Group #20273116 — subgroups

Subgroups — #20273116

Order 6784.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16811177	C₂	1	1	yes	maximal, characteristic
2	#16810674	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810355	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810353	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₅₃	—	2	1	no	—
9	#13133216	C₄	1	1	yes	characteristic
10	#13133215	C₄	1	1	yes	characteristic
11	#13133207	—	2	1	no	—
12	#13133197	C₄	1	1	yes	characteristic
13	#13133195	—	2	1	no	—
14	#13133193	C₄	1	1	yes	characteristic
15	C₄×Dic₁₀₆	—	2	1	no	—
16	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	#13133174	—	2	1	no	—
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	#13133146	—	2	1	no	—
21	#13133146	—	2	1	no	—
22	#13133129	—	2	1	no	—
23	C₄×#88322	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
31	C₂×Dic₁₀₆	—	2	1	no	—
32	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	—	2	1	no	—
34	C₂×C₈×C₅₃	—	2	1	no	—
35	C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₅₃	—	2	1	no	—
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#11719921	—	2	1	no	—
40	#11719921	—	2	1	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	#11719919	—	2	1	no	—
44	#11719919	—	2	1	no	—
45	#11719919	—	2	1	no	—
46	#11719919	—	2	1	no	—
47	#11719919	—	2	1	no	—
48	C₄×Dic₅₃	—	2	1	no	—
49	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₅₃	D₄	1	1	yes	characteristic
51	C₄×Dic₅₃	D₄	1	1	yes	characteristic
52	C₂×#88322	—	2	1	no	—
53	C₂×#88322	—	2	1	no	—
54	C₈×D₅₃	—	4	2	no	—
55	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	Q₈×C₅₃	—	2	1	no	—
57	Q₈×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₅₃	—	2	1	no	—
64	C₂×Dic₅₃	—	2	1	no	—
65	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄×D₅₃	—	2	1	no	—
69	C₄×D₅₃	—	2	1	no	—
70	C₄×D₅₃	—	4	1	no	—
71	C₄×D₅₃	SD₁₆	1	2	yes	—
72	C₄×D₅₃	Q₁₆	1	2	yes	—
73	Dic₁₀₆	—	2	1	no	—
74	Dic₁₀₆	—	2	1	no	—
75	Dic₁₀₆	—	4	2	no	—
76	C₈×C₅₃	—	4	1	no	—
77	C₈×C₅₃	—	4	1	no	—
78	#88322	—	4	1	no	—
79	#88322	—	4	1	no	—
80	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×D₅₃	—	2	1	no	—
82	C₂×D₅₃	—	2	1	no	—
83	C₂×D₅₃	#103	1	2	yes	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	4	1	no	—
86	C₄×C₅₃	—	4	1	no	—
87	C₄×C₅₃	#131	1	1	yes	characteristic
88	C₄×C₅₃	#131	1	1	yes	characteristic
89	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
91	Dic₅₃	—	2	1	no	—
92	Dic₅₃	—	2	1	no	—
93	Dic₅₃	—	4	1	no	—
94	Dic₅₃	—	4	1	no	—
95	Dic₅₃	#103	1	2	yes	—
96	#1481	—	53	1	no	maximal, Sylow
97	C₂×C₅₃	#408	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
99	C₂×C₅₃	#421	1	1	yes	characteristic
100	D₅₃	—	2	2	no	—
101	#533	—	53	1	no	—
102	#431	—	53	1	no	—
103	C₂×C₄⋊C₈	—	53	1	no	—
104	Q₈⋊C₈	—	53	1	no	—
105	Q₈⋊C₈	—	53	1	no	—
106	Q₈⋊C₈	—	53	1	no	—
107	Q₈⋊C₈	D₅₃	1	1	yes	characteristic, complemented
108	C₅₃	#1481	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	106	1	no	—
110	C₄⋊Q₈	—	53	1	no	—
111	C₄⋊Q₈	—	53	1	no	—
112	#125	—	106	1	no	—
113	C₂²⋊Q₈	—	53	1	no	—
114	C₂²⋊Q₈	—	53	1	no	—
115	C₂²⋊Q₈	—	106	1	no	—
116	C₄×Q₈	—	53	1	no	—
117	C₄×Q₈	—	106	1	no	—
118	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
119	#117	—	106	1	no	—
120	C₂×C₄²	—	53	1	no	—
121	C₄⋊C₈	—	53	1	no	—
122	C₄⋊C₈	—	106	1	no	—
123	C₄⋊C₈	—	106	1	no	—
124	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂²⋊C₈	—	106	1	no	—
126	C₄×C₈	—	53	1	no	—
127	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
128	C₂×Q₈	—	53	1	no	—
129	C₂×Q₈	—	106	1	no	—
130	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
131	C₂²×C₄	—	53	1	no	—
132	C₂²×C₄	—	53	1	no	—
133	C₂²×C₄	—	53	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	106	1	no	—
137	C₂×C₈	—	106	1	no	—
138	C₂×C₈	—	212	2	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	53	1	no	—
141	C₄⋊C₄	—	106	1	no	—
142	C₄⋊C₄	—	106	1	no	—
143	C₄⋊C₄	—	106	1	no	—
144	C₄⋊C₄	—	106	1	no	—
145	C₄⋊C₄	—	106	1	no	—
146	C₄⋊C₄	—	106	1	no	—
147	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
148	C₂²⋊C₄	—	106	1	no	—
149	C₂²⋊C₄	—	106	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	53	1	no	—
152	C₄²	—	53	1	no	—
153	C₄²	—	53	1	no	—
154	C₄²	—	106	1	no	—
155	C₄²	C₂²×D₅₃	1	1	yes	characteristic
156	C₂³	—	53	1	no	—
157	Q₈	—	2	1	no	—
158	Q₈	—	2	1	no	—
159	Q₈	—	106	1	no	—
160	Q₈	—	106	1	no	—
161	Q₈	—	212	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	53	1	no	—
165	C₂×C₄	—	53	2	no	—
166	C₂×C₄	—	53	2	no	—
167	C₂×C₄	—	53	1	no	—
168	C₂×C₄	—	53	1	no	—
169	C₂×C₄	—	106	1	no	—
170	C₂×C₄	—	106	1	no	—
171	C₂×C₄	—	106	1	no	—
172	C₂×C₄	—	106	1	no	—
173	C₂×C₄	—	212	1	no	—
174	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
176	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
177	C₈	—	4	1	no	—
178	C₈	—	4	1	no	—
179	C₈	—	212	1	no	—
180	C₈	—	212	1	no	—
181	C₂²	—	53	2	no	—
182	C₂²	—	106	1	no	—
183	C₂²	—	106	1	no	—
184	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	53	2	no	—
189	C₄	—	106	1	no	—
190	C₄	—	106	1	no	—
191	C₄	—	212	1	no	—
192	C₄	—	212	1	no	—
193	C₄	#13133223	1	1	yes	characteristic
194	C₄	#13133225	1	1	yes	characteristic
195	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
196	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
197	C₂	—	106	2	no	—
198	C₂	#16810588	1	1	yes	characteristic
199	C₂	#103×D₅₃	1	1	yes	characteristic
200	C₂	#16810638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811177

C₂

yes

maximal, characteristic

#16810674

C₂

yes

maximal, characteristic

C₄⋊C₈×D₅₃

C₂

yes

maximal, characteristic

#16810355

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, characteristic, complemented

#16810353

C₂

yes

maximal, characteristic, complemented

C₅₃×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

—

#13133216

C₄

yes

characteristic

#13133215

C₄

yes

characteristic

#13133207

—

#13133197

C₄

yes

characteristic

#13133195

—

#13133193

C₄

yes

characteristic

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133174

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

—

—

—

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₈×D₅₃

—

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×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×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

Q₁₆

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#103

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#103

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

100

D₅₃

—

101

#533

—

102

#431

—

103

C₂×C₄⋊C₈

—

104

Q₈⋊C₈

—

105

Q₈⋊C₈

—

106

Q₈⋊C₈

—

107

Q₈⋊C₈

D₅₃

yes

characteristic, complemented

108

C₅₃

#1481

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

106

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

#125

—

106

—

113

C₂²⋊Q₈

—

114

C₂²⋊Q₈

—

115

C₂²⋊Q₈

—

106

—

116

C₄×Q₈

—

117

C₄×Q₈

—

106

—

118

C₄×Q₈

C₂×D₅₃

yes

characteristic

119

#117

—

106

—

120

C₂×C₄²

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

106

—

123

C₄⋊C₈

—

106

—

124

C₄⋊C₈

C₂×D₅₃

yes

characteristic

125

C₂²⋊C₈

—

106

—

126

C₄×C₈

—

127

C₄×C₈

C₂×D₅₃

yes

characteristic

128

C₂×Q₈

—

129

C₂×Q₈

—

106

—

130

C₂×Q₈

C₄×D₅₃

yes

characteristic

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

106

—

137

C₂×C₈

—

106

—

138

C₂×C₈

—

212

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

106

—

142

C₄⋊C₄

—

106

—

143

C₄⋊C₄

—

106

—

144

C₄⋊C₄

—

106

—

145

C₄⋊C₄

—

106

—

146

C₄⋊C₄

—

106

—

147

C₄⋊C₄

C₄×D₅₃

yes

characteristic

148

C₂²⋊C₄

—

106

—

149

C₂²⋊C₄

—

106

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

106

—

155

C₄²

C₂²×D₅₃

yes

characteristic

156

C₂³

—

157

Q₈

—

158

Q₈

—

159

Q₈

—

106

—

160

Q₈

—

106

—

161

Q₈

—

212

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

106

—

170

C₂×C₄

—

106

—

171

C₂×C₄

—

106

—

172

C₂×C₄

—

106

—

173

C₂×C₄

—

212

—

174

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

175

C₂×C₄

D₄×D₅₃

yes

characteristic

176

C₂×C₄

D₄×D₅₃

yes

characteristic

177

C₈

—

178

C₈

—

179

C₈

—

212

—

180

C₈

—

212

—

181

C₂²

—

182

C₂²

—

106

—

183

C₂²

—

106

—

184

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

106

—

190

C₄

—

106

—

191

C₄

—

212

—

192

C₄

—

212

—

193

C₄

#13133223

yes

characteristic

194

C₄

#13133225

yes

characteristic

195

C₄

SD₁₆×D₅₃

yes

characteristic

196

C₄

Q₁₆×D₅₃

yes

characteristic

197

C₂

—

106

—

198

C₂

#16810588

yes

characteristic

199

C₂

#103×D₅₃

yes

characteristic

200

C₂

#16810638

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×Q₈⋊C₈	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	Q₈⋊C₈	D₅₃
hypocenter	C₅₃	#1481
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×Q₈⋊C₈

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

Q₈⋊C₈

D₅₃

hypocenter

C₅₃

#1481

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
4. #16810355	C₂	semidirect
5. Q₈⋊#88322	C₂	semidirect
6. #16810353	C₂	semidirect
7. C₅₃×Q₈⋊C₈	C₂	semidirect
107. Q₈⋊C₈	D₅₃	semidirect
108. C₅₃	#1481	semidirect

H ≅ G/N

type

4. #16810355

C₂

semidirect

5. Q₈⋊#88322

C₂

semidirect

6. #16810353

C₂

semidirect

7. C₅₃×Q₈⋊C₈

C₂

semidirect

107. Q₈⋊C₈

D₅₃

semidirect

108. C₅₃

#1481

semidirect

Subgroups — #20273116

Conjugacy classes of subgroups

Complements