Group #24999259 — subgroups

Subgroups — #24999259

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	#19145723	C₂	1	2	yes	maximal
2	#19135423	C₂	1	1	yes	maximal, characteristic
3	#19133280	C₂	1	2	yes	maximal, complemented
4	#19132428	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583449	C₂²	1	1	yes	characteristic
7	#14583295	—	2	2	no	—
8	#14583247	—	2	2	no	—
9	#14583217	—	2	4	no	—
10	#14583186	—	2	4	no	—
11	#14582952	C₂²	1	2	yes	—
12	#14582923	C₂²	1	1	yes	characteristic
13	C₂×#13063253	—	2	2	no	—
14	#14582645	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
16	C₄₃×#391	—	2	4	no	—
17	C₄₃×#382	—	2	2	no	—
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×#10621898	—	2	1	no	—
20	C₂×#10621877	—	2	1	no	—
21	#13063325	D₄	1	1	yes	characteristic
22	#13063322	D₄	1	1	yes	characteristic
23	#13063313	—	2	1	no	—
24	#13063299	—	2	2	no	—
25	C₄×D₁₇₂	—	2	2	no	—
26	#13063256	—	2	8	no	—
27	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
29	#13063253	—	4	4	no	—
30	#13063245	—	2	8	no	—
31	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
32	C₄×C₄₃⋊C₈	—	2	2	no	—
33	C₂²×Q₈×C₄₃	—	2	2	no	—
34	C₄⋊Q₈×C₄₃	—	2	2	no	—
35	C₄⋊Q₈×C₄₃	—	2	2	no	—
36	C₄⋊Q₈×C₄₃	—	2	2	no	—
37	C₄⋊Q₈×C₄₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₃	D₄	1	4	yes	—
40	C₂×C₄²×C₄₃	—	2	2	no	—
41	C₂×C₄²×C₄₃	D₄	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	#95×C₄₃	—	2	8	no	—
44	C₂×Q₈×C₄₃	—	4	8	no	—
45	C₂×Q₈×C₄₃	—	4	2	no	—
46	C₂×Q₈×C₄₃	—	4	2	no	—
47	C₂²×C₄×C₄₃	—	2	2	no	—
48	C₂²×C₄×C₄₃	—	2	4	no	—
49	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₄₃	—	4	1	no	—
52	#10621898	—	4	2	no	—
53	#10621898	—	4	2	no	—
54	C₂×D₁₇₂	—	2	2	no	—
55	C₂×D₁₇₂	—	2	1	no	—
56	C₂×C₄×D₄₃	—	4	1	no	—
57	C₂×Dic₈₆	—	2	1	no	—
58	C₄⋊C₄×C₄₃	—	2	4	no	—
59	C₄⋊C₄×C₄₃	—	2	4	no	—
60	C₄⋊C₄×C₄₃	—	4	2	no	—
61	C₄⋊C₄×C₄₃	—	4	2	no	—
62	C₄²×C₄₃	—	2	2	no	—
63	C₄²×C₄₃	—	4	4	no	—
64	C₄²×C₄₃	SD₁₆	1	4	yes	—
65	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₄₃	C₂×D₄	1	2	yes	—
67	C₄²×C₄₃	#41	1	1	yes	characteristic
68	(C₂×D₄₃)⋊C₄	—	4	2	no	—
69	C₄⋊Dic₄₃	—	2	2	no	—
70	#10621877	—	4	2	no	—
71	C₂×C₄₃⋊C₈	—	2	4	no	—
72	C₂×C₄₃⋊C₈	—	4	1	no	—
73	C₂³×C₄₃	#120	1	1	yes	characteristic
74	C₂²×D₄₃	—	4	1	no	—
75	Q₈×C₄₃	—	8	4	no	—
76	Q₈×C₄₃	—	8	4	no	—
77	C₂×C₄×C₄₃	—	2	4	no	—
78	C₂×C₄×C₄₃	—	2	4	no	—
79	C₂×C₄×C₄₃	—	4	4	no	—
80	C₂×C₄×C₄₃	—	4	8	no	—
81	C₂×C₄×C₄₃	—	4	4	no	—
82	C₂×C₄×C₄₃	#120	1	2	yes	—
83	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#121	1	2	yes	—
85	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₄₃	#137	1	2	yes	—
88	C₂²⋊D₄₃	—	8	2	no	—
89	C₂×Dic₄₃	—	4	1	no	—
90	D₁₇₂	—	4	4	no	—
91	D₁₇₂	—	4	2	no	—
92	C₄×D₄₃	—	8	1	no	—
93	C₄×D₄₃	—	8	1	no	—
94	Dic₈₆	—	4	2	no	—
95	C₄₃⋊C₈	—	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	—	8	4	no	—
106	C₄×C₄₃	#454	1	4	yes	—
107	C₄×C₄₃	#461	1	2	yes	—
108	C₄×C₄₃	#492	1	1	yes	characteristic
109	C₄×C₄₃	#497	1	1	yes	characteristic
110	Dic₄₃	—	8	1	no	—
111	#3227	—	43	2	no	—
112	#2862	D₄₃	1	1	yes	characteristic, complemented
113	#1962	—	43	1	no	—
114	#1549	—	43	2	no	—
115	#1276	—	43	1	no	—
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	C₂×D₄₃	1	2	yes	—
122	#498	—	86	2	no	—
123	#486	—	86	2	no	—
124	#481	—	86	4	no	—
125	#476	—	86	4	no	—
126	#431	—	43	2	no	—
127	#423	—	43	1	no	—
128	C₂×#104	—	86	2	no	—
129	#391	—	2	4	no	—
130	#382	—	2	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	—	2	2	no	—
136	C₂×M₁₆	—	86	1	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	43	1	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	—	2	2	no	—
147	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	4	yes	—
148	C₂×C₄²	—	2	2	no	—
149	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
150	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
151	#106	—	86	8	no	—
152	C₄⋊C₈	—	43	4	no	—
153	C₄⋊C₈	—	86	2	no	—
154	#104	—	172	4	no	—
155	#102	—	86	8	no	—
156	C₂²⋊C₈	—	86	2	no	—
157	C₄×C₈	—	86	2	no	—
158	#95	—	2	8	no	—
159	#41	—	172	2	no	—
160	#41	—	172	2	no	—
161	C₂×Q₈	—	4	8	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	86	1	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	—	2	4	no	—
170	C₂²×C₄	—	172	1	no	—
171	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
173	M₁₆	—	172	2	no	—
174	C₂×C₈	—	86	4	no	—
175	C₂×C₈	—	172	1	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	86	2	no	—
181	C₂²⋊C₄	—	172	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	4	4	no	—
184	C₄²	#10621884	1	4	yes	—
185	C₄²	D₄×D₄₃	1	1	yes	characteristic
186	C₄²	D₄×D₄₃	1	2	yes	—
187	C₄²	#10621900	1	1	yes	characteristic
188	C₂³	—	172	1	no	—
189	C₂³	#13063294	1	1	yes	characteristic
190	Q₈	—	8	4	no	—
191	Q₈	—	8	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	—	4	8	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	344	1	no	—
202	C₂×C₄	—	344	1	no	—
203	C₂×C₄	—	172	1	no	—
204	C₂×C₄	#13063293	1	2	yes	—
205	C₂×C₄	#13063296	1	1	yes	characteristic
206	C₂×C₄	#13063297	1	2	yes	—
207	C₂×C₄	#13063321	1	1	yes	characteristic
208	C₂×C₄	C₂×#10621884	1	2	yes	—
209	C₂×C₄	#13063356	1	2	yes	—
210	C₈	—	344	1	no	—
211	C₈	—	172	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	344	2	no	—
214	C₂²	—	344	1	no	—
215	C₂²	#14582807	1	1	yes	characteristic
216	C₂²	#14583038	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	4	no	—
220	C₄	—	344	1	no	—
221	C₄	#14583071	1	4	yes	—
222	C₄	#14583101	1	2	yes	—
223	C₄	#14583273	1	1	yes	characteristic
224	C₄	#14583291	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	688	1	no	—
227	C₂	#19135617	1	2	yes	—
228	C₂	#19135678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145723

C₂

yes

maximal

#19135423

C₂

yes

maximal, characteristic

#19133280

C₂

yes

maximal, complemented

#19132428

C₂

yes

maximal, characteristic, complemented

C₄₃×#2862

C₂

yes

maximal, characteristic, complemented

#14583449

C₂²

yes

—

—

—

—

C₂²

yes

—

#14582923

C₂²

yes

characteristic

C₂×#13063253

—

#14582645

—

C₂×C₄⋊Q₈×C₄₃

C₂²

yes

—

C₄₃×#391

—

C₄₃×#382

—

C₄³×C₄₃

C₂²

yes

—

—

D₄

yes

characteristic

#13063322

D₄

yes

characteristic

#13063313

—

#13063299

—

C₄×D₁₇₂

—

#13063256

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

#13063253

—

#13063245

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂²×Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#95×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₄₃⋊C₈

—

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

—

106

C₄×C₄₃

#454

yes

—

107

C₄×C₄₃

#461

yes

—

108

C₄×C₄₃

#492

yes

characteristic

109

C₄×C₄₃

#497

yes

characteristic

110

Dic₄₃

—

111

#3227

—

112

#2862

D₄₃

yes

characteristic, complemented

113

#1962

—

114

#1549

—

115

#1276

—

116

C₂×C₄₃

—

117

C₂×C₄₃

#2003

yes

—

118

C₂×C₄₃

#2013

yes

characteristic

119

D₄₃

—

120

#532

—

121

C₂×C₄⋊Q₈

C₂×D₄₃

yes

—

122

#498

—

123

#486

—

124

#481

—

125

#476

—

126

#431

—

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

C₂²⋊D₄₃

yes

—

148

C₂×C₄²

—

149

C₂×C₄²

C₂²⋊D₄₃

yes

—

150

C₂×C₄²

C₂²×D₄₃

yes

characteristic

151

#106

—

152

C₄⋊C₈

—

153

C₄⋊C₈

—

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

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

172

—

168

C₂²×C₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

172

—

171

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

172

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

173

M₁₆

—

172

—

174

C₂×C₈

—

175

C₂×C₈

—

172

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₂²⋊C₄

—

172

—

182

C₄²

—

183

C₄²

—

184

C₄²

#10621884

yes

—

185

C₄²

D₄×D₄₃

yes

characteristic

186

C₄²

D₄×D₄₃

yes

—

187

C₄²

#10621900

yes

characteristic

188

C₂³

—

172

—

189

C₂³

#13063294

yes

characteristic

190

Q₈

—

191

Q₈

—

192

Q₈

—

172

—

193

D₄

—

344

—

194

D₄

—

172

—

195

D₄

—

172

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

344

—

202

C₂×C₄

—

344

—

203

C₂×C₄

—

172

—

204

C₂×C₄

#13063293

yes

—

205

C₂×C₄

#13063296

yes

characteristic

206

C₂×C₄

#13063297

yes

—

207

C₂×C₄

#13063321

yes

characteristic

208

C₂×C₄

C₂×#10621884

yes

—

209

C₂×C₄

#13063356

yes

—

210

C₈

—

344

—

211

C₈

—

172

—

212

C₂²

—

213

C₂²

—

344

—

214

C₂²

—

344

—

215

C₂²

#14582807

yes

characteristic

216

C₂²

#14583038

yes

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

344

—

221

C₄

#14583071

yes

—

222

C₄

#14583101

yes

—

223

C₄

#14583273

yes

characteristic

224

C₄

#14583291

yes

characteristic

225

C₂

—

226

C₂

—

688

—

227

C₂

#19135617

yes

—

228

C₂

#19135678

yes

characteristic

	subgroup	G/H
center	C₂²	#14582807
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2862	C₂
socle	C₂²×C₄₃	#392
prime radical	#14583449	C₂²
hypercenter	#2862	D₄₃
hypocenter	C₄₃	#10285
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2862

subgroup

G/H

center

C₂²

#14582807

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#2862

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2862

N	H ≅ G/N	type
3. #19133280	C₂	semidirect
4. #19132428	C₂	semidirect
5. C₄₃×#2862	C₂	semidirect
112. #2862	D₄₃	semidirect
133. C₄₃	#10285	semidirect

H ≅ G/N

type

3. #19133280

C₂

semidirect

4. #19132428

C₂

semidirect

5. C₄₃×#2862

C₂

semidirect

112. #2862

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
#19145723	2
#19135423	2
#19133280	2
#19132428	2
C₄₃×#2862	2
#10285	43

subgroup

index

C₄₃×#2862

Subgroups — #24999259

Conjugacy classes of subgroups

Complements

Maximal subgroups