Group #24994209 — subgroups

Subgroups — #24994209

Order 11008.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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	#19144971	C₂	1	1	yes	maximal, characteristic
2	#19143697	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133264	C₂	1	2	yes	maximal
4	#19132484	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1561	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063322	C₂²	1	1	yes	characteristic
7	#14583265	C₄	1	4	yes	—
8	#14583247	—	2	2	no	—
9	#14583030	—	2	2	no	—
10	Q₈×C₄₃⋊C₈	—	2	2	no	—
11	#14582960	—	2	2	no	—
12	#14582952	C₂²	1	2	yes	—
13	#14582903	—	2	2	no	—
14	C₄×#10621877	C₂²	1	1	yes	characteristic
15	#14582642	—	2	2	no	—
16	#14582642	—	2	2	no	—
17	C₄₃×#533	C₂²	1	1	yes	characteristic
18	C₄₃×#423	C₂²	1	2	yes	—
19	C₄₃×#332	—	2	2	no	—
20	C₂²×D₁₇₂	—	2	1	no	—
21	C₂²×D₁₇₂	—	2	2	no	—
22	C₂×#10621877	—	2	1	no	—
23	#13063322	—	2	2	no	—
24	#13063322	—	2	1	no	—
25	#13063322	C₂×C₄	1	2	yes	—
26	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
27	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
28	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
29	#13063245	—	2	8	no	—
30	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
31	#13063236	—	2	2	no	—
32	#13063236	C₂×C₄	1	2	yes	—
33	C₄×C₄₃⋊C₈	—	2	2	no	—
34	C₄×C₄₃⋊C₈	—	2	1	no	—
35	C₄×C₄₃⋊C₈	—	2	2	no	—
36	C₂×M₁₆×C₄₃	—	2	2	no	—
37	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
38	#125×C₄₃	C₂×C₄	1	2	yes	—
39	C₂²⋊Q₈×C₄₃	—	2	2	no	—
40	C₄×Q₈×C₄₃	—	2	2	no	—
41	#117×C₄₃	D₄	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₃	—	2	2	no	—
44	C₄⋊C₈×C₄₃	—	2	2	no	—
45	C₂³×D₄₃	—	4	1	no	—
46	C₂×Q₈×C₄₃	—	2	2	no	—
47	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	#41	1	2	yes	—
49	C₂×D₁₇₂	—	2	2	no	—
50	C₂×D₁₇₂	—	2	4	no	—
51	C₂×D₁₇₂	—	4	1	no	—
52	C₂×D₁₇₂	—	4	2	no	—
53	C₂×D₁₇₂	—	4	4	no	—
54	C₂×D₁₇₂	—	8	2	no	—
55	C₂×D₁₇₂	—	8	2	no	—
56	M₁₆×C₄₃	—	8	2	no	—
57	C₂×C₈×C₄₃	—	4	2	no	—
58	C₄⋊C₄×C₄₃	—	2	4	no	—
59	C₄⋊C₄×C₄₃	—	2	2	no	—
60	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
61	C₂²⋊C₄×C₄₃	—	2	2	no	—
62	C₄²×C₄₃	—	2	2	no	—
63	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₄₃	C₂×D₄	1	2	yes	—
65	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
66	#10621877	—	4	2	no	—
67	C₂×C₄₃⋊C₈	—	2	2	no	—
68	C₂×C₄₃⋊C₈	—	2	2	no	—
69	C₂×C₄₃⋊C₈	—	2	2	no	—
70	C₂³×C₄₃	#123	1	1	yes	characteristic
71	C₂²×D₄₃	—	4	2	no	—
72	C₂²×D₄₃	—	8	2	no	—
73	C₂²×D₄₃	—	8	4	no	—
74	Q₈×C₄₃	—	4	2	no	—
75	Q₈×C₄₃	—	4	2	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	—	2	2	no	—
79	C₂×C₄×C₄₃	—	2	4	no	—
80	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
83	C₂×C₄×C₄₃	#121	1	2	yes	—
84	D₁₇₂	—	4	4	no	—
85	D₁₇₂	—	8	4	no	—
86	D₁₇₂	—	8	2	no	—
87	D₁₇₂	—	8	2	no	—
88	D₁₇₂	—	8	2	no	—
89	C₈×C₄₃	—	8	2	no	—
90	C₄₃⋊C₈	—	4	2	no	—
91	C₄₃⋊C₈	—	4	2	no	—
92	C₄₃⋊C₈	—	4	2	no	—
93	#9438	—	43	1	no	maximal, Sylow
94	C₂²×C₄₃	—	2	1	no	—
95	C₂²×C₄₃	—	2	2	no	—
96	C₂²×C₄₃	#386	1	1	yes	characteristic
97	C₂×D₄₃	—	8	2	no	—
98	C₂×D₄₃	—	8	4	no	—
99	C₂×D₄₃	—	16	4	no	—
100	C₄×C₄₃	—	2	1	no	—
101	C₄×C₄₃	—	2	2	no	—
102	C₄×C₄₃	—	4	4	no	—
103	C₄×C₄₃	#454	1	2	yes	—
104	C₄×C₄₃	#471	1	2	yes	—
105	C₄×C₄₃	#472	1	2	yes	—
106	#3138	—	43	1	no	—
107	#2986	—	43	1	no	—
108	#1561	D₄₃	1	1	yes	characteristic, complemented
109	#1545	—	43	2	no	—
110	#1299	—	43	2	no	—
111	C₂×C₄₃	—	4	1	no	—
112	C₂×C₄₃	#1888	1	1	yes	characteristic
113	C₂×C₄₃	#1907	1	2	yes	—
114	D₄₃	—	16	2	no	—
115	#533	C₂×D₄₃	1	1	yes	characteristic
116	C₂×#127	—	43	1	no	—
117	#490	—	43	4	no	—
118	#486	—	86	2	no	—
119	#446	—	86	2	no	—
120	C₈×Q₈	—	86	2	no	—
121	#433	—	86	2	no	—
122	#431	—	43	2	no	—
123	#423	C₂×D₄₃	1	2	yes	—
124	#418	—	86	2	no	—
125	C₄×M₁₆	—	43	1	no	—
126	#332	—	2	2	no	—
127	#331	—	86	2	no	—
128	#331	—	86	2	no	—
129	C₄₃	#9438	1	1	yes	characteristic, complemented, Sylow
130	C₂²×D₄	—	86	1	no	—
131	C₂²×D₄	—	86	2	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	86	1	no	—
134	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
135	#127	—	43	2	no	—
136	#127	—	86	2	no	—
137	#127	—	86	1	no	—
138	#125	C₄×D₄₃	1	2	yes	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	#117	D₁₇₂	1	2	yes	—
142	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	43	2	no	—
146	C₄⋊C₈	—	43	2	no	—
147	C₄⋊C₈	—	86	4	no	—
148	#102	—	86	8	no	—
149	C₂²⋊C₈	—	86	2	no	—
150	#97	—	43	2	no	—
151	#97	—	86	2	no	—
152	C₄×C₈	—	86	2	no	—
153	C₄×C₈	—	86	2	no	—
154	C₄×C₈	—	86	1	no	—
155	C₂⁴	—	172	1	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×D₄	—	86	2	no	—
158	C₂×D₄	—	86	4	no	—
159	C₂×D₄	—	344	2	no	—
160	C₂×D₄	—	344	2	no	—
161	C₂×D₄	—	172	4	no	—
162	C₂×D₄	—	172	1	no	—
163	C₂×D₄	—	172	2	no	—
164	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
165	C₂²×C₄	#10621900	1	2	yes	—
166	M₁₆	—	8	2	no	—
167	M₁₆	—	172	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	86	2	no	—
170	C₂×C₈	—	86	2	no	—
171	C₂×C₈	—	86	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
178	C₄²	D₄×D₄₃	1	2	yes	—
179	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
180	C₂³	—	344	4	no	—
181	C₂³	—	344	2	no	—
182	C₂³	—	172	2	no	—
183	C₂³	#13063309	1	1	yes	characteristic
184	Q₈	—	4	2	no	—
185	Q₈	—	4	2	no	—
186	D₄	—	344	4	no	—
187	D₄	—	344	2	no	—
188	D₄	—	344	2	no	—
189	D₄	—	344	2	no	—
190	D₄	—	172	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
196	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
197	C₂×C₄	#13063295	1	1	yes	characteristic
198	C₂×C₄	#13063297	1	2	yes	—
199	C₈	—	8	2	no	—
200	C₈	—	172	2	no	—
201	C₈	—	172	2	no	—
202	C₈	—	172	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	344	2	no	—
206	C₂²	—	344	4	no	—
207	C₂²	—	688	4	no	—
208	C₂²	#14582776	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	4	4	no	—
212	C₄	#14583074	1	2	yes	—
213	C₄	#14583164	1	2	yes	—
214	C₄	#14583167	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	688	2	no	—
217	C₂	#19134996	1	1	yes	characteristic
218	C₂	#19135101	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19144971

C₂

yes

maximal, characteristic

#19143697

C₂

yes

maximal, characteristic, complemented

#19133264

C₂

yes

maximal

#19132484

C₂

yes

maximal, complemented

C₄₃×#1561

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂²

yes

characteristic

#14583265

C₄

yes

—

#14583247

—

#14583030

—

Q₈×C₄₃⋊C₈

—

#14582960

—

#14582952

C₂²

yes

—

#14582903

—

C₄×#10621877

C₂²

yes

characteristic

#14582642

—

#14582642

—

C₄₃×#533

C₂²

yes

characteristic

C₄₃×#423

C₂²

yes

—

C₄₃×#332

—

C₂²×D₁₇₂

—

C₂²×D₁₇₂

—

—

—

—

C₂×C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

#13063245

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

#13063236

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₂×M₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

D₄

yes

—

#125×C₄₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₂³×D₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#123

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#9438

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#386

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

100

C₄×C₄₃

—

101

C₄×C₄₃

—

102

C₄×C₄₃

—

103

C₄×C₄₃

#454

yes

—

104

C₄×C₄₃

#471

yes

—

105

C₄×C₄₃

#472

yes

—

106

#3138

—

107

#2986

—

108

#1561

D₄₃

yes

characteristic, complemented

109

#1545

—

110

#1299

—

111

C₂×C₄₃

—

112

C₂×C₄₃

#1888

yes

characteristic

113

C₂×C₄₃

#1907

yes

—

114

D₄₃

—

115

#533

C₂×D₄₃

yes

characteristic

116

C₂×#127

—

117

#490

—

118

#486

—

119

#446

—

120

C₈×Q₈

—

121

#433

—

122

#431

—

123

#423

C₂×D₄₃

yes

—

124

#418

—

125

C₄×M₁₆

—

126

#332

—

127

#331

—

128

#331

—

129

C₄₃

#9438

yes

characteristic, complemented, Sylow

130

C₂²×D₄

—

131

C₂²×D₄

—

132

C₂×M₁₆

—

133

C₂×M₁₆

—

134

C₄⋊Q₈

C₂²⋊D₄₃

yes

—

135

#127

—

136

#127

—

137

#127

—

138

#125

C₄×D₄₃

yes

—

139

C₂²⋊Q₈

—

140

C₄×Q₈

—

141

#117

D₁₇₂

yes

—

142

C₂×C₄²

C₂²×D₄₃

yes

characteristic

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

#102

—

149

C₂²⋊C₈

—

150

#97

—

151

#97

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₂⁴

—

172

—

156

C₂×Q₈

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂×D₄

—

344

—

160

C₂×D₄

—

344

—

161

C₂×D₄

—

172

—

162

C₂×D₄

—

172

—

163

C₂×D₄

—

172

—

164

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

165

C₂²×C₄

#10621900

yes

—

166

M₁₆

—

167

M₁₆

—

172

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

(C₂×D₄₃)⋊C₄

yes

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

C₂×C₄×D₄₃

yes

characteristic

178

C₄²

D₄×D₄₃

yes

—

179

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

180

C₂³

—

344

—

181

C₂³

—

344

—

182

C₂³

—

172

—

183

C₂³

#13063309

yes

characteristic

184

Q₈

—

185

Q₈

—

186

D₄

—

344

—

187

D₄

—

344

—

188

D₄

—

344

—

189

D₄

—

344

—

190

D₄

—

172

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

196

C₂×C₄

(C₂²⋊D₄₃)⋊C₄

yes

—

197

C₂×C₄

#13063295

yes

characteristic

198

C₂×C₄

#13063297

yes

—

199

C₈

—

200

C₈

—

172

—

201

C₈

—

172

—

202

C₈

—

172

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

344

—

206

C₂²

—

344

—

207

C₂²

—

688

—

208

C₂²

#14582776

yes

characteristic

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

#14583074

yes

—

213

C₄

#14583164

yes

—

214

C₄

#14583167

yes

—

215

C₂

—

216

C₂

—

688

—

217

C₂

#19134996

yes

characteristic

218

C₂

#19135101

yes

—

	subgroup	G/H
center	C₂²	#14582776
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1561	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂×#13063322	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1561	D₄₃
hypocenter	C₄₃	#9438
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

subgroup

G/H

center

C₂²

#14582776

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1561

C₂

C₂²×C₄₃

C₂²

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

N	H ≅ G/N	type
2. #19143697	C₂	semidirect
4. #19132484	C₂	semidirect
5. C₄₃×#1561	C₂	semidirect
108. #1561	D₄₃	semidirect
129. C₄₃	#9438	semidirect

H ≅ G/N

type

2. #19143697

C₂

semidirect

4. #19132484

C₂

semidirect

5. C₄₃×#1561

C₂

semidirect

108. #1561

D₄₃

semidirect

129. C₄₃

#9438

semidirect

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

subgroup	index
#19144971	2
#19143697	2
#19133264	2
#19132484	2
C₄₃×#1561	2
#9438	43

subgroup

index

C₄₃×#1561

Subgroups — #24994209

Conjugacy classes of subgroups

Complements

Maximal subgroups