Group #20275355 — subgroups

Subgroups — #20275355

Order 6784.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13133142	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133140	C₂	1	1	yes	maximal, characteristic
3	#16810528	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810385	C₂	1	1	yes	maximal, characteristic
6	#16810377	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	—	2	1	no	—
12	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
13	#13133142	C₄	1	4	yes	complemented
14	#13133140	—	2	2	no	—
15	#13133124	—	2	1	no	—
16	#13133124	—	2	1	no	—
17	#13133124	—	2	1	no	—
18	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	#107×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	—	2	1	no	—
23	C₂²×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	2	1	no	—
26	C₂²×Dic₅₃	—	2	1	no	—
27	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
28	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₆	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
42	C₄⋊Dic₅₃	—	2	1	no	—
43	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
44	C₄×Dic₅₃	—	2	1	no	—
45	C₄×Dic₅₃	—	2	1	no	—
46	C₄×Dic₅₃	—	4	2	no	—
47	C₂×#88322	—	2	1	no	—
48	C₂×#88322	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	—	4	1	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	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂×Dic₅₃	—	2	1	no	—
64	C₂×Dic₅₃	—	2	1	no	—
65	C₂×Dic₅₃	—	2	2	no	—
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	2	1	no	—
68	C₂×Dic₅₃	—	2	1	no	—
69	C₂×Dic₅₃	—	2	1	no	—
70	C₂×Dic₅₃	—	4	1	no	—
71	C₂×Dic₅₃	—	4	1	no	—
72	C₂×Dic₅₃	—	4	1	no	—
73	C₂×Dic₅₃	—	4	1	no	—
74	C₂×Dic₅₃	Q₁₆	1	2	yes	—
75	Dic₁₀₆	—	2	2	no	—
76	Dic₁₀₆	—	2	2	no	—
77	Dic₁₀₆	—	4	1	no	—
78	Dic₁₀₆	—	4	1	no	—
79	C₈×C₅₃	—	2	2	no	—
80	#88322	—	4	1	no	—
81	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
82	C₂²×C₅₃	#121	1	1	yes	characteristic
83	C₂²×C₅₃	#123	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
85	C₂²×C₅₃	#135	1	1	yes	characteristic
86	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂²×C₅₃	#137	1	1	yes	characteristic
88	C₄×C₅₃	—	4	2	no	—
89	C₄×C₅₃	—	8	1	no	—
90	C₄×C₅₃	#117	1	1	yes	characteristic
91	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₄×C₅₃	#124	1	1	yes	characteristic
93	C₄×C₅₃	#126	1	1	yes	characteristic
94	Dic₅₃	—	2	2	no	—
95	Dic₅₃	—	4	2	no	—
96	Dic₅₃	—	4	2	no	—
97	Dic₅₃	—	4	1	no	—
98	#1928	—	53	1	no	maximal, Sylow
99	C₂×C₅₃	#388	1	1	yes	characteristic
100	C₂×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
101	C₂×C₅₃	#440	1	1	yes	characteristic
102	C₂×C₅₃	#462	1	1	yes	characteristic
103	C₂×C₅₃	#463	1	1	yes	characteristic
104	C₂×C₅₃	#484	1	1	yes	characteristic
105	C₂×C₅₃	#485	1	1	yes	characteristic
106	C₂×#107	D₅₃	1	1	yes	characteristic
107	C₂×#103	—	53	1	no	—
108	C₂×#103	—	53	1	no	—
109	#391	—	53	1	no	—
110	C₄×C₄⋊C₄	—	53	1	no	—
111	#340	—	53	1	no	—
112	#336	—	53	1	no	—
113	C₅₃	#1928	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	53	1	no	—
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	53	1	no	—
118	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
120	C₂×C₄²	—	53	1	no	—
121	C₂×C₄²	—	106	1	no	—
122	#107	—	2	2	no	—
123	#103	—	53	4	no	—
124	#103	—	106	2	no	—
125	#95	—	106	1	no	—
126	#95	—	106	1	no	—
127	#95	—	106	1	no	—
128	C₂×Q₈	—	53	2	no	—
129	C₂×Q₈	—	106	1	no	—
130	C₂×Q₈	—	106	1	no	—
131	C₂×Q₈	—	106	1	no	—
132	C₂×Q₈	—	106	1	no	—
133	C₂×Q₈	—	106	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	53	1	no	—
137	C₂²×C₄	—	53	1	no	—
138	C₂²×C₄	—	106	1	no	—
139	C₂²×C₄	—	106	1	no	—
140	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	106	1	no	—
144	C₂×C₈	—	106	1	no	—
145	C₂×C₈	—	106	1	no	—
146	C₂×C₈	C₄×D₅₃	1	2	yes	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	53	2	no	—
151	C₄⋊C₄	—	106	1	no	—
152	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
153	C₄²	—	106	1	no	—
154	C₄²	—	106	1	no	—
155	C₄²	—	212	2	no	—
156	C₂³	D₄×D₅₃	1	1	yes	characteristic
157	Q₈	—	106	2	no	—
158	Q₈	—	106	2	no	—
159	Q₈	—	212	1	no	—
160	Q₈	—	212	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	53	2	no	—
168	C₂×C₄	—	106	1	no	—
169	C₂×C₄	—	106	1	no	—
170	C₂×C₄	—	106	2	no	—
171	C₂×C₄	—	106	2	no	—
172	C₂×C₄	—	106	1	no	—
173	C₂×C₄	—	106	1	no	—
174	C₂×C₄	—	106	1	no	—
175	C₂×C₄	—	212	1	no	—
176	C₂×C₄	—	212	1	no	—
177	C₂×C₄	—	212	1	no	—
178	C₂×C₄	—	212	1	no	—
179	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄	#11719945	1	1	yes	characteristic
181	C₂×C₄	#11719947	1	1	yes	characteristic
182	C₂×C₄	#11719949	1	1	yes	characteristic
183	C₂×C₄	#11719949	1	1	yes	characteristic
184	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	212	1	no	—
187	C₂²	#13133179	1	1	yes	characteristic
188	C₂²	#13133189	1	1	yes	characteristic
189	C₂²	#13133202	1	1	yes	characteristic
190	C₂²	Q₁₆×D₅₃	1	1	yes	characteristic
191	C₂²	#13133236	1	1	yes	characteristic
192	C₂²	#13133241	1	1	yes	characteristic
193	C₂²	#13133249	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	8	1	no	—
196	C₄	—	106	2	no	—
197	C₄	—	212	2	no	—
198	C₄	—	212	1	no	—
199	C₄	—	212	2	no	—
200	C₄	#13133175	1	1	yes	characteristic
201	C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₄	#13133206	1	1	yes	characteristic
203	C₄	#13133212	1	1	yes	characteristic
204	C₂	#16810513	1	1	yes	characteristic
205	C₂	#16810717	1	1	yes	characteristic
206	C₂	#16810722	1	1	yes	characteristic
207	C₂	#16810831	1	1	yes	characteristic
208	C₂	#16810836	1	1	yes	characteristic
209	C₂	#16810960	1	1	yes	characteristic
210	C₂	#16810966	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133142

C₂

yes

maximal, characteristic

C₂×#13133140

C₂

yes

maximal, characteristic

#16810528

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₅₃

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal, characteristic

C₂×#107×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133142

C₄

yes

—

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

#107×C₅₃

—

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

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

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

Q₁₆

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

100

C₂×C₅₃

C₄×Q₁₆

yes

characteristic

101

C₂×C₅₃

#440

yes

characteristic

102

C₂×C₅₃

#462

yes

characteristic

103

C₂×C₅₃

#463

yes

characteristic

104

C₂×C₅₃

#484

yes

characteristic

105

C₂×C₅₃

#485

yes

characteristic

106

C₂×#107

D₅₃

yes

characteristic

107

C₂×#103

—

108

C₂×#103

—

109

#391

—

110

C₄×C₄⋊C₄

—

111

#340

—

112

#336

—

113

C₅₃

#1928

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

115

C₂²×C₈

—

116

C₂²×C₈

C₂×D₅₃

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

120

C₂×C₄²

—

121

C₂×C₄²

—

106

—

122

#107

—

123

#103

—

124

#103

—

106

—

125

#95

—

106

—

126

#95

—

106

—

127

#95

—

106

—

128

C₂×Q₈

—

129

C₂×Q₈

—

106

—

130

C₂×Q₈

—

106

—

131

C₂×Q₈

—

106

—

132

C₂×Q₈

—

106

—

133

C₂×Q₈

—

106

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

106

—

139

C₂²×C₄

—

106

—

140

C₂²×C₄

C₂²×D₅₃

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

106

—

144

C₂×C₈

—

106

—

145

C₂×C₈

—

106

—

146

C₂×C₈

C₄×D₅₃

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

106

—

152

C₄⋊C₄

C₂²⋊D₅₃

yes

—

153

C₄²

—

106

—

154

C₄²

—

106

—

155

C₄²

—

212

—

156

C₂³

D₄×D₅₃

yes

characteristic

157

Q₈

—

106

—

158

Q₈

—

106

—

159

Q₈

—

212

—

160

Q₈

—

212

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

106

—

169

C₂×C₄

—

106

—

170

C₂×C₄

—

106

—

171

C₂×C₄

—

106

—

172

C₂×C₄

—

106

—

173

C₂×C₄

—

106

—

174

C₂×C₄

—

106

—

175

C₂×C₄

—

212

—

176

C₂×C₄

—

212

—

177

C₂×C₄

—

212

—

178

C₂×C₄

—

212

—

179

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

180

C₂×C₄

#11719945

yes

characteristic

181

C₂×C₄

#11719947

yes

characteristic

182

C₂×C₄

#11719949

yes

characteristic

183

C₂×C₄

#11719949

yes

characteristic

184

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

185

C₈

—

186

C₈

—

212

—

187

C₂²

#13133179

yes

characteristic

188

C₂²

#13133189

yes

characteristic

189

C₂²

#13133202

yes

characteristic

190

C₂²

Q₁₆×D₅₃

yes

characteristic

191

C₂²

#13133236

yes

characteristic

192

C₂²

#13133241

yes

characteristic

193

C₂²

#13133249

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

106

—

197

C₄

—

212

—

198

C₄

—

212

—

199

C₄

—

212

—

200

C₄

#13133175

yes

characteristic

201

C₄

C₄×C₂²⋊D₅₃

yes

characteristic

202

C₄

#13133206

yes

characteristic

203

C₄

#13133212

yes

characteristic

204

C₂

#16810513

yes

characteristic

205

C₂

#16810717

yes

characteristic

206

C₂

#16810722

yes

characteristic

207

C₂

#16810831

yes

characteristic

208

C₂

#16810836

yes

characteristic

209

C₂

#16810960

yes

characteristic

210

C₂

#16810966

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	#1928
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
13. #13133142	C₄	semidirect
113. C₅₃	#1928	semidirect

H ≅ G/N

type

13. #13133142

C₄

semidirect

113. C₅₃

#1928

semidirect

Subgroups — #20275355

Conjugacy classes of subgroups

Complements