Group #24989622 — subgroups

Subgroups — #24989622

Order 11008.

404 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 101 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	#19143306	C₂	1	1	yes	maximal, characteristic
2	#19133353	C₂	1	2	yes	maximal, complemented
3	#19133328	C₂	1	2	yes	maximal
4	C₄₃⋊#1309	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063333	C₄	1	2	yes	—
7	#14583449	C₂²	1	1	yes	characteristic
8	#14582976	—	2	2	no	—
9	#14582964	—	2	2	no	—
10	#14582923	C₂²	1	2	yes	—
11	#14582867	C₄	1	2	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	#14582651	—	2	2	no	—
14	#14582651	—	2	2	no	—
15	#14582647	—	2	2	no	—
16	#14582642	—	2	2	no	—
17	C₄₃×#423	C₂²	1	2	yes	—
18	C₄₃×#423	C₂²	1	1	yes	characteristic
19	C₂×#10621898	C₂×C₄	1	2	yes	—
20	#13063333	—	4	4	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621872	—	2	2	no	—
23	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₄₃	—	2	2	no	—
26	#13063325	D₄	1	1	yes	characteristic
27	#13063322	D₄	1	1	yes	characteristic
28	#13063313	Q₈	1	2	yes	—
29	#13063299	—	2	2	no	—
30	C₄×D₁₇₂	—	2	2	no	—
31	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
33	(C₂×D₄₃)⋊C₈	—	2	4	no	—
34	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
35	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
36	C₂×M₁₆×C₄₃	—	2	2	no	—
37	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₃	—	2	2	no	—
40	C₄⋊C₈×C₄₃	—	2	2	no	—
41	C₄⋊C₈×C₄₃	—	2	2	no	—
42	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	#41	1	2	yes	—
44	C₂×C₂²⋊D₄₃	—	2	2	no	—
45	#10621898	—	4	2	no	—
46	#10621898	—	4	2	no	—
47	C₂×D₁₇₂	—	2	2	no	—
48	C₂×D₁₇₂	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×D₄₃	—	2	2	no	—
50	C₂×Dic₈₆	C₄⋊C₄	1	2	yes	—
51	M₁₆×C₄₃	—	4	2	no	—
52	M₁₆×C₄₃	—	8	2	no	—
53	C₂×C₈×C₄₃	—	2	2	no	—
54	C₂×C₈×C₄₃	—	4	2	no	—
55	C₄²×C₄₃	C₂×D₄	1	2	yes	—
56	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
58	(C₂×D₄₃)⋊C₄	—	2	4	no	—
59	C₄⋊Dic₄₃	—	2	2	no	—
60	#10621877	—	8	2	no	—
61	C₂×C₄₃⋊C₈	—	2	2	no	—
62	C₂×C₄₃⋊C₈	—	4	1	no	—
63	C₂×C₄₃⋊C₈	—	4	2	no	—
64	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
67	#10621872	—	4	4	no	—
68	C₈×D₄₃	—	4	4	no	—
69	C₂³×C₄₃	#117	1	1	yes	characteristic
70	C₂²×D₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	2	1	no	—
72	C₂×C₄×C₄₃	—	2	1	no	—
73	C₂×C₄×C₄₃	—	2	2	no	—
74	C₂×C₄×C₄₃	—	2	2	no	—
75	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#121	1	2	yes	—
78	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
79	C₂²⋊D₄₃	—	4	4	no	—
80	C₂×Dic₄₃	—	2	2	no	—
81	D₁₇₂	—	4	2	no	—
82	D₁₇₂	—	4	2	no	—
83	D₁₇₂	—	4	2	no	—
84	C₄×D₄₃	—	4	2	no	—
85	C₄×D₄₃	—	4	2	no	—
86	Dic₈₆	—	4	2	no	—
87	C₈×C₄₃	—	4	2	no	—
88	C₈×C₄₃	—	8	2	no	—
89	C₄₃⋊C₈	—	2	1	no	—
90	C₄₃⋊C₈	—	2	2	no	—
91	C₄₃⋊C₈	—	8	2	no	—
92	C₄₃⋊C₈	#100	1	2	yes	—
93	#8638	—	43	1	no	maximal, Sylow
94	C₂²×C₄₃	—	2	1	no	—
95	C₂²×C₄₃	—	2	2	no	—
96	C₂²×C₄₃	#380	1	1	yes	characteristic
97	C₂×D₄₃	—	4	2	no	—
98	C₂×D₄₃	—	4	4	no	—
99	C₄×C₄₃	—	2	1	no	—
100	C₄×C₄₃	—	2	2	no	—
101	C₄×C₄₃	C₂×#100	1	1	yes	characteristic
102	C₄×C₄₃	#413	1	1	yes	characteristic
103	C₄×C₄₃	#472	1	2	yes	—
104	C₄×C₄₃	#473	1	2	yes	—
105	Dic₄₃	—	4	2	no	—
106	#2931	—	43	1	no	—
107	#1567	—	43	2	no	—
108	#1560	—	43	2	no	—
109	#1309	—	43	1	no	—
110	#1309	D₄₃	1	1	yes	characteristic, complemented
111	C₂×C₄₃	—	4	1	no	—
112	C₂×C₄₃	#1800	1	1	yes	characteristic
113	C₂×C₄₃	#1816	1	2	yes	—
114	D₄₃	—	8	2	no	—
115	C₂×#131	—	43	2	no	—
116	#532	—	43	1	no	—
117	#436	—	86	2	no	—
118	C₈×D₄	—	86	2	no	—
119	#423	—	43	2	no	—
120	#423	C₂×D₄₃	1	2	yes	—
121	#423	C₂×D₄₃	1	1	yes	characteristic
122	#408	—	43	2	no	—
123	C₂×C₄×C₈	—	43	1	no	—
124	#335	—	86	2	no	—
125	#335	—	86	2	no	—
126	#333	—	86	2	no	—
127	#331	—	86	2	no	—
128	C₄₃	#8638	1	1	yes	characteristic, complemented, Sylow
129	C₂×#41	—	43	2	no	—
130	#131	—	172	4	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	86	2	no	—
133	C₂×M₁₆	—	86	2	no	—
134	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
135	C₂²×C₈	—	43	1	no	—
136	C₂²×C₈	—	43	1	no	—
137	C₂²×C₈	—	86	2	no	—
138	C₄⋊Q₈	—	43	1	no	—
139	#127	—	43	1	no	—
140	#124	—	43	2	no	—
141	#121	—	86	2	no	—
142	C₄×D₄	—	86	2	no	—
143	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	86	2	no	—
148	C₄⋊C₈	—	86	2	no	—
149	C₂²⋊C₈	—	86	4	no	—
150	C₄×C₈	—	43	2	no	—
151	C₄×C₈	—	43	2	no	—
152	#41	—	172	2	no	—
153	#41	—	172	2	no	—
154	C₂×Q₈	—	43	2	no	—
155	C₂×D₄	—	43	2	no	—
156	C₂×D₄	—	86	2	no	—
157	C₂×D₄	—	86	2	no	—
158	C₂²×C₄	—	86	2	no	—
159	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
160	C₂²×C₄	#10621898	1	2	yes	—
161	M₁₆	—	4	2	no	—
162	M₁₆	—	8	2	no	—
163	M₁₆	—	344	2	no	—
164	M₁₆	—	172	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	43	1	no	—
168	C₂×C₈	—	43	2	no	—
169	C₂×C₈	—	43	1	no	—
170	C₂×C₈	—	86	2	no	—
171	C₂×C₈	—	172	4	no	—
172	C₂×C₈	—	172	1	no	—
173	C₂×C₈	—	172	2	no	—
174	C₄⋊C₄	—	86	2	no	—
175	C₂²⋊C₄	—	86	4	no	—
176	C₄²	D₄×D₄₃	1	2	yes	—
177	C₄²	D₄×D₄₃	1	1	yes	characteristic
178	C₄²	Q₈×D₄₃	1	1	yes	characteristic
179	C₂³	—	86	2	no	—
180	C₂³	#13063282	1	1	yes	characteristic
181	Q₈	—	172	2	no	—
182	D₄	—	172	2	no	—
183	D₄	—	172	4	no	—
184	D₄	—	172	2	no	—
185	D₄	—	172	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	86	2	no	—
191	C₂×C₄	—	172	2	no	—
192	C₂×C₄	—	172	2	no	—
193	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
194	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
195	C₂×C₄	#13063298	1	2	yes	—
196	C₂×C₄	#13063303	1	2	yes	—
197	C₈	—	4	2	no	—
198	C₈	—	8	2	no	—
199	C₈	—	43	2	no	—
200	C₈	—	86	1	no	—
201	C₈	—	86	2	no	—
202	C₈	—	344	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	172	2	no	—
206	C₂²	—	172	4	no	—
207	C₂²	#14582758	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	2	no	—
210	C₄	—	172	2	no	—
211	C₄	#100×D₄₃	1	1	yes	characteristic
212	C₄	#14582882	1	1	yes	characteristic
213	C₄	#14583168	1	2	yes	—
214	C₄	#14583173	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	344	2	no	—
217	C₂	#19134586	1	1	yes	characteristic
218	C₂	#19134658	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19143306

C₂

yes

maximal, characteristic

#19133353

C₂

yes

maximal, complemented

#19133328

C₂

yes

maximal

C₄₃⋊#1309

C₂

yes

maximal, characteristic, complemented

C₄₃×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#13063333

C₄

yes

—

#14583449

C₂²

yes

—

—

C₂²

yes

—

#14582867

C₄

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

—

—

—

C₄₃×#423

C₂²

yes

—

C₄₃×#423

C₂²

yes

characteristic

C₂×#10621898

C₂×C₄

yes

—

#13063333

—

C₂×#10621877

—

C₂×#10621872

—

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈×D₄₃

—

#13063325

D₄

yes

characteristic

#13063322

D₄

yes

characteristic

#13063313

Q₈

yes

—

#13063299

—

C₄×D₁₇₂

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

(C₂×D₄₃)⋊C₈

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂×M₁₆×C₄₃

—

C₂×M₁₆×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₄⋊C₄

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

C₄⋊C₄

yes

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

#10621872

—

C₈×D₄₃

—

C₂³×C₄₃

#117

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

#100

yes

—

#8638

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#380

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

100

C₄×C₄₃

—

101

C₄×C₄₃

C₂×#100

yes

characteristic

102

C₄×C₄₃

#413

yes

characteristic

103

C₄×C₄₃

#472

yes

—

104

C₄×C₄₃

#473

yes

—

105

Dic₄₃

—

106

#2931

—

107

#1567

—

108

#1560

—

109

#1309

—

110

#1309

D₄₃

yes

characteristic, complemented

111

C₂×C₄₃

—

112

C₂×C₄₃

#1800

yes

characteristic

113

C₂×C₄₃

#1816

yes

—

114

D₄₃

—

115

C₂×#131

—

116

#532

—

117

#436

—

118

C₈×D₄

—

119

#423

—

120

#423

C₂×D₄₃

yes

—

121

#423

C₂×D₄₃

yes

characteristic

122

#408

—

123

C₂×C₄×C₈

—

124

#335

—

125

#335

—

126

#333

—

127

#331

—

128

C₄₃

#8638

yes

characteristic, complemented, Sylow

129

C₂×#41

—

130

#131

—

172

—

131

C₂×M₁₆

—

132

C₂×M₁₆

—

133

C₂×M₁₆

—

134

C₂×M₁₆

C₄×D₄₃

yes

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

138

C₄⋊Q₈

—

139

#127

—

140

#124

—

141

#121

—

142

C₄×D₄

—

143

C₂×C₄²

C₂²×D₄₃

yes

characteristic

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

149

C₂²⋊C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

#41

—

172

—

153

#41

—

172

—

154

C₂×Q₈

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂²×C₄

—

159

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

160

C₂²×C₄

#10621898

yes

—

161

M₁₆

—

162

M₁₆

—

163

M₁₆

—

344

—

164

M₁₆

—

172

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

—

172

C₂×C₈

—

172

—

173

C₂×C₈

—

172

—

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

D₄×D₄₃

yes

—

177

C₄²

D₄×D₄₃

yes

characteristic

178

C₄²

Q₈×D₄₃

yes

characteristic

179

C₂³

—

180

C₂³

#13063282

yes

characteristic

181

Q₈

—

172

—

182

D₄

—

172

—

183

D₄

—

172

—

184

D₄

—

172

—

185

D₄

—

172

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

172

—

192

C₂×C₄

—

172

—

193

C₂×C₄

C₂²⋊C₄×D₄₃

yes

characteristic

194

C₂×C₄

C₄⋊C₄×D₄₃

yes

characteristic

195

C₂×C₄

#13063298

yes

—

196

C₂×C₄

#13063303

yes

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

344

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

172

—

206

C₂²

—

172

—

207

C₂²

#14582758

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

172

—

211

C₄

#100×D₄₃

yes

characteristic

212

C₄

#14582882

yes

characteristic

213

C₄

#14583168

yes

—

214

C₄

#14583173

yes

—

215

C₂

—

216

C₂

—

344

—

217

C₂

#19134586

yes

characteristic

218

C₂

#19134658

yes

—

	subgroup	G/H
center	C₂²	#14582758
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1309	C₂
socle	C₂²×C₄₃	#380
prime radical	#14583449	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1309	D₄₃
hypocenter	C₄₃	#8638
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#14582758

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1309

C₂

C₂²×C₄₃

C₂²

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
2. #19133353	C₂	semidirect
4. C₄₃⋊#1309	C₂	semidirect
5. C₄₃×#1309	C₂	semidirect
110. #1309	D₄₃	semidirect
128. C₄₃	#8638	semidirect

H ≅ G/N

type

2. #19133353

C₂

semidirect

4. C₄₃⋊#1309

C₂

semidirect

5. C₄₃×#1309

C₂

semidirect

110. #1309

D₄₃

semidirect

128. C₄₃

#8638

semidirect

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

subgroup	index
#19143306	2
#19133353	2
#19133328	2
C₄₃⋊#1309	2
C₄₃×#1309	2
#8638	43

subgroup

index

#19143306

#19133353

#19133328

C₄₃⋊#1309

C₄₃×#1309

#8638

Subgroups — #24989622

Conjugacy classes of subgroups

Complements

Maximal subgroups