Group #25015086 — subgroups

Subgroups — #25015086

Order 11008.

566 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 143 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	#19136351	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14582712	C₂	1	4	yes	maximal, complemented
3	C₄×#13063261	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063265	C₂²	1	2	yes	—
6	C₂×#13063261	C₂²	1	2	yes	—
7	C₄×#10621877	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
9	#14582712	—	2	8	no	—
10	#14582664	—	2	1	no	—
11	C₂²×D₈×C₄₃	—	2	1	no	—
12	C₂²×D₈×C₄₃	C₂²	1	2	yes	—
13	C₂×#127×C₄₃	—	2	1	no	—
14	C₄₃×#493	—	2	2	no	—
15	C₄₃×#493	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#10621877	—	2	2	no	—
18	#13063265	—	2	4	no	—
19	C₂×C₄₃⋊C₁₆	—	2	2	no	—
20	#13063261	—	2	2	no	—
21	#13063261	D₄	1	4	yes	complemented
22	#13063236	—	4	1	no	—
23	C₄×C₄₃⋊C₈	—	4	1	no	—
24	C₂²×D₄×C₄₃	—	2	2	no	—
25	C₂²×D₄×C₄₃	—	4	2	no	—
26	C₂×D₈×C₄₃	—	2	4	no	—
27	C₂×D₈×C₄₃	—	2	4	no	—
28	C₂×D₈×C₄₃	—	2	2	no	—
29	C₂×D₈×C₄₃	—	2	4	no	—
30	C₂×D₈×C₄₃	—	4	4	no	—
31	C₂×D₈×C₄₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
34	#127×C₄₃	—	2	4	no	—
35	#127×C₄₃	—	4	1	no	—
36	#127×C₄₃	—	4	1	no	—
37	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₃	D₄	1	2	yes	—
40	C₂⁴×C₄₃	—	4	2	no	—
41	C₂×D₄×C₄₃	—	2	4	no	—
42	C₂×D₄×C₄₃	—	4	4	no	—
43	C₂×D₄×C₄₃	—	4	4	no	—
44	C₂×D₄×C₄₃	—	4	2	no	—
45	C₂×D₄×C₄₃	—	4	4	no	—
46	C₂×D₄×C₄₃	—	4	4	no	—
47	C₂×D₄×C₄₃	—	8	8	no	—
48	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	D₈×C₄₃	—	2	8	no	—
51	D₈×C₄₃	—	4	8	no	—
52	D₈×C₄₃	—	4	2	no	—
53	D₈×C₄₃	—	4	2	no	—
54	C₂×C₈×C₄₃	—	2	2	no	—
55	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₃	#41	1	2	yes	—
60	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₄₃	D₈	1	1	yes	characteristic
62	C₄²×C₄₃	D₈	1	1	yes	characteristic
63	#10621877	—	2	4	no	—
64	#10621877	—	4	2	no	—
65	C₂×C₄₃⋊C₈	—	4	2	no	—
66	C₄₃⋊C₁₆	—	4	2	no	—
67	C₂³×C₄₃	—	4	4	no	—
68	C₂³×C₄₃	—	8	8	no	—
69	C₂³×C₄₃	—	8	4	no	—
70	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
71	D₄×C₄₃	—	4	4	no	—
72	D₄×C₄₃	—	4	4	no	—
73	D₄×C₄₃	—	8	8	no	—
74	D₄×C₄₃	—	8	8	no	—
75	D₄×C₄₃	—	8	4	no	—
76	C₂×C₄×C₄₃	—	2	4	no	—
77	C₂×C₄×C₄₃	#102	1	2	yes	—
78	C₂×C₄×C₄₃	#102	1	2	yes	—
79	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#121	1	2	yes	—
81	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₈×C₄₃	—	2	2	no	—
84	C₈×C₄₃	C₄×D₄	1	2	yes	—
85	C₈×C₄₃	#124	1	1	yes	characteristic
86	C₈×C₄₃	#127	1	1	yes	characteristic
87	C₄₃⋊C₈	—	4	4	no	—
88	#13257	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	—	2	2	no	—
90	C₂²×C₄₃	—	8	8	no	—
91	C₂²×C₄₃	—	8	4	no	—
92	C₂²×C₄₃	—	16	8	no	—
93	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
94	C₂²×C₄₃	#460	1	2	yes	—
95	C₄×C₄₃	—	2	2	no	—
96	C₄×C₄₃	#361	1	4	yes	complemented
97	C₄×C₄₃	#390	1	1	yes	characteristic
98	C₄×C₄₃	#418	1	1	yes	characteristic
99	C₄×C₄₃	#459	1	2	yes	—
100	C₂×#493	D₄₃	1	1	yes	characteristic
101	#2150	—	43	1	no	—
102	C₂×#361	—	43	4	no	—
103	C₄×#108	—	43	1	no	—
104	C₂×C₄₃	—	2	2	no	—
105	C₂×C₄₃	—	16	4	no	—
106	C₂×C₄₃	#1982	1	1	yes	characteristic
107	C₂×C₄₃	C₂×#361	1	2	yes	—
108	C₂²×D₈	—	2	1	no	—
109	C₂²×D₈	C₂×D₄₃	1	2	yes	—
110	C₂×#127	—	2	1	no	—
111	C₂×M₃₂	—	43	2	no	—
112	#493	—	2	2	no	—
113	#493	Dic₄₃	1	4	yes	—
114	C₂×#108	—	43	2	no	—
115	C₄×M₁₆	—	86	1	no	—
116	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
117	C₄⋊C₁₆	—	86	2	no	—
118	#361	—	86	8	no	—
119	#341	—	86	1	no	—
120	C₄₃	#13257	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₄	—	2	2	no	—
122	C₂²×D₄	—	4	2	no	—
123	C₂×D₈	—	2	4	no	—
124	C₂×D₈	—	2	4	no	—
125	C₂×D₈	—	2	4	no	—
126	C₂×D₈	—	2	2	no	—
127	C₂×D₈	—	4	4	no	—
128	C₂×D₈	C₂×Dic₄₃	1	4	yes	—
129	C₂×M₁₆	—	86	2	no	—
130	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
132	#127	—	2	4	no	—
133	#127	—	4	1	no	—
134	#127	—	4	1	no	—
135	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
136	M₃₂	—	86	4	no	—
137	C₂×C₁₆	—	86	2	no	—
138	#108	—	43	4	no	—
139	#108	—	86	2	no	—
140	#97	—	172	1	no	—
141	C₄×C₈	—	172	1	no	—
142	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
143	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
144	C₂⁴	—	4	2	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	4	4	no	—
147	C₂×D₄	—	4	4	no	—
148	C₂×D₄	—	4	4	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	8	8	no	—
152	C₂²×C₄	#10621883	1	2	yes	—
153	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
154	D₈	—	2	8	no	—
155	D₈	—	4	8	no	—
156	D₈	—	4	2	no	—
157	D₈	—	4	2	no	—
158	M₁₆	—	86	4	no	—
159	M₁₆	—	172	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	172	2	no	—
162	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
163	C₂×C₈	D₄×D₄₃	1	2	yes	—
164	C₂×C₈	#10621900	1	2	yes	—
165	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
167	C₄²	#10621882	1	1	yes	characteristic
168	C₄²	#10621882	1	1	yes	characteristic
169	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
170	C₁₆	—	172	2	no	—
171	C₂³	—	4	4	no	—
172	C₂³	—	8	8	no	—
173	C₂³	—	8	4	no	—
174	C₂³	C₂×#10621883	1	1	yes	characteristic
175	D₄	—	4	4	no	—
176	D₄	—	4	4	no	—
177	D₄	—	8	8	no	—
178	D₄	—	8	4	no	—
179	D₄	—	8	8	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	#13063246	1	2	yes	—
182	C₂×C₄	#13063246	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
184	C₂×C₄	#13063296	1	2	yes	—
185	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
186	C₂×C₄	#13063351	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	172	4	no	—
189	C₈	D₄×Dic₄₃	1	2	yes	—
190	C₈	#13063312	1	1	yes	characteristic
191	C₈	#13063321	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	8	4	no	—
194	C₂²	—	8	8	no	—
195	C₂²	—	16	8	no	—
196	C₂²	C₂×#13063246	1	1	yes	characteristic
197	C₂²	#14583096	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	#14582712	1	4	yes	complemented
200	C₄	#14582802	1	1	yes	characteristic
201	C₄	#14582904	1	1	yes	characteristic
202	C₄	#14583091	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	16	4	no	—
205	C₂	#19135523	1	1	yes	characteristic
206	C₂	C₂×#14582712	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136351

C₂

yes

maximal, characteristic, complemented

C₂×#14582712

C₂

yes

maximal, complemented

C₄×#13063261

C₂

yes

maximal, characteristic, complemented

C₂×C₄₃×#493

C₂

yes

maximal, characteristic

C₂×#13063265

C₂²

yes

—

C₂×#13063261

C₂²

yes

—

C₄×#10621877

—

C₄⋊(C₄₃⋊C₁₆)

—

#14582712

—

#14582664

—

C₂²×D₈×C₄₃

—

C₂²×D₈×C₄₃

C₂²

yes

—

C₂×#127×C₄₃

—

C₄₃×#493

—

C₄₃×#493

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

#13063261

—

#13063261

D₄

yes

complemented

#13063236

—

C₄×C₄₃⋊C₈

—

C₂²×D₄×C₄₃

—

C₂²×D₄×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

C₂×C₄

yes

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

#127×C₄₃

—

#127×C₄₃

—

#127×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂⁴×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

SD₁₆

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

D₈×C₄₃

—

D₈×C₄₃

—

D₈×C₄₃

—

D₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

D₈

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂×SD₁₆

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₄×D₄

yes

—

C₈×C₄₃

yes

C₈×C₄₃

yes

C₄₃⋊C₈

—

#13257

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#102

yes

characteristic

C₂²×C₄₃

#460

yes

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

100

C₂×#493

D₄₃

yes

characteristic

101

#2150

—

102

C₂×#361

—

103

C₄×#108

—

104

C₂×C₄₃

—

105

C₂×C₄₃

—

106

C₂×C₄₃

#1982

yes

characteristic

107

C₂×C₄₃

C₂×#361

yes

—

108

C₂²×D₈

—

109

C₂²×D₈

C₂×D₄₃

yes

—

110

C₂×#127

—

111

C₂×M₃₂

—

112

#493

—

113

#493

Dic₄₃

yes

—

114

C₂×#108

—

115

C₄×M₁₆

—

116

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

117

C₄⋊C₁₆

—

118

#361

—

119

#341

—

120

C₄₃

#13257

yes

characteristic, complemented, Sylow

121

C₂²×D₄

—

122

C₂²×D₄

—

123

C₂×D₈

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×D₈

—

127

C₂×D₈

—

128

C₂×D₈

C₂×Dic₄₃

yes

—

129

C₂×M₁₆

—

130

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

131

C₂²×C₈

C₂²×D₄₃

yes

characteristic

132

#127

—

133

#127

—

134

#127

—

135

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

136

M₃₂

—

137

C₂×C₁₆

—

138

#108

—

139

#108

—

140

#97

—

172

—

141

C₄×C₈

—

172

—

142

C₄×C₈

C₂×Dic₄₃

yes

—

143

C₄×C₈

C₂²⋊D₄₃

yes

—

144

C₂⁴

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂²×C₄

#10621883

yes

—

153

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

154

D₈

—

155

D₈

—

156

D₈

—

157

D₈

—

158

M₁₆

—

159

M₁₆

—

172

—

160

C₂×C₈

—

161

C₂×C₈

—

172

—

162

C₂×C₈

C₂²⋊Dic₄₃

yes

—

163

C₂×C₈

D₄×D₄₃

yes

—

164

C₂×C₈

#10621900

yes

—

165

C₂×C₈

C₂²×Dic₄₃

yes

characteristic

166

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

167

C₄²

#10621882

yes

characteristic

168

C₄²

#10621882

yes

characteristic

169

C₄²

C₂²⋊Dic₄₃

yes

—

170

C₁₆

—

172

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

C₂×#10621883

yes

characteristic

175

D₄

—

176

D₄

—

177

D₄

—

178

D₄

—

179

D₄

—

180

C₂×C₄

—

181

C₂×C₄

#13063246

yes

—

182

C₂×C₄

#13063246

yes

—

183

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

184

C₂×C₄

#13063296

yes

—

185

C₂×C₄

C₂×#10621882

yes

characteristic

186

C₂×C₄

#13063351

yes

—

187

C₈

—

188

C₈

—

172

—

189

C₈

D₄×Dic₄₃

yes

—

190

C₈

#13063312

yes

characteristic

191

C₈

#13063321

yes

characteristic

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

C₂×#13063246

yes

characteristic

197

C₂²

#14583096

yes

—

198

C₄

—

199

C₄

#14582712

yes

complemented

200

C₄

#14582802

yes

characteristic

201

C₄

#14582904

yes

characteristic

202

C₄

#14583091

yes

—

203

C₂

—

204

C₂

—

205

C₂

#19135523

yes

characteristic

206

C₂

C₂×#14582712

yes

—

	subgroup	G/H
center	C₂²	C₂×#13063246
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting, prime radical	C₂×C₄₃×#493	C₂
socle	C₂²×C₄₃	C₂×#102
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#493	D₄₃
hypocenter	C₄₃	#13257
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂²

C₂×#13063246

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting, prime radical

C₂×C₄₃×#493

C₂

socle

C₂²×C₄₃

C₂×#102

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

N	H ≅ G/N	type
1. #19136351	C₂	semidirect
2. C₂×#14582712	C₂	semidirect
3. C₄×#13063261	C₂	semidirect
21. #13063261	D₄	semidirect
96. C₄×C₄₃	#361	semidirect
120. C₄₃	#13257	semidirect
199. C₄	#14582712	semidirect

H ≅ G/N

type

1. #19136351

C₂

semidirect

2. C₂×#14582712

C₂

semidirect

3. C₄×#13063261

C₂

semidirect

21. #13063261

D₄

semidirect

96. C₄×C₄₃

#361

semidirect

120. C₄₃

#13257

semidirect

199. C₄

#14582712

semidirect

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

subgroup	index
#19136351	2
C₂×#14582712	2
C₄×#13063261	2
C₂×C₄₃×#493	2
#13257	43

subgroup

index

C₂×C₄₃×#493

Subgroups — #25015086

Conjugacy classes of subgroups

Complements

Maximal subgroups