Group #25016698 — subgroups

Subgroups — #25016698

Order 11008.

346 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14582721	C₂	1	2	yes	maximal, complemented
2	C₂×#14582708	C₂	1	2	yes	maximal
3	#19135483	C₂	1	1	yes	maximal, characteristic
4	#19132677	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063347	C₂²	1	1	yes	characteristic
7	C₂×#13063265	C₂²	1	2	yes	—
8	C₂×#13063261	C₂²	1	1	yes	characteristic
9	C₂×#13063253	—	2	1	no	—
10	#14582865	—	2	1	no	—
11	#14582721	—	2	4	no	—
12	#14582708	—	2	4	no	—
13	C₂×#106×C₄₃	—	2	1	no	—
14	C₂×#106×C₄₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#384	—	2	1	no	—
17	C₂×#10621898	—	2	1	no	—
18	#13063347	—	2	2	no	—
19	#13063347	—	2	2	no	—
20	C₂×Dic₁₇₂	Q₈	1	1	yes	characteristic
21	C₂×#10621873	—	2	1	no	—
22	C₂×D₃₄₄	Q₈	1	1	yes	characteristic
23	C₂×#10621877	—	2	1	no	—
24	#13063265	—	2	2	no	—
25	#13063265	D₄	1	4	yes	—
26	C₂×C₄₃⋊C₁₆	—	2	2	no	—
27	#13063261	—	2	2	no	—
28	#13063253	—	4	2	no	—
29	(C₂×D₄₃)⋊C₈	—	4	2	no	—
30	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
34	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
35	#106×C₄₃	—	2	2	no	—
36	#106×C₄₃	—	2	2	no	—
37	#106×C₄₃	—	4	2	no	—
38	C₄×C₈×C₄₃	—	2	2	no	—
39	C₂²×C₄×C₄₃	—	2	1	no	—
40	C₂²×C₄×C₄₃	—	4	2	no	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₄₃	—	4	1	no	—
43	#10621898	—	4	1	no	—
44	#10621898	—	4	1	no	—
45	#10621898	—	4	1	no	—
46	#10621898	—	4	1	no	—
47	C₂×D₁₇₂	—	2	1	no	—
48	C₂×C₄×D₄₃	—	4	1	no	—
49	C₂×Dic₈₆	—	2	1	no	—
50	C₂×C₈×C₄₃	—	2	1	no	—
51	C₂×C₈×C₄₃	—	2	1	no	—
52	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
53	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₄₃	#41	1	2	yes	—
56	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₄₃	—	4	2	no	—
58	C₄⋊C₄×C₄₃	—	4	2	no	—
59	C₄⋊C₄×C₄₃	—	8	4	no	—
60	C₄²×C₄₃	—	2	1	no	—
61	C₄²×C₄₃	SD₁₆	1	2	yes	—
62	#10621877	—	4	1	no	—
63	#10621877	—	4	1	no	—
64	C₂×C₄₃⋊C₈	—	4	1	no	—
65	Dic₁₇₂	—	2	2	no	—
66	D₃₄₄	—	2	2	no	—
67	#10621873	—	4	2	no	—
68	C₄₃⋊C₁₆	—	2	4	no	—
69	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂²×D₄₃	—	4	1	no	—
71	C₂×C₄×C₄₃	—	2	4	no	—
72	C₂×C₄×C₄₃	—	2	2	no	—
73	C₂×C₄×C₄₃	—	8	4	no	—
74	C₂×C₄×C₄₃	—	8	2	no	—
75	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²⋊D₄₃	—	8	1	no	—
82	C₂²⋊D₄₃	—	8	1	no	—
83	C₂×Dic₄₃	—	4	1	no	—
84	D₁₇₂	—	4	1	no	—
85	D₁₇₂	—	4	1	no	—
86	C₄×D₄₃	—	8	1	no	—
87	C₄×D₄₃	—	8	1	no	—
88	Dic₈₆	—	4	1	no	—
89	Dic₈₆	—	4	1	no	—
90	C₈×C₄₃	—	2	2	no	—
91	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
92	C₈×C₄₃	#123	1	2	yes	—
93	C₄₃⋊C₈	—	8	1	no	—
94	#13533	—	43	1	no	maximal, Sylow
95	C₂²×C₄₃	—	2	2	no	—
96	C₂²×C₄₃	#476	1	1	yes	characteristic
97	C₂²×C₄₃	#481	1	1	yes	characteristic
98	C₂²×C₄₃	#496	1	1	yes	characteristic
99	C₂×D₄₃	—	8	1	no	—
100	C₂×D₄₃	—	8	2	no	—
101	C₄×C₄₃	—	2	4	no	—
102	C₄×C₄₃	—	16	2	no	—
103	C₄×C₄₃	#399	1	1	yes	characteristic
104	C₄×C₄₃	#476	1	1	yes	characteristic
105	C₄×C₄₃	#481	1	1	yes	characteristic
106	C₄×C₄₃	#492	1	1	yes	characteristic
107	Dic₄₃	—	8	1	no	—
108	C₂×#365	—	43	2	no	—
109	C₂×#360	—	43	2	no	—
110	#1974	—	43	1	no	—
111	#1844	D₄₃	1	1	yes	characteristic, complemented
112	#1376	—	43	1	no	—
113	C₂×C₄₃	—	2	1	no	—
114	C₂×C₄₃	—	2	1	no	—
115	C₂×C₄₃	#2094	1	1	yes	characteristic
116	C₂×C₄₃	#2230	1	2	yes	—
117	D₄₃	—	16	1	no	—
118	C₂×#135	—	43	1	no	—
119	C₂×M₃₂	—	43	2	no	—
120	C₂×#108	—	43	1	no	—
121	C₂×#106	—	2	1	no	—
122	C₂×#106	C₂×D₄₃	1	2	yes	—
123	C₂×#104	—	86	1	no	—
124	#408	—	86	1	no	—
125	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
126	#384	—	2	1	no	—
127	#365	—	86	4	no	—
128	#360	—	86	4	no	—
129	C₄₃	#13533	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	—	86	1	no	—
131	#135	—	86	2	no	—
132	#135	—	86	2	no	—
133	C₂×Q₁₆	—	43	1	no	—
134	C₂×SD₁₆	—	86	1	no	—
135	C₂×D₈	—	43	1	no	—
136	C₂×M₁₆	—	86	1	no	—
137	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	4	2	no	—
141	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
142	M₃₂	—	43	4	no	—
143	M₃₂	—	86	2	no	—
144	C₂×C₁₆	—	86	2	no	—
145	#108	—	86	2	no	—
146	#106	—	2	2	no	—
147	#106	—	2	2	no	—
148	#106	—	4	2	no	—
149	#104	—	172	2	no	—
150	C₂²⋊C₈	—	172	2	no	—
151	C₄×C₈	—	2	2	no	—
152	#41	—	172	1	no	—
153	#41	—	172	1	no	—
154	#41	—	172	1	no	—
155	#41	—	172	1	no	—
156	C₂×Q₈	—	86	1	no	—
157	C₂×D₄	—	86	1	no	—
158	C₂×D₄	—	172	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	—	172	1	no	—
162	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
163	Q₁₆	—	86	2	no	—
164	SD₁₆	—	172	2	no	—
165	D₈	—	86	2	no	—
166	M₁₆	—	172	1	no	—
167	M₁₆	—	172	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	172	1	no	—
171	C₂×C₈	#10621884	1	2	yes	—
172	C₂×C₈	#10621898	1	2	yes	—
173	C₂×C₈	D₄×D₄₃	1	2	yes	—
174	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
175	C₂×C₈	#10621902	1	1	yes	characteristic
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	8	4	no	—
179	C₄²	—	2	1	no	—
180	C₄²	#10621884	1	2	yes	—
181	C₁₆	—	86	4	no	—
182	C₂³	—	172	1	no	—
183	C₂³	#13063351	1	1	yes	characteristic
184	Q₈	—	172	1	no	—
185	Q₈	—	172	1	no	—
186	D₄	—	344	1	no	—
187	D₄	—	344	1	no	—
188	D₄	—	172	1	no	—
189	D₄	—	172	1	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	—	344	1	no	—
195	C₂×C₄	—	344	1	no	—
196	C₂×C₄	—	172	1	no	—
197	C₂×C₄	#13063301	1	1	yes	characteristic
198	C₂×C₄	#13063308	1	1	yes	characteristic
199	C₂×C₄	#13063321	1	1	yes	characteristic
200	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
201	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
202	C₂×C₄	#13063351	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	344	1	no	—
205	C₈	#13063303	1	2	yes	—
206	C₈	#13063310	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	344	1	no	—
209	C₂²	—	344	2	no	—
210	C₂²	#14583186	1	1	yes	characteristic
211	C₂²	#14583217	1	1	yes	characteristic
212	C₂²	#14583286	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	16	2	no	—
215	C₄	—	344	1	no	—
216	C₄	#14582834	1	1	yes	characteristic
217	C₄	#14583186	1	1	yes	characteristic
218	C₄	#14583217	1	1	yes	characteristic
219	C₄	#14583273	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	—	688	1	no	—
223	C₂	#19136129	1	1	yes	characteristic
224	C₂	#19136763	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582721

C₂

yes

maximal, complemented

C₂×#14582708

C₂

yes

maximal

#19135483

C₂

yes

maximal, characteristic

#19132677

C₂

yes

maximal, characteristic, complemented

C₄₃×#1844

C₂

yes

maximal, characteristic, complemented

C₂×#13063347

C₂²

yes

characteristic

C₂×#13063265

C₂²

yes

—

C₂×#13063261

C₂²

yes

—

—

—

—

C₂×#106×C₄₃

—

C₂×#106×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#384

—

C₂×#10621898

—

#13063347

—

#13063347

—

C₂×Dic₁₇₂

Q₈

yes

characteristic

C₂×#10621873

—

C₂×D₃₄₄

Q₈

yes

—

—

D₄

yes

—

C₂×C₄₃⋊C₁₆

—

#13063261

—

#13063253

—

(C₂×D₄₃)⋊C₈

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

#106×C₄₃

—

#106×C₄₃

—

#106×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

—

—

—

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

D₃₄₄

—

#10621873

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

#123

yes

—

C₄₃⋊C₈

—

#13533

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂×D₄₃

—

100

C₂×D₄₃

—

101

C₄×C₄₃

—

102

C₄×C₄₃

—

103

C₄×C₄₃

#399

yes

characteristic

104

C₄×C₄₃

#476

yes

characteristic

105

C₄×C₄₃

#481

yes

characteristic

106

C₄×C₄₃

#492

yes

characteristic

107

Dic₄₃

—

108

C₂×#365

—

109

C₂×#360

—

110

#1974

—

111

#1844

D₄₃

yes

characteristic, complemented

112

#1376

—

113

C₂×C₄₃

—

114

C₂×C₄₃

—

115

C₂×C₄₃

#2094

yes

characteristic

116

C₂×C₄₃

#2230

yes

—

117

D₄₃

—

118

C₂×#135

—

119

C₂×M₃₂

—

120

C₂×#108

—

121

C₂×#106

—

122

C₂×#106

C₂×D₄₃

yes

—

123

C₂×#104

—

124

#408

—

125

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

126

#384

—

127

#365

—

128

#360

—

129

C₄₃

#13533

yes

characteristic, complemented, Sylow

130

C₂×#41

—

131

#135

—

132

#135

—

133

C₂×Q₁₆

—

134

C₂×SD₁₆

—

135

C₂×D₈

—

136

C₂×M₁₆

—

137

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

138

C₂²×C₈

C₂²×D₄₃

yes

characteristic

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

142

M₃₂

—

143

M₃₂

—

144

C₂×C₁₆

—

145

#108

—

146

#106

—

147

#106

—

148

#106

—

149

#104

—

172

—

150

C₂²⋊C₈

—

172

—

151

C₄×C₈

—

152

#41

—

172

—

153

#41

—

172

—

154

#41

—

172

—

155

#41

—

172

—

156

C₂×Q₈

—

157

C₂×D₄

—

158

C₂×D₄

—

172

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

—

172

—

162

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

163

Q₁₆

—

164

SD₁₆

—

172

—

165

D₈

—

166

M₁₆

—

172

—

167

M₁₆

—

172

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

172

—

171

C₂×C₈

#10621884

yes

—

172

C₂×C₈

#10621898

yes

—

173

C₂×C₈

D₄×D₄₃

yes

—

174

C₂×C₈

Q₈×D₄₃

yes

characteristic

175

C₂×C₈

#10621902

yes

characteristic

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄²

—

180

C₄²

#10621884

yes

—

181

C₁₆

—

182

C₂³

—

172

—

183

C₂³

#13063351

yes

characteristic

184

Q₈

—

172

—

185

Q₈

—

172

—

186

D₄

—

344

—

187

D₄

—

344

—

188

D₄

—

172

—

189

D₄

—

172

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

344

—

195

C₂×C₄

—

344

—

196

C₂×C₄

—

172

—

197

C₂×C₄

#13063301

yes

characteristic

198

C₂×C₄

#13063308

yes

characteristic

199

C₂×C₄

#13063321

yes

characteristic

200

C₂×C₄

C₂×#10621884

yes

characteristic

201

C₂×C₄

C₂×#10621884

yes

characteristic

202

C₂×C₄

#13063351

yes

characteristic

203

C₈

—

204

C₈

—

344

—

205

C₈

#13063303

yes

—

206

C₈

#13063310

yes

—

207

C₂²

—

208

C₂²

—

344

—

209

C₂²

—

344

—

210

C₂²

#14583186

yes

characteristic

211

C₂²

#14583217

yes

characteristic

212

C₂²

#14583286

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

344

—

216

C₄

#14582834

yes

characteristic

217

C₄

#14583186

yes

characteristic

218

C₄

#14583217

yes

characteristic

219

C₄

#14583273

yes

characteristic

220

C₂

—

221

C₂

—

222

C₂

—

688

—

223

C₂

#19136129

yes

characteristic

224

C₂

#19136763

yes

—

	subgroup	G/H
center	C₂²	#14583186
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1844	C₂
socle	C₂²×C₄₃	#476
prime radical	C₂×#13063347	C₂²
hypercenter	#1844	D₄₃
hypocenter	C₄₃	#13533
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

subgroup

G/H

center

C₂²

#14583186

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1844

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

N	H ≅ G/N	type
1. C₂×#14582721	C₂	semidirect
4. #19132677	C₂	semidirect
5. C₄₃×#1844	C₂	semidirect
111. #1844	D₄₃	semidirect
129. C₄₃	#13533	semidirect

H ≅ G/N

type

1. C₂×#14582721

C₂

semidirect

4. #19132677

C₂

semidirect

5. C₄₃×#1844

C₂

semidirect

111. #1844

D₄₃

semidirect

129. C₄₃

#13533

semidirect

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

subgroup	index
C₂×#14582721	2
C₂×#14582708	2
#19135483	2
#19132677	2
C₄₃×#1844	2
#13533	43

subgroup

index

C₄₃×#1844

Subgroups — #25016698

Conjugacy classes of subgroups

Complements

Maximal subgroups