Group #20275573 — subgroups

Subgroups — #20275573

Order 6784.

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

normal subgroups only

This group has 312 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	C₂²×#11719925	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133150	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133142	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810530	C₂	1	1	yes	maximal, characteristic
6	#16810528	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂×#11719925	—	2	2	no	—
10	C₂×#11719925	—	2	2	no	—
11	C₂×#11719925	C₄	1	4	yes	complemented
12	C₂²×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	#13133150	—	2	2	no	—
16	#13133142	—	2	2	no	—
17	C₈×Dic₅₃	—	2	2	no	—
18	#13133124	—	2	2	no	—
19	#13133124	—	2	2	no	—
20	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	#103×C₅₃	—	2	2	no	—
24	C₂×Q₈×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	—	2	1	no	—
27	C₂×Q₈×C₅₃	—	2	1	no	—
28	C₂×Q₈×C₅₃	—	2	1	no	—
29	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	—	2	1	no	—
31	C₂²×C₄×C₅₃	—	2	1	no	—
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₅₃	—	2	1	no	—
34	C₂²×Dic₅₃	—	2	1	no	—
35	C₂²×Dic₅₃	D₄	1	2	yes	—
36	C₂×Dic₁₀₆	—	2	1	no	—
37	C₂×Dic₁₀₆	—	2	1	no	—
38	C₂×Dic₁₀₆	—	2	1	no	—
39	C₂×Dic₁₀₆	—	2	1	no	—
40	C₂×Dic₁₀₆	—	2	1	no	—
41	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	D₄	1	2	yes	—
45	C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₄⋊C₄×C₅₃	—	2	1	no	—
47	#11719925	—	2	8	no	—
48	C₄⋊Dic₅₃	—	2	1	no	—
49	C₄⋊Dic₅₃	—	2	1	no	—
50	C₄×Dic₅₃	—	2	1	no	—
51	C₄×Dic₅₃	—	2	1	no	—
52	C₂×#88322	C₂×C₄	1	2	yes	—
53	C₂×#88322	D₄	1	2	yes	—
54	C₂×#88322	D₄	1	2	yes	—
55	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₅₃	—	2	2	no	—
57	Q₈×C₅₃	—	2	2	no	—
58	Q₈×C₅₃	—	4	1	no	—
59	Q₈×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	4	1	no	—
62	C₂×C₄×C₅₃	—	4	1	no	—
63	C₂×C₄×C₅₃	—	4	1	no	—
64	C₂×C₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	—	2	2	no	—
73	C₂×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	—	4	1	no	—
75	C₂×Dic₅₃	—	4	1	no	—
76	C₂×Dic₅₃	—	4	1	no	—
77	C₂×Dic₅₃	—	4	2	no	—
78	C₂×Dic₅₃	Q₁₆	1	4	yes	complemented
79	Dic₁₀₆	—	2	2	no	—
80	Dic₁₀₆	—	2	2	no	—
81	Dic₁₀₆	—	4	1	no	—
82	Dic₁₀₆	—	4	1	no	—
83	C₈×C₅₃	—	2	2	no	—
84	#88322	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₅₃	#121	1	1	yes	characteristic
87	C₂²×C₅₃	#127	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
89	C₂²×C₅₃	#135	1	2	yes	—
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	—	8	1	no	—
92	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₄×C₅₃	#121	1	1	yes	characteristic
95	C₄×C₅₃	#124	1	1	yes	characteristic
96	Dic₅₃	—	2	4	no	—
97	Dic₅₃	—	4	2	no	—
98	Dic₅₃	—	8	1	no	—
99	#1969	—	53	1	no	maximal, Sylow
100	C₂×C₅₃	#390	1	1	yes	characteristic
101	C₂×C₅₃	C₄×Q₁₆	1	2	yes	—
102	C₂×C₅₃	#467	1	2	yes	—
103	C₂×C₅₃	#494	1	1	yes	characteristic
104	C₂×C₅₃	#495	1	1	yes	characteristic
105	C₂²×Q₁₆	—	53	1	no	—
106	C₂×#107	—	53	1	no	—
107	C₂×#103	—	53	1	no	—
108	C₂×#103	D₅₃	1	1	yes	characteristic
109	C₂×C₄×C₈	—	53	1	no	—
110	#391	—	53	1	no	—
111	#391	—	53	1	no	—
112	C₅₃	#1969	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	53	1	no	—
114	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
115	C₂×Q₁₆	—	53	4	no	—
116	C₂×Q₁₆	—	106	2	no	—
117	C₂×Q₁₆	—	106	2	no	—
118	C₂²×C₈	—	53	1	no	—
119	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	53	1	no	—
121	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
122	C₂×C₄²	—	53	1	no	—
123	#107	—	106	2	no	—
124	#103	—	2	2	no	—
125	#103	—	106	2	no	—
126	C₄×C₈	—	106	2	no	—
127	#95	—	106	2	no	—
128	#95	—	106	2	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	53	2	no	—
135	C₂×Q₈	—	106	1	no	—
136	C₂×Q₈	—	106	1	no	—
137	C₂×Q₈	—	106	1	no	—
138	C₂×Q₈	—	106	1	no	—
139	C₂×Q₈	—	106	1	no	—
140	C₂×Q₈	C₄×D₅₃	1	2	yes	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	53	2	no	—
144	C₂²×C₄	—	106	1	no	—
145	C₂²×C₄	—	106	1	no	—
146	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
147	Q₁₆	—	106	8	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	53	2	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	106	1	no	—
157	C₄⋊C₄	—	106	1	no	—
158	C₄²	—	106	1	no	—
159	C₄²	—	106	1	no	—
160	C₂³	D₄×D₅₃	1	1	yes	characteristic
161	Q₈	—	2	2	no	—
162	Q₈	—	2	2	no	—
163	Q₈	—	4	1	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	106	2	no	—
166	Q₈	—	106	2	no	—
167	Q₈	—	212	1	no	—
168	Q₈	—	212	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	53	4	no	—
175	C₂×C₄	—	106	2	no	—
176	C₂×C₄	—	106	2	no	—
177	C₂×C₄	—	106	2	no	—
178	C₂×C₄	—	212	1	no	—
179	C₂×C₄	—	212	2	no	—
180	C₂×C₄	—	212	1	no	—
181	C₂×C₄	—	212	1	no	—
182	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₂×C₄	#11719945	1	1	yes	characteristic
184	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
185	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
186	C₂×C₄	#11719947	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	53	4	no	—
190	C₂²	#13133177	1	1	yes	characteristic
191	C₂²	#13133190	1	1	yes	characteristic
192	C₂²	#13133213	1	1	yes	characteristic
193	C₂²	Q₁₆×D₅₃	1	2	yes	—
194	C₂²	#13133238	1	2	yes	—
195	C₄	—	4	2	no	—
196	C₄	—	8	1	no	—
197	C₄	—	106	4	no	—
198	C₄	—	424	1	no	—
199	C₄	—	212	2	no	—
200	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
201	C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₄	#13133189	1	1	yes	characteristic
203	C₄	#13133203	1	1	yes	characteristic
204	C₂	#16810523	1	1	yes	characteristic
205	C₂	#16810715	1	2	yes	—
206	C₂	#16810863	1	2	yes	—
207	C₂	#16811001	1	1	yes	characteristic
208	C₂	#16811004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719925

C₂

yes

maximal, characteristic

C₂×#13133150

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal, characteristic

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810530

C₂

yes

maximal, characteristic

#16810528

C₂

yes

maximal, characteristic

C₂×#103×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133150

—

#13133142

—

C₈×Dic₅₃

—

#13133124

—

#13133124

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

D₄

yes

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

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

Q₁₆

yes

complemented

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

C₄×D₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

#135

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1969

—

maximal, Sylow

100

C₂×C₅₃

#390

yes

characteristic

101

C₂×C₅₃

C₄×Q₁₆

yes

—

102

C₂×C₅₃

#467

yes

—

103

C₂×C₅₃

#494

yes

characteristic

104

C₂×C₅₃

#495

yes

characteristic

105

C₂²×Q₁₆

—

106

C₂×#107

—

107

C₂×#103

—

108

C₂×#103

D₅₃

yes

characteristic

109

C₂×C₄×C₈

—

110

#391

—

111

#391

—

112

C₅₃

#1969

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

114

C₂²×Q₈

C₂×D₅₃

yes

characteristic

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

106

—

117

C₂×Q₁₆

—

106

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂×D₅₃

yes

characteristic

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

122

C₂×C₄²

—

123

#107

—

106

—

124

#103

—

125

#103

—

106

—

126

C₄×C₈

—

106

—

127

#95

—

106

—

128

#95

—

106

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

106

—

136

C₂×Q₈

—

106

—

137

C₂×Q₈

—

106

—

138

C₂×Q₈

—

106

—

139

C₂×Q₈

—

106

—

140

C₂×Q₈

C₄×D₅₃

yes

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

106

—

145

C₂²×C₄

—

106

—

146

C₂²×C₄

C₂²×D₅₃

yes

characteristic

147

Q₁₆

—

106

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²⋊D₅₃

yes

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

106

—

157

C₄⋊C₄

—

106

—

158

C₄²

—

106

—

159

C₄²

—

106

—

160

C₂³

D₄×D₅₃

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

106

—

166

Q₈

—

106

—

167

Q₈

—

212

—

168

Q₈

—

212

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

106

—

177

C₂×C₄

—

106

—

178

C₂×C₄

—

212

—

179

C₂×C₄

—

212

—

180

C₂×C₄

—

212

—

181

C₂×C₄

—

212

—

182

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

183

C₂×C₄

#11719945

yes

characteristic

184

C₂×C₄

D₄×D₅₃

yes

characteristic

185

C₂×C₄

D₄×D₅₃

yes

characteristic

186

C₂×C₄

#11719947

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

188

C₈

—

189

C₈

—

190

C₂²

#13133177

yes

characteristic

191

C₂²

#13133190

yes

characteristic

192

C₂²

#13133213

yes

characteristic

193

C₂²

Q₁₆×D₅₃

yes

—

194

C₂²

#13133238

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

106

—

198

C₄

—

424

—

199

C₄

—

212

—

200

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

201

C₄

C₄×C₂²⋊D₅₃

yes

characteristic

202

C₄

#13133189

yes

characteristic

203

C₄

#13133203

yes

characteristic

204

C₂

#16810523

yes

characteristic

205

C₂

#16810715

yes

—

206

C₂

#16810863

yes

—

207

C₂

#16811001

yes

characteristic

208

C₂

#16811004

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#103×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#103	D₅₃
hypocenter	C₅₃	#1969
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#103×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
11. C₂×#11719925	C₄	semidirect
78. C₂×Dic₅₃	Q₁₆	semidirect
112. C₅₃	#1969	semidirect

H ≅ G/N

type

11. C₂×#11719925

C₄

semidirect

78. C₂×Dic₅₃

Q₁₆

semidirect

112. C₅₃

#1969

semidirect

Subgroups — #20275573

Conjugacy classes of subgroups

Complements