Group #20275865 — subgroups

Subgroups — #20275865

Order 6784.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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×#13133208	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133142	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133140	C₂	1	1	yes	maximal, characteristic
4	#16810528	C₂	1	1	yes	maximal, characteristic
5	#16810385	C₂	1	1	yes	maximal, characteristic
6	#16810377	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#340	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	#13133208	—	2	2	no	—
11	#13133208	—	2	2	no	—
12	C₂×C₄⋊Dic₅₃	—	2	1	no	—
13	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×#11719919	—	2	1	no	—
15	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
16	#13133142	—	2	2	no	—
17	#13133140	—	2	2	no	—
18	#13133124	—	2	1	no	—
19	#13133124	—	2	1	no	—
20	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₄×C₅₃	—	2	1	no	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₃	—	2	1	no	—
27	C₂²×Dic₅₃	—	2	1	no	—
28	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
29	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
30	C₂×Dic₁₀₆	—	2	1	no	—
31	C₂×Dic₁₀₆	—	2	1	no	—
32	C₂×Dic₁₀₆	—	2	1	no	—
33	C₂×Dic₁₀₆	—	2	1	no	—
34	C₂×Dic₁₀₆	—	2	1	no	—
35	C₂×Dic₁₀₆	Q₈	1	2	yes	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₄⋊C₄×C₅₃	—	2	1	no	—
42	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
43	C₄⋊Dic₅₃	—	2	4	no	—
44	C₄⋊Dic₅₃	—	2	1	no	—
45	C₄⋊Dic₅₃	D₄	1	2	yes	—
46	#11719919	—	2	4	no	—
47	C₄×Dic₅₃	—	2	1	no	—
48	C₄×Dic₅₃	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂×#88322	—	2	1	no	—
52	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	—	4	1	no	—
56	C₂×C₄×C₅₃	—	4	1	no	—
57	C₂×C₄×C₅₃	—	4	1	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
65	C₂×Dic₅₃	—	2	2	no	—
66	C₂×Dic₅₃	—	2	1	no	—
67	C₂×Dic₅₃	—	2	1	no	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	1	no	—
70	C₂×Dic₅₃	—	2	1	no	—
71	C₂×Dic₅₃	—	2	1	no	—
72	C₂×Dic₅₃	—	2	1	no	—
73	C₂×Dic₅₃	—	4	1	no	—
74	C₂×Dic₅₃	—	4	1	no	—
75	C₂×Dic₅₃	—	4	1	no	—
76	C₂×Dic₅₃	—	4	1	no	—
77	Dic₁₀₆	—	2	2	no	—
78	Dic₁₀₆	—	2	2	no	—
79	Dic₁₀₆	—	4	1	no	—
80	Dic₁₀₆	—	4	1	no	—
81	C₈×C₅₃	—	4	1	no	—
82	#88322	—	4	1	no	—
83	C₂²×C₅₃	#120	1	1	yes	characteristic
84	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂²×C₅₃	#124	1	1	yes	characteristic
86	C₂²×C₅₃	#135	1	1	yes	characteristic
87	C₂²×C₅₃	#135	1	1	yes	characteristic
88	C₂²×C₅₃	#137	1	1	yes	characteristic
89	C₂²×C₅₃	#137	1	1	yes	characteristic
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	—	8	1	no	—
92	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₄×C₅₃	#124	1	1	yes	characteristic
95	C₄×C₅₃	#124	1	1	yes	characteristic
96	Dic₅₃	—	4	2	no	—
97	Dic₅₃	—	4	1	no	—
98	Dic₅₃	—	4	1	no	—
99	Dic₅₃	—	4	2	no	—
100	#2026	—	53	1	no	maximal, Sylow
101	C₂×C₅₃	#393	1	1	yes	characteristic
102	C₂×C₅₃	#452	1	1	yes	characteristic
103	C₂×C₅₃	#452	1	1	yes	characteristic
104	C₂×C₅₃	#479	1	1	yes	characteristic
105	C₂×C₅₃	#479	1	1	yes	characteristic
106	C₂×C₅₃	#488	1	1	yes	characteristic
107	C₂×C₅₃	#491	1	1	yes	characteristic
108	C₂×#125	—	53	1	no	—
109	C₂×#103	—	53	1	no	—
110	C₂×#103	—	53	1	no	—
111	#391	—	53	1	no	—
112	#340	—	53	1	no	—
113	#340	D₅₃	1	1	yes	characteristic
114	#336	—	53	1	no	—
115	C₅₃	#2026	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	53	1	no	—
117	C₂²×C₈	—	53	1	no	—
118	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
119	#125	—	106	2	no	—
120	#125	—	106	2	no	—
121	C₂×C₄⋊C₄	—	53	1	no	—
122	C₂×C₄⋊C₄	—	106	1	no	—
123	C₂×C₄⋊C₄	—	106	1	no	—
124	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
126	C₂×C₄²	—	53	1	no	—
127	#103	—	106	2	no	—
128	#103	—	106	2	no	—
129	#95	—	106	1	no	—
130	#95	—	106	1	no	—
131	C₂×Q₈	—	53	2	no	—
132	C₂×Q₈	—	106	1	no	—
133	C₂×Q₈	—	106	1	no	—
134	C₂×Q₈	—	106	1	no	—
135	C₂×Q₈	—	106	1	no	—
136	C₂×Q₈	—	106	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	53	1	no	—
140	C₂²×C₄	—	53	1	no	—
141	C₂²×C₄	—	106	1	no	—
142	C₂²×C₄	—	106	1	no	—
143	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	106	1	no	—
148	C₂×C₈	—	106	1	no	—
149	C₂×C₈	—	106	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	53	2	no	—
154	C₄⋊C₄	—	106	4	no	—
155	C₄⋊C₄	—	106	4	no	—
156	C₄⋊C₄	—	106	1	no	—
157	C₄⋊C₄	D₂₁₂	1	2	yes	—
158	C₄²	—	106	1	no	—
159	C₄²	—	106	1	no	—
160	C₂³	D₄×D₅₃	1	1	yes	characteristic
161	Q₈	—	106	2	no	—
162	Q₈	—	106	2	no	—
163	Q₈	—	212	1	no	—
164	Q₈	—	212	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	106	1	no	—
172	C₂×C₄	—	106	1	no	—
173	C₂×C₄	—	106	2	no	—
174	C₂×C₄	—	106	2	no	—
175	C₂×C₄	—	106	1	no	—
176	C₂×C₄	—	106	1	no	—
177	C₂×C₄	—	106	1	no	—
178	C₂×C₄	—	106	1	no	—
179	C₂×C₄	—	212	1	no	—
180	C₂×C₄	—	212	1	no	—
181	C₂×C₄	—	212	1	no	—
182	C₂×C₄	—	212	1	no	—
183	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
184	C₂×C₄	#11719945	1	1	yes	characteristic
185	C₂×C₄	#11719945	1	1	yes	characteristic
186	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
187	C₂×C₄	#11719947	1	1	yes	characteristic
188	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
189	C₈	—	4	1	no	—
190	C₈	—	212	1	no	—
191	C₂²	#13133187	1	1	yes	characteristic
192	C₂²	#13133195	1	1	yes	characteristic
193	C₂²	#13133203	1	1	yes	characteristic
194	C₂²	#13133236	1	1	yes	characteristic
195	C₂²	#13133238	1	1	yes	characteristic
196	C₂²	#13133246	1	1	yes	characteristic
197	C₂²	#13133249	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	212	1	no	—
201	C₄	—	212	1	no	—
202	C₄	—	212	2	no	—
203	C₄	—	212	2	no	—
204	C₄	#13133195	1	1	yes	characteristic
205	C₄	#13133197	1	1	yes	characteristic
206	C₄	#13133203	1	1	yes	characteristic
207	C₄	#13133205	1	1	yes	characteristic
208	C₂	#16810536	1	1	yes	characteristic
209	C₂	#16810786	1	1	yes	characteristic
210	C₂	#16810788	1	1	yes	characteristic
211	C₂	#16810929	1	1	yes	characteristic
212	C₂	#16810933	1	1	yes	characteristic
213	C₂	#16810976	1	1	yes	characteristic
214	C₂	#16810990	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133208

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal, characteristic

C₂×#13133140

C₂

yes

maximal, characteristic

#16810528

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal, characteristic

C₅₃×#340

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133208

—

#13133208

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

100

#2026

—

maximal, Sylow

101

C₂×C₅₃

#393

yes

characteristic

102

C₂×C₅₃

#452

yes

characteristic

103

C₂×C₅₃

#452

yes

characteristic

104

C₂×C₅₃

#479

yes

characteristic

105

C₂×C₅₃

#479

yes

characteristic

106

C₂×C₅₃

#488

yes

characteristic

107

C₂×C₅₃

#491

yes

characteristic

108

C₂×#125

—

109

C₂×#103

—

110

C₂×#103

—

111

#391

—

112

#340

—

113

#340

D₅₃

yes

characteristic

114

#336

—

115

C₅₃

#2026

yes

characteristic, complemented, Sylow

116

C₂²×Q₈

—

117

C₂²×C₈

—

118

C₂²×C₈

C₂×D₅₃

yes

characteristic

119

#125

—

106

—

120

#125

—

106

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

106

—

123

C₂×C₄⋊C₄

—

106

—

124

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

125

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

126

C₂×C₄²

—

127

#103

—

106

—

128

#103

—

106

—

129

#95

—

106

—

130

#95

—

106

—

131

C₂×Q₈

—

132

C₂×Q₈

—

106

—

133

C₂×Q₈

—

106

—

134

C₂×Q₈

—

106

—

135

C₂×Q₈

—

106

—

136

C₂×Q₈

—

106

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

106

—

142

C₂²×C₄

—

106

—

143

C₂²×C₄

C₂²×D₅₃

yes

characteristic

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

106

—

148

C₂×C₈

—

106

—

149

C₂×C₈

—

106

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

106

—

155

C₄⋊C₄

—

106

—

156

C₄⋊C₄

—

106

—

157

C₄⋊C₄

D₂₁₂

yes

—

158

C₄²

—

106

—

159

C₄²

—

106

—

160

C₂³

D₄×D₅₃

yes

characteristic

161

Q₈

—

106

—

162

Q₈

—

106

—

163

Q₈

—

212

—

164

Q₈

—

212

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

106

—

172

C₂×C₄

—

106

—

173

C₂×C₄

—

106

—

174

C₂×C₄

—

106

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

106

—

177

C₂×C₄

—

106

—

178

C₂×C₄

—

106

—

179

C₂×C₄

—

212

—

180

C₂×C₄

—

212

—

181

C₂×C₄

—

212

—

182

C₂×C₄

—

212

—

183

C₂×C₄

C₂×D₂₁₂

yes

characteristic

184

C₂×C₄

#11719945

yes

characteristic

185

C₂×C₄

#11719945

yes

characteristic

186

C₂×C₄

D₄×D₅₃

yes

characteristic

187

C₂×C₄

#11719947

yes

characteristic

188

C₂×C₄

Q₈×D₅₃

yes

characteristic

189

C₈

—

190

C₈

—

212

—

191

C₂²

#13133187

yes

characteristic

192

C₂²

#13133195

yes

characteristic

193

C₂²

#13133203

yes

characteristic

194

C₂²

#13133236

yes

characteristic

195

C₂²

#13133238

yes

characteristic

196

C₂²

#13133246

yes

characteristic

197

C₂²

#13133249

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

212

—

201

C₄

—

212

—

202

C₄

—

212

—

203

C₄

—

212

—

204

C₄

#13133195

yes

characteristic

205

C₄

#13133197

yes

characteristic

206

C₄

#13133203

yes

characteristic

207

C₄

#13133205

yes

characteristic

208

C₂

#16810536

yes

characteristic

209

C₂

#16810786

yes

characteristic

210

C₂

#16810788

yes

characteristic

211

C₂

#16810929

yes

characteristic

212

C₂

#16810933

yes

characteristic

213

C₂

#16810976

yes

characteristic

214

C₂

#16810990

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#340	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#340	D₅₃
hypocenter	C₅₃	#2026
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#340

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
115. C₅₃	#2026	semidirect

H ≅ G/N

type

115. C₅₃

#2026

semidirect

Subgroups — #20275865

Conjugacy classes of subgroups

Complements