Group #24999261 — subgroups

Subgroups — #24999261

Order 11008.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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	#19145725	C₂	1	2	yes	maximal
2	#19142973	C₂	1	1	yes	maximal, characteristic, complemented
3	#19135423	C₂	1	1	yes	maximal, characteristic
4	#19133279	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583449	C₂²	1	1	yes	characteristic
7	C₂×#13063325	C₂²	1	2	yes	—
8	#14583297	—	2	2	no	—
9	#14583246	—	2	2	no	—
10	#14583218	—	2	4	no	—
11	#14583190	—	2	4	no	—
12	#14582923	C₂²	1	1	yes	characteristic
13	C₂×#13063253	—	2	2	no	—
14	#14582805	—	2	4	no	—
15	#14582765	—	2	2	no	—
16	#14582646	—	2	4	no	—
17	C₄₃×#431	C₂²	1	2	yes	—
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×#10621898	—	2	1	no	—
20	C₂²×Dic₈₆	—	2	2	no	—
21	C₂×#10621877	—	2	1	no	—
22	#13063325	—	2	2	no	—
23	#13063325	—	2	2	no	—
24	#13063325	—	2	2	no	—
25	#13063325	—	2	2	no	—
26	#13063325	D₄	1	1	yes	characteristic
27	#13063322	D₄	1	1	yes	characteristic
28	#13063313	—	2	1	no	—
29	#13063299	—	2	2	no	—
30	C₄×D₁₇₂	—	2	2	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	D₄	1	4	yes	—
33	#13063257	—	2	8	no	—
34	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
35	#13063253	—	4	4	no	—
36	#13063247	—	2	8	no	—
37	C₄₃⋊#95	—	2	8	no	—
38	C₂×C₄²×C₄₃	—	2	2	no	—
39	C₂×C₄²×C₄₃	D₄	1	2	yes	—
40	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₃	D₄	1	4	yes	—
42	C₂²⋊C₈×C₄₃	—	2	2	no	—
43	C₄×C₈×C₄₃	—	2	2	no	—
44	C₂²×C₄×C₄₃	—	2	2	no	—
45	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
47	C₂×C₂²⋊D₄₃	—	4	1	no	—
48	C₂²×Dic₄₃	—	2	4	no	—
49	#10621898	—	4	2	no	—
50	#10621898	—	4	2	no	—
51	C₂×D₁₇₂	—	2	2	no	—
52	C₂×D₁₇₂	—	2	1	no	—
53	C₂×C₄×D₄₃	—	4	1	no	—
54	C₂×Dic₈₆	—	2	1	no	—
55	C₂×Dic₈₆	—	4	8	no	—
56	C₂×Dic₈₆	—	4	2	no	—
57	C₂×Dic₈₆	—	4	2	no	—
58	C₂×C₈×C₄₃	—	2	4	no	—
59	C₄²×C₄₃	—	2	2	no	—
60	C₄²×C₄₃	—	4	4	no	—
61	C₄²×C₄₃	SD₁₆	1	4	yes	—
62	C₄²×C₄₃	C₂×D₄	1	2	yes	—
63	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₄₃	#41	1	1	yes	characteristic
65	(C₂×D₄₃)⋊C₄	—	4	2	no	—
66	C₄⋊Dic₄₃	—	2	4	no	—
67	C₄⋊Dic₄₃	—	2	2	no	—
68	C₄⋊Dic₄₃	—	2	4	no	—
69	C₄⋊Dic₄₃	—	4	2	no	—
70	C₄⋊Dic₄₃	—	4	2	no	—
71	#10621877	—	4	2	no	—
72	C₂×C₄₃⋊C₈	—	4	1	no	—
73	C₂³×C₄₃	#120	1	1	yes	characteristic
74	C₂²×D₄₃	—	4	1	no	—
75	C₂×C₄×C₄₃	—	2	4	no	—
76	C₂×C₄×C₄₃	—	2	4	no	—
77	C₂×C₄×C₄₃	—	4	4	no	—
78	C₂×C₄×C₄₃	#120	1	2	yes	—
79	C₂×C₄×C₄₃	#121	1	2	yes	—
80	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
83	C₂×C₄×C₄₃	#137	1	2	yes	—
84	C₂²⋊D₄₃	—	8	2	no	—
85	C₂×Dic₄₃	—	4	8	no	—
86	C₂×Dic₄₃	—	4	4	no	—
87	C₂×Dic₄₃	—	4	1	no	—
88	D₁₇₂	—	4	4	no	—
89	D₁₇₂	—	4	2	no	—
90	C₄×D₄₃	—	8	1	no	—
91	C₄×D₄₃	—	8	1	no	—
92	Dic₈₆	—	4	2	no	—
93	Dic₈₆	—	8	4	no	—
94	Dic₈₆	—	8	4	no	—
95	C₈×C₄₃	—	4	4	no	—
96	C₄₃⋊C₈	—	8	1	no	—
97	#10285	—	43	1	no	maximal, Sylow
98	C₂²×C₄₃	—	2	2	no	—
99	C₂²×C₄₃	#392	1	1	yes	characteristic
100	C₂²×C₄₃	#448	1	2	yes	—
101	C₂×D₄₃	—	8	2	no	—
102	C₂×D₄₃	—	8	1	no	—
103	C₄×C₄₃	—	2	2	no	—
104	C₄×C₄₃	—	4	4	no	—
105	C₄×C₄₃	#454	1	4	yes	—
106	C₄×C₄₃	#461	1	2	yes	—
107	C₄×C₄₃	#492	1	1	yes	characteristic
108	C₄×C₄₃	#497	1	1	yes	characteristic
109	Dic₄₃	—	8	4	no	—
110	Dic₄₃	—	8	1	no	—
111	#3227	—	43	2	no	—
112	#2862	—	43	1	no	—
113	#1962	—	43	1	no	—
114	#1549	—	43	2	no	—
115	#1276	D₄₃	1	1	yes	characteristic, complemented
116	C₂×C₄₃	—	2	2	no	—
117	C₂×C₄₃	#2003	1	2	yes	—
118	C₂×C₄₃	#2013	1	1	yes	characteristic
119	D₄₃	—	16	1	no	—
120	#532	—	43	1	no	—
121	C₂×C₄⋊Q₈	—	43	2	no	—
122	#498	—	86	2	no	—
123	#486	—	86	2	no	—
124	#481	—	86	4	no	—
125	#476	—	86	4	no	—
126	#431	C₂×D₄₃	1	2	yes	—
127	#423	—	43	1	no	—
128	C₂×#104	—	86	2	no	—
129	#391	—	86	4	no	—
130	#382	—	86	2	no	—
131	C₄³	C₂×D₄₃	1	1	yes	characteristic
132	#332	—	86	4	no	—
133	C₄₃	#10285	1	1	yes	characteristic, complemented, Sylow
134	C₂×#41	—	86	1	no	—
135	C₂²×Q₈	—	86	2	no	—
136	C₂×M₁₆	—	86	1	no	—
137	C₄⋊Q₈	—	43	1	no	—
138	C₄⋊Q₈	—	86	2	no	—
139	C₄⋊Q₈	—	86	2	no	—
140	C₄⋊Q₈	—	86	2	no	—
141	C₄⋊Q₈	—	86	2	no	—
142	#127	—	43	1	no	—
143	#124	—	86	1	no	—
144	#121	—	86	2	no	—
145	C₄×D₄	—	86	2	no	—
146	C₂×C₄⋊C₄	—	43	4	no	—
147	C₂×C₄⋊C₄	—	86	2	no	—
148	C₂×C₄²	—	2	2	no	—
149	C₂×C₄²	D₁₇₂	1	2	yes	—
150	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
151	#106	—	86	8	no	—
152	C₄⋊C₈	—	86	2	no	—
153	C₄⋊C₈	D₁₇₂	1	4	yes	—
154	#104	—	172	4	no	—
155	#102	—	86	8	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	#95	—	86	8	no	—
159	#41	—	172	2	no	—
160	#41	—	172	2	no	—
161	C₂×Q₈	—	86	1	no	—
162	C₂×Q₈	—	172	8	no	—
163	C₂×Q₈	—	172	2	no	—
164	C₂×Q₈	—	172	2	no	—
165	C₂×D₄	—	86	2	no	—
166	C₂×D₄	—	86	1	no	—
167	C₂×D₄	—	172	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	86	4	no	—
170	C₂²×C₄	—	172	1	no	—
171	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
172	C₂²×C₄	D₄×D₄₃	1	2	yes	—
173	M₁₆	—	172	2	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	172	1	no	—
176	C₄⋊C₄	—	86	4	no	—
177	C₄⋊C₄	—	86	2	no	—
178	C₄⋊C₄	—	86	4	no	—
179	C₄⋊C₄	—	172	2	no	—
180	C₄⋊C₄	—	172	2	no	—
181	C₂²⋊C₄	—	172	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	4	4	no	—
184	C₄²	#10621873	1	4	yes	—
185	C₄²	C₂×D₁₇₂	1	2	yes	—
186	C₄²	D₄×D₄₃	1	1	yes	characteristic
187	C₄²	#10621902	1	1	yes	characteristic
188	C₂³	—	172	1	no	—
189	C₂³	#13063295	1	1	yes	characteristic
190	Q₈	—	344	4	no	—
191	Q₈	—	344	4	no	—
192	Q₈	—	172	2	no	—
193	D₄	—	344	2	no	—
194	D₄	—	172	4	no	—
195	D₄	—	172	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	344	1	no	—
200	C₂×C₄	—	344	1	no	—
201	C₂×C₄	—	172	8	no	—
202	C₂×C₄	—	172	1	no	—
203	C₂×C₄	—	172	4	no	—
204	C₂×C₄	#13063295	1	2	yes	—
205	C₂×C₄	#13063300	1	2	yes	—
206	C₂×C₄	#13063300	1	1	yes	characteristic
207	C₂×C₄	#13063322	1	1	yes	characteristic
208	C₂×C₄	C₂×#10621873	1	2	yes	—
209	C₂×C₄	#13063358	1	2	yes	—
210	C₈	—	4	4	no	—
211	C₈	—	344	1	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	344	2	no	—
214	C₂²	—	344	1	no	—
215	C₂²	#14582809	1	1	yes	characteristic
216	C₂²	#14583040	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	4	4	no	—
219	C₄	—	344	4	no	—
220	C₄	—	344	1	no	—
221	C₄	#14583073	1	4	yes	—
222	C₄	#14583105	1	2	yes	—
223	C₄	#14583272	1	1	yes	characteristic
224	C₄	#14583293	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	688	1	no	—
227	C₂	#19135620	1	2	yes	—
228	C₂	#19135680	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145725

C₂

yes

maximal

#19142973

C₂

yes

maximal, characteristic, complemented

#19135423

C₂

yes

maximal, characteristic

#19133279

C₂

yes

maximal, complemented

C₄₃×#1276

C₂

yes

maximal, characteristic, complemented

#14583449

C₂²

yes

characteristic

C₂×#13063325

C₂²

yes

—

—

—

—

—

C₂²

yes

—

—

—

—

C₄₃×#431

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂×#10621898

—

C₂²×Dic₈₆

—

—

—

—

—

—

D₄

yes

characteristic

#13063322

D₄

yes

characteristic

#13063313

—

#13063299

—

C₄×D₁₇₂

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

D₄

yes

—

#13063257

—

C₄⋊(C₄₃⋊C₈)

—

#13063253

—

#13063247

—

C₄₃⋊#95

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

D₄

yes

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₂²⋊D₄₃

—

C₂²×Dic₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10285

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#392

yes

characteristic

100

C₂²×C₄₃

#448

yes

—

101

C₂×D₄₃

—

102

C₂×D₄₃

—

103

C₄×C₄₃

—

104

C₄×C₄₃

—

105

C₄×C₄₃

#454

yes

—

106

C₄×C₄₃

#461

yes

—

107

C₄×C₄₃

#492

yes

characteristic

108

C₄×C₄₃

#497

yes

characteristic

109

Dic₄₃

—

110

Dic₄₃

—

111

#3227

—

112

#2862

—

113

#1962

—

114

#1549

—

115

#1276

D₄₃

yes

characteristic, complemented

116

C₂×C₄₃

—

117

C₂×C₄₃

#2003

yes

—

118

C₂×C₄₃

#2013

yes

characteristic

119

D₄₃

—

120

#532

—

121

C₂×C₄⋊Q₈

—

122

#498

—

123

#486

—

124

#481

—

125

#476

—

126

#431

C₂×D₄₃

yes

—

127

#423

—

128

C₂×#104

—

129

#391

—

130

#382

—

131

C₄³

C₂×D₄₃

yes

characteristic

132

#332

—

133

C₄₃

#10285

yes

characteristic, complemented, Sylow

134

C₂×#41

—

135

C₂²×Q₈

—

136

C₂×M₁₆

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

—

142

#127

—

143

#124

—

144

#121

—

145

C₄×D₄

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄²

—

149

C₂×C₄²

D₁₇₂

yes

—

150

C₂×C₄²

C₂²×D₄₃

yes

characteristic

151

#106

—

152

C₄⋊C₈

—

153

C₄⋊C₈

D₁₇₂

yes

—

154

#104

—

172

—

155

#102

—

156

C₂²⋊C₈

—

157

C₄×C₈

—

158

#95

—

159

#41

—

172

—

160

#41

—

172

—

161

C₂×Q₈

—

162

C₂×Q₈

—

172

—

163

C₂×Q₈

—

172

—

164

C₂×Q₈

—

172

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

172

—

168

C₂²×C₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

172

—

171

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

172

C₂²×C₄

D₄×D₄₃

yes

—

173

M₁₆

—

172

—

174

C₂×C₈

—

175

C₂×C₈

—

172

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

172

—

180

C₄⋊C₄

—

172

—

181

C₂²⋊C₄

—

172

—

182

C₄²

—

183

C₄²

—

184

C₄²

#10621873

yes

—

185

C₄²

C₂×D₁₇₂

yes

—

186

C₄²

D₄×D₄₃

yes

characteristic

187

C₄²

#10621902

yes

characteristic

188

C₂³

—

172

—

189

C₂³

#13063295

yes

characteristic

190

Q₈

—

344

—

191

Q₈

—

344

—

192

Q₈

—

172

—

193

D₄

—

344

—

194

D₄

—

172

—

195

D₄

—

172

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

344

—

200

C₂×C₄

—

344

—

201

C₂×C₄

—

172

—

202

C₂×C₄

—

172

—

203

C₂×C₄

—

172

—

204

C₂×C₄

#13063295

yes

—

205

C₂×C₄

#13063300

yes

—

206

C₂×C₄

#13063300

yes

characteristic

207

C₂×C₄

#13063322

yes

characteristic

208

C₂×C₄

C₂×#10621873

yes

—

209

C₂×C₄

#13063358

yes

—

210

C₈

—

211

C₈

—

344

—

212

C₂²

—

213

C₂²

—

344

—

214

C₂²

—

344

—

215

C₂²

#14582809

yes

characteristic

216

C₂²

#14583040

yes

—

217

C₄

—

218

C₄

—

219

C₄

—

344

—

220

C₄

—

344

—

221

C₄

#14583073

yes

—

222

C₄

#14583105

yes

—

223

C₄

#14583272

yes

characteristic

224

C₄

#14583293

yes

characteristic

225

C₂

—

226

C₂

—

688

—

227

C₂

#19135620

yes

—

228

C₂

#19135680

yes

characteristic

	subgroup	G/H
center	C₂²	#14582809
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1276	C₂
socle	C₂²×C₄₃	#392
prime radical	#14583449	C₂²
hypercenter	#1276	D₄₃
hypocenter	C₄₃	#10285
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#14582809

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1276

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
2. #19142973	C₂	semidirect
4. #19133279	C₂	semidirect
5. C₄₃×#1276	C₂	semidirect
115. #1276	D₄₃	semidirect
133. C₄₃	#10285	semidirect

H ≅ G/N

type

2. #19142973

C₂

semidirect

4. #19133279

C₂

semidirect

5. C₄₃×#1276

C₂

semidirect

115. #1276

D₄₃

semidirect

133. C₄₃

#10285

semidirect

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

subgroup	index
#19145725	2
#19142973	2
#19135423	2
#19133279	2
C₄₃×#1276	2
#10285	43

subgroup

index

C₄₃×#1276

Subgroups — #24999261

Conjugacy classes of subgroups

Complements

Maximal subgroups