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	—	4	2	no	—
74	C₂×C₄×C₄₃	—	4	2	no	—
75	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#123	1	2	yes	—
77	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#126	1	2	yes	—
79	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
83	C₂²⋊D₄₃	—	8	1	no	—
84	C₂²⋊D₄₃	—	8	1	no	—
85	C₂×Dic₄₃	—	4	1	no	—
86	D₁₇₂	—	4	1	no	—
87	D₁₇₂	—	4	1	no	—
88	D₁₇₂	—	8	2	no	—
89	C₄×D₄₃	—	8	1	no	—
90	C₄×D₄₃	—	8	1	no	—
91	Dic₈₆	—	4	1	no	—
92	Dic₈₆	—	4	1	no	—
93	C₈×C₄₃	—	8	2	no	—
94	C₄₃⋊C₈	—	8	2	no	—
95	C₄₃⋊C₈	—	8	1	no	—
96	#10627	—	43	1	no	maximal, Sylow
97	C₂²×C₄₃	—	2	2	no	—
98	C₂²×C₄₃	#397	1	1	yes	characteristic
99	C₂²×C₄₃	#480	1	1	yes	characteristic
100	C₂²×C₄₃	#481	1	1	yes	characteristic
101	C₂×D₄₃	—	8	1	no	—
102	C₂×D₄₃	—	8	2	no	—
103	C₄×C₄₃	—	2	2	no	—
104	C₄×C₄₃	—	2	2	no	—
105	C₄×C₄₃	—	4	2	no	—
106	C₄×C₄₃	—	4	2	no	—
107	C₄×C₄₃	#459	1	1	yes	characteristic
108	C₄×C₄₃	#461	1	1	yes	characteristic
109	C₄×C₄₃	#486	1	1	yes	characteristic
110	C₄×C₄₃	#489	1	1	yes	characteristic
111	Dic₄₃	—	8	1	no	—
112	#1963	—	43	1	no	—
113	#1843	D₄₃	1	1	yes	characteristic, complemented
114	#1540	—	43	2	no	—
115	#1301	—	43	2	no	—
116	#1274	—	43	1	no	—
117	C₂×C₄₃	—	2	1	no	—
118	C₂×C₄₃	—	2	1	no	—
119	C₂×C₄₃	#2071	1	1	yes	characteristic
120	C₂×C₄₃	#2080	1	2	yes	—
121	D₄₃	—	16	1	no	—
122	#532	—	43	1	no	—
123	#431	—	43	2	no	—
124	#423	—	43	1	no	—
125	C₂×C₄⋊C₈	—	2	1	no	—
126	C₂×C₄⋊C₈	—	2	1	no	—
127	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
128	C₂×#104	—	86	1	no	—
129	C₂×#104	—	86	1	no	—
130	#376	C₂×D₄₃	1	1	yes	characteristic
131	#332	—	86	2	no	—
132	#331	—	86	2	no	—
133	C₄₃	#10627	1	1	yes	characteristic, complemented, Sylow
134	C₂×#41	—	86	1	no	—
135	C₂×M₁₆	—	86	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₄⋊Q₈	—	43	1	no	—
138	#127	—	43	1	no	—
139	#124	—	86	1	no	—
140	#121	—	86	2	no	—
141	C₄×D₄	—	86	2	no	—
142	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	4	2	no	—
148	C₄⋊C₈	—	4	2	no	—
149	C₄⋊C₈	—	86	2	no	—
150	C₄⋊C₈	—	86	2	no	—
151	#104	—	172	2	no	—
152	#104	—	172	2	no	—
153	C₂²⋊C₈	—	86	2	no	—
154	C₄×C₈	—	86	2	no	—
155	#95	—	2	2	no	—
156	#41	—	172	1	no	—
157	#41	—	172	1	no	—
158	#41	—	172	1	no	—
159	#41	—	172	1	no	—
160	C₂×Q₈	—	86	1	no	—
161	C₂×D₄	—	86	1	no	—
162	C₂×D₄	—	86	2	no	—
163	C₂×D₄	—	172	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	172	1	no	—
167	C₂²×C₄	#10621898	1	2	yes	—
168	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
169	M₁₆	—	172	1	no	—
170	M₁₆	—	172	1	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	172	2	no	—
175	C₂×C₈	—	172	1	no	—
176	C₄⋊C₄	—	86	2	no	—
177	C₂²⋊C₄	—	172	2	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	#10621882	1	2	yes	—
181	C₄²	#10621884	1	2	yes	—
182	C₄²	D₄×D₄₃	1	1	yes	characteristic
183	C₄²	#10621900	1	1	yes	characteristic
184	C₄²	#10621902	1	2	yes	—
185	C₂³	—	172	1	no	—
186	C₂³	#13063308	1	1	yes	characteristic
187	Q₈	—	172	1	no	—
188	Q₈	—	172	1	no	—
189	D₄	—	344	2	no	—
190	D₄	—	344	1	no	—
191	D₄	—	344	1	no	—
192	D₄	—	172	1	no	—
193	D₄	—	172	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	344	1	no	—
200	C₂×C₄	—	344	1	no	—
201	C₂×C₄	—	172	1	no	—
202	C₂×C₄	#13063296	1	1	yes	characteristic
203	C₂×C₄	#13063310	1	2	yes	—
204	C₂×C₄	#13063314	1	1	yes	characteristic
205	C₂×C₄	#13063320	1	2	yes	—
206	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
207	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
208	C₂×C₄	#13063351	1	1	yes	characteristic
209	C₂×C₄	#13063356	1	1	yes	characteristic
210	C₈	—	8	2	no	—
211	C₈	—	344	2	no	—
212	C₈	—	344	1	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	344	2	no	—
215	C₂²	—	344	1	no	—
216	C₂²	#14582825	1	1	yes	characteristic
217	C₂²	#14583212	1	1	yes	characteristic
218	C₂²	#14583217	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	344	1	no	—
224	C₄	#14583091	1	1	yes	characteristic
225	C₄	#14583101	1	1	yes	characteristic
226	C₄	#14583247	1	1	yes	characteristic
227	C₄	#14583261	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	—	688	1	no	—
231	C₂	#19135999	1	1	yes	characteristic
232	C₂	#19136046	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135427

C₂

yes

maximal, characteristic

#19133245

C₂

yes

maximal

#19132492

C₂

yes

maximal, complemented

#19132424

C₂

yes

maximal, characteristic, complemented

C₄₃×#1843

C₂

yes

maximal, characteristic, complemented

#14583449

C₂²

yes

characteristic

#14582952

C₂²

yes

—

#14582923

C₂²

yes

—

—

—

—

C₂×C₄⋊C₈×C₄₃

—

C₂×C₄⋊C₈×C₄₃

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

—

C₄₃×#376

C₂²

yes

—

—

D₄

yes

characteristic

#13063322

D₄

yes

characteristic

#13063313

—

#13063299

—

C₄×D₁₇₂

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063253

—

#13063253

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×C₂²⋊D₄₃

—

—

—

—

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₈

yes

—

C₄²×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

C₂×D₄

yes

C₄²×C₄₃

yes

C₄²×C₄₃

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#123

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

#126

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10627

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

100

C₂²×C₄₃

#481

yes

characteristic

101

C₂×D₄₃

—

102

C₂×D₄₃

—

103

C₄×C₄₃

—

104

C₄×C₄₃

—

105

C₄×C₄₃

—

106

C₄×C₄₃

—

107

C₄×C₄₃

#459

yes

characteristic

108

C₄×C₄₃

#461

yes

characteristic

109

C₄×C₄₃

#486

yes

characteristic

110

C₄×C₄₃

#489

yes

characteristic

111

Dic₄₃

—

112

#1963

—

113

#1843

D₄₃

yes

characteristic, complemented

114

#1540

—

115

#1301

—

116

#1274

—

117

C₂×C₄₃

—

118

C₂×C₄₃

—

119

C₂×C₄₃

#2071

yes

characteristic

120

C₂×C₄₃

#2080

yes

—

121

D₄₃

—

122

#532

—

123

#431

—

124

#423

—

125

C₂×C₄⋊C₈

—

126

C₂×C₄⋊C₈

—

127

C₂×C₄⋊C₈

C₂×D₄₃

yes

—

128

C₂×#104

—

129

C₂×#104

—

130

#376

C₂×D₄₃

yes

characteristic

131

#332

—

132

#331

—

133

C₄₃

#10627

yes

characteristic, complemented, Sylow

134

C₂×#41

—

135

C₂×M₁₆

—

136

C₂²×C₈

—

137

C₄⋊Q₈

—

138

#127

—

139

#124

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

143

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

144

C₂×C₄²

C₂²×D₄₃

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

149

C₄⋊C₈

—

150

C₄⋊C₈

—

151

#104

—

172

—

152

#104

—

172

—

153

C₂²⋊C₈

—

154

C₄×C₈

—

155

#95

—

156

#41

—

172

—

157

#41

—

172

—

158

#41

—

172

—

159

#41

—

172

—

160

C₂×Q₈

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂×D₄

—

172

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

—

172

—

167

C₂²×C₄

#10621898

yes

—

168

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

169

M₁₆

—

172

—

170

M₁₆

—

172

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

172

—

175

C₂×C₈

—

172

—

176

C₄⋊C₄

—

177

C₂²⋊C₄

—

172

—

178

C₄²

—

179

C₄²

—

180

C₄²

#10621882

yes

—

181

C₄²

#10621884

yes

—

182

C₄²

D₄×D₄₃

yes

characteristic

183

C₄²

#10621900

yes

characteristic

184

C₄²

#10621902

yes

—

185

C₂³

—

172

—

186

C₂³

#13063308

yes

characteristic

187

Q₈

—

172

—

188

Q₈

—

172

—

189

D₄

—

344

—

190

D₄

—

344

—

191

D₄

—

344

—

192

D₄

—

172

—

193

D₄

—

172

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

344

—

200

C₂×C₄

—

344

—

201

C₂×C₄

—

172

—

202

C₂×C₄

#13063296

yes

characteristic

203

C₂×C₄

#13063310

yes

—

204

C₂×C₄

#13063314

yes

characteristic

205

C₂×C₄

#13063320

yes

—

206

C₂×C₄

C₂×#10621882

yes

characteristic

207

C₂×C₄

C₂×#10621884

yes

characteristic

208

C₂×C₄

#13063351

yes

characteristic

209

C₂×C₄

#13063356

yes

characteristic

210

C₈

—

211

C₈

—

344

—

212

C₈

—

344

—

213

C₂²

—

214

C₂²

—

344

—

215

C₂²

—

344

—

216

C₂²

#14582825

yes

characteristic

217

C₂²

#14583212

yes

characteristic

218

C₂²

#14583217

yes

characteristic

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

344

—

224

C₄

#14583091

yes

characteristic

225

C₄

#14583101

yes

characteristic

226

C₄

#14583247

yes

characteristic

227

C₄

#14583261

yes

characteristic

228

C₂

—

229

C₂

—

230

C₂

—

688

—

231

C₂

#19135999

yes

characteristic

232

C₂

#19136046

yes

—

	subgroup	G/H
center	C₂×C₄	#13063314
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1843	C₂
socle	C₂²×C₄₃	#397
prime radical	#14583449	C₂²
hypercenter	#1843	D₄₃
hypocenter	C₄₃	#10627
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#13063314

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1843

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
3. #19132492	C₂	semidirect
4. #19132424	C₂	semidirect
5. C₄₃×#1843	C₂	semidirect
113. #1843	D₄₃	semidirect
133. C₄₃	#10627	semidirect

H ≅ G/N

type

3. #19132492

C₂

semidirect

4. #19132424

C₂

semidirect

5. C₄₃×#1843

C₂

semidirect

113. #1843

D₄₃

semidirect

133. C₄₃

#10627

semidirect

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

subgroup	index
#19135427	2
#19133245	2
#19132492	2
#19132424	2
C₄₃×#1843	2
#10627	43

subgroup

index

C₄₃×#1843

Subgroups — #25001366

Conjugacy classes of subgroups

Complements

Maximal subgroups