Group #25016454 — subgroups

Subgroups — #25016454

Order 11008.

342 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 105 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×(C₂×D₄₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#19134621	C₂	1	2	yes	maximal
3	#19132687	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132686	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#13063259	C₂²	1	2	yes	—
9	#14582756	—	2	2	no	—
10	(C₂×D₄₃)⋊C₁₆	—	2	2	no	—
11	#14582661	—	2	2	no	—
12	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×#107×C₄₃	C₂²	1	2	yes	—
14	C₄₃×#366	—	2	2	no	—
15	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₈×D₄₃	—	2	1	no	—
18	C₂×C₈×D₄₃	—	2	2	no	—
19	C₂×C₈×D₄₃	—	2	1	no	—
20	C₂×C₈×D₄₃	—	2	2	no	—
21	C₂×C₄⋊Dic₄₃	—	2	2	no	—
22	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
23	C₂×C₄₃⋊C₁₆	—	2	1	no	—
24	C₂×C₄₃⋊C₁₆	—	2	2	no	—
25	#13063259	—	2	2	no	—
26	#13063259	—	2	2	no	—
27	C₄₃⋊#95	—	4	2	no	—
28	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂×C₁₆×C₄₃	—	2	2	no	—
31	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
32	#107×C₄₃	—	2	2	no	—
33	#107×C₄₃	—	2	2	no	—
34	C₂³×D₄₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	—	4	2	no	—
36	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₃	—	4	2	no	—
38	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
39	C₂×C₄×D₄₃	—	2	1	no	—
40	C₂×C₄×D₄₃	—	2	2	no	—
41	C₂×C₄×D₄₃	—	2	1	no	—
42	C₂×C₄×D₄₃	—	2	2	no	—
43	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₄₃	#41	1	2	yes	—
45	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
46	C₂×C₈×C₄₃	#41	1	2	yes	—
47	C₄⋊C₄×C₄₃	—	4	2	no	—
48	C₄⋊C₄×C₄₃	—	4	2	no	—
49	(C₂×D₄₃)⋊C₄	—	8	4	no	—
50	C₄⋊Dic₄₃	—	4	2	no	—
51	C₄⋊Dic₄₃	—	4	2	no	—
52	C₂×C₄₃⋊C₈	—	2	1	no	—
53	C₂×C₄₃⋊C₈	—	2	2	no	—
54	C₈×D₄₃	—	4	1	no	—
55	C₈×D₄₃	—	4	1	no	—
56	C₈×D₄₃	—	4	2	no	—
57	C₁₆×C₄₃	—	2	2	no	—
58	C₄₃⋊C₁₆	—	4	1	no	—
59	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
60	C₂²×D₄₃	—	2	1	no	—
61	C₂²×D₄₃	—	2	2	no	—
62	C₂²×D₄₃	—	2	1	no	—
63	C₂²×D₄₃	—	2	2	no	—
64	C₂²×D₄₃	D₁₆	1	2	yes	—
65	C₂×C₄×C₄₃	—	8	2	no	—
66	C₂×C₄×C₄₃	—	8	2	no	—
67	C₂×C₄×C₄₃	—	8	2	no	—
68	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#123	1	2	yes	—
70	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
71	C₂×C₄×C₄₃	#137	1	2	yes	—
72	C₂×Dic₄₃	—	2	1	no	—
73	C₂×Dic₄₃	—	2	2	no	—
74	C₂×Dic₄₃	—	8	2	no	—
75	C₂×Dic₄₃	—	8	2	no	—
76	C₂×Dic₄₃	—	8	2	no	—
77	C₄×D₄₃	—	4	1	no	—
78	C₄×D₄₃	—	4	1	no	—
79	C₄×D₄₃	—	4	2	no	—
80	C₈×C₄₃	#123	1	1	yes	characteristic
81	C₈×C₄₃	#124	1	1	yes	characteristic
82	C₈×C₄₃	#126	1	2	yes	—
83	C₄₃⋊C₈	—	4	1	no	—
84	#13492	—	43	1	no	maximal, Sylow
85	C₂²×C₄₃	#466	1	1	yes	characteristic
86	C₂²×C₄₃	#480	1	2	yes	—
87	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
88	C₂²×C₄₃	#508	1	1	yes	characteristic
89	C₂²×C₄₃	#510	1	2	yes	—
90	C₂×D₄₃	—	2	2	no	—
91	C₂×D₄₃	—	2	4	no	—
92	C₂×D₄₃	—	4	1	no	—
93	C₂×D₄₃	—	4	1	no	—
94	C₂×D₄₃	—	4	2	no	—
95	C₄×C₄₃	—	16	2	no	—
96	C₄×C₄₃	#397	1	1	yes	characteristic
97	C₄×C₄₃	#470	1	1	yes	characteristic
98	C₄×C₄₃	#485	1	2	yes	—
99	Dic₄₃	—	4	1	no	—
100	Dic₄₃	—	16	2	no	—
101	C₂×#366	D₄₃	1	1	yes	characteristic, complemented
102	C₂×C₂²⋊C₁₆	—	43	1	no	—
103	#1808	—	43	2	no	—
104	#1380	—	43	1	no	—
105	#1380	—	43	2	no	—
106	C₂×C₄₃	#2075	1	1	yes	characteristic
107	C₂×C₄₃	#2215	1	1	yes	characteristic
108	C₂×C₄₃	#2219	1	1	yes	characteristic
109	C₂×C₄₃	#2232	1	2	yes	—
110	C₂×C₄₃	#2237	1	2	yes	—
111	D₄₃	—	4	2	no	—
112	C₂³×C₈	—	43	1	no	—
113	C₂²×C₁₆	—	43	1	no	—
114	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
115	C₂×#107	—	43	2	no	—
116	C₂×#107	C₂×D₄₃	1	2	yes	—
117	#380	—	86	2	no	—
118	#366	—	2	2	no	—
119	C₂²⋊C₁₆	—	86	2	no	—
120	#340	—	86	2	no	—
121	C₄₃	#13492	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	—	43	1	no	—
123	C₂²×C₈	—	43	1	no	—
124	C₂²×C₈	—	86	2	no	—
125	C₂²×C₈	—	86	1	no	—
126	C₂²×C₈	—	86	2	no	—
127	C₂²×C₈	—	86	1	no	—
128	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	86	2	no	—
131	C₂×C₂²⋊C₄	—	172	2	no	—
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	86	2	no	—
134	C₂×C₁₆	—	86	1	no	—
135	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	—	86	2	no	—
139	#107	—	86	2	no	—
140	#95	—	172	2	no	—
141	C₂⁴	—	43	1	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	43	1	no	—
144	C₂²×C₄	—	86	1	no	—
145	C₂²×C₄	—	86	2	no	—
146	C₂²×C₄	—	86	1	no	—
147	C₂²×C₄	—	86	2	no	—
148	C₂²×C₄	—	172	2	no	—
149	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
150	C₂×C₈	—	86	1	no	—
151	C₂×C₈	—	86	2	no	—
152	C₂×C₈	—	172	1	no	—
153	C₂×C₈	—	172	1	no	—
154	C₂×C₈	—	172	2	no	—
155	C₂×C₈	#10621898	1	2	yes	—
156	C₂×C₈	#10621900	1	1	yes	characteristic
157	C₂×C₈	#10621902	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	172	2	no	—
162	C₄⋊C₄	—	172	2	no	—
163	C₂²⋊C₄	—	344	4	no	—
164	C₁₆	—	2	2	no	—
165	C₁₆	—	172	1	no	—
166	C₂³	—	43	2	no	—
167	C₂³	—	86	1	no	—
168	C₂³	—	86	2	no	—
169	C₂³	—	86	1	no	—
170	C₂³	—	86	2	no	—
171	C₂³	D₈×D₄₃	1	1	yes	characteristic
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	86	1	no	—
176	C₂×C₄	—	86	2	no	—
177	C₂×C₄	—	344	2	no	—
178	C₂×C₄	—	344	2	no	—
179	C₂×C₄	—	344	2	no	—
180	C₂×C₄	—	172	1	no	—
181	C₂×C₄	—	172	1	no	—
182	C₂×C₄	—	172	2	no	—
183	C₂×C₄	#13063296	1	1	yes	characteristic
184	C₂×C₄	#13063310	1	2	yes	—
185	C₂×C₄	#13063344	1	1	yes	characteristic
186	C₂×C₄	#13063357	1	2	yes	—
187	C₈	—	172	1	no	—
188	C₈	#13063308	1	1	yes	characteristic
189	C₈	#13063314	1	1	yes	characteristic
190	C₈	#13063320	1	2	yes	—
191	C₂²	—	86	4	no	—
192	C₂²	—	86	2	no	—
193	C₂²	—	172	1	no	—
194	C₂²	—	172	2	no	—
195	C₂²	—	172	1	no	—
196	C₂²	#14583132	1	1	yes	characteristic
197	C₂²	#14583214	1	2	yes	—
198	C₂²	D₁₆×D₄₃	1	1	yes	characteristic
199	C₂²	#14583328	1	1	yes	characteristic
200	C₂²	#14583341	1	2	yes	—
201	C₄	—	16	2	no	—
202	C₄	—	172	1	no	—
203	C₄	—	688	2	no	—
204	C₄	#14582825	1	1	yes	characteristic
205	C₄	#14583154	1	1	yes	characteristic
206	C₄	#14583242	1	2	yes	—
207	C₂	—	172	2	no	—
208	C₂	#19136019	1	1	yes	characteristic
209	C₂	#19136678	1	1	yes	characteristic
210	C₂	#19136700	1	1	yes	characteristic
211	C₂	#19136777	1	2	yes	—
212	C₂	#19136805	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×(C₂×D₄₃)⋊C₁₆

C₂

yes

maximal, characteristic

#19134621

C₂

yes

maximal

#19132687

C₂

yes

maximal, characteristic, complemented

#19132686

C₂

yes

maximal, complemented

C₂×C₄₃×#366

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₂×#13063259

C₂²

yes

—

#14582756

—

(C₂×D₄₃)⋊C₁₆

—

#14582661

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#107×C₄₃

C₂²

yes

—

C₄₃×#366

—

C₂²×C₄×D₄₃

D₄

yes

characteristic

C₂²×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₈×D₄₃

—

C₂×C₈×D₄₃

—

C₂×C₈×D₄₃

—

C₂×C₈×D₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×(C₂×D₄₃)⋊C₄

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

#13063259

—

#13063259

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

—

#107×C₄₃

—

#107×C₄₃

—

C₂³×D₄₃

D₈

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₈

yes

characteristic

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₈×D₄₃

—

C₈×D₄₃

—

C₈×D₄₃

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

D₁₆

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

#135

yes

characteristic

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

yes

C₈×C₄₃

yes

C₈×C₄₃

yes

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

C₂×D₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

100

Dic₄₃

—

101

C₂×#366

D₄₃

yes

characteristic, complemented

102

C₂×C₂²⋊C₁₆

—

103

#1808

—

104

#1380

—

105

#1380

—

106

C₂×C₄₃

#2075

yes

characteristic

107

C₂×C₄₃

#2215

yes

characteristic

108

C₂×C₄₃

#2219

yes

characteristic

109

C₂×C₄₃

#2232

yes

—

110

C₂×C₄₃

#2237

yes

—

111

D₄₃

—

112

C₂³×C₈

—

113

C₂²×C₁₆

—

114

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

115

C₂×#107

—

116

C₂×#107

C₂×D₄₃

yes

—

117

#380

—

118

#366

—

119

C₂²⋊C₁₆

—

120

#340

—

121

C₄₃

#13492

yes

characteristic, complemented, Sylow

122

C₂³×C₄

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²×D₄₃

yes

characteristic

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₂²⋊C₄

—

172

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₂×C₁₆

C₂²⋊D₄₃

yes

—

136

#107

—

137

#107

—

138

#107

—

139

#107

—

140

#95

—

172

—

141

C₂⁴

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

172

—

149

C₂²×C₄

D₄×D₄₃

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

172

—

153

C₂×C₈

—

172

—

154

C₂×C₈

—

172

—

155

C₂×C₈

#10621898

yes

—

156

C₂×C₈

#10621900

yes

characteristic

157

C₂×C₈

#10621902

yes

—

158

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

172

—

162

C₄⋊C₄

—

172

—

163

C₂²⋊C₄

—

344

—

164

C₁₆

—

165

C₁₆

—

172

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

D₈×D₄₃

yes

characteristic

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

344

—

178

C₂×C₄

—

344

—

179

C₂×C₄

—

344

—

180

C₂×C₄

—

172

—

181

C₂×C₄

—

172

—

182

C₂×C₄

—

172

—

183

C₂×C₄

#13063296

yes

characteristic

184

C₂×C₄

#13063310

yes

—

185

C₂×C₄

#13063344

yes

characteristic

186

C₂×C₄

#13063357

yes

—

187

C₈

—

172

—

188

C₈

#13063308

yes

characteristic

189

C₈

#13063314

yes

characteristic

190

C₈

#13063320

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

172

—

194

C₂²

—

172

—

195

C₂²

—

172

—

196

C₂²

#14583132

yes

characteristic

197

C₂²

#14583214

yes

—

198

C₂²

D₁₆×D₄₃

yes

characteristic

199

C₂²

#14583328

yes

characteristic

200

C₂²

#14583341

yes

—

201

C₄

—

202

C₄

—

172

—

203

C₄

—

688

—

204

C₄

#14582825

yes

characteristic

205

C₄

#14583154

yes

characteristic

206

C₄

#14583242

yes

—

207

C₂

—

172

—

208

C₂

#19136019

yes

characteristic

209

C₂

#19136678

yes

characteristic

210

C₂

#19136700

yes

characteristic

211

C₂

#19136777

yes

—

212

C₂

#19136805

yes

—

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#366	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂³×D₄₃	D₈
hypercenter	C₂×#366	D₄₃
hypocenter	C₄₃	#13492
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₄₃

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₄₃×#366

C₂

socle

C₂³×C₄₃

C₂×D₈

prime radical

C₂³×D₄₃

D₈

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
3. #19132687	C₂	semidirect
4. #19132686	C₂	semidirect
5. C₂×C₄₃×#366	C₂	semidirect
101. C₂×#366	D₄₃	semidirect
121. C₄₃	#13492	semidirect

H ≅ G/N

type

3. #19132687

C₂

semidirect

4. #19132686

C₂

semidirect

5. C₂×C₄₃×#366

C₂

semidirect

101. C₂×#366

D₄₃

semidirect

121. C₄₃

#13492

semidirect

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

subgroup	index
C₂×(C₂×D₄₃)⋊C₁₆	2
#19134621	2
#19132687	2
#19132686	2
C₂×C₄₃×#366	2
#13492	43

subgroup

index

C₂×(C₂×D₄₃)⋊C₁₆

C₂×C₄₃×#366

Subgroups — #25016454

Conjugacy classes of subgroups

Complements

Maximal subgroups