Group #27287195 — subgroups

Subgroups — #27287195

Order 13568.

400 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20283078	C₂	1	1	yes	maximal, characteristic
2	#20273279	C₂	1	2	yes	maximal
3	#20272575	C₂	1	2	yes	maximal
4	#20272548	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1994	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133209	—	2	1	no	—
7	#16810676	C₂²	1	2	yes	—
8	#16810521	—	2	4	no	—
9	#16810493	—	2	1	no	—
10	#16810489	C₂²	1	2	yes	—
11	#16810360	—	2	4	no	—
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₅₃×#423	C₂²	1	1	yes	characteristic
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#339	—	2	1	no	—
16	C₅₃×#339	—	2	1	no	—
17	#13133209	—	4	2	no	—
18	#13133209	—	4	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	4	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×#11719919	—	2	4	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₂²⋊#88322	—	2	2	no	—
24	C₄×#88322	—	2	2	no	—
25	#13133124	—	2	2	no	—
26	#13133124	D₄	1	4	yes	—
27	C₂×M₁₆×C₅₃	—	2	1	no	—
28	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	—
29	#125×C₅₃	—	2	1	no	—
30	C₂²⋊Q₈×C₅₃	—	2	2	no	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	#117×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₂×Q₈×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	—	2	1	no	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	2	4	no	—
44	M₁₆×C₅₃	—	4	1	no	—
45	M₁₆×C₅₃	—	4	1	no	—
46	C₂×C₈×C₅₃	—	4	1	no	—
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₂²⋊C₄×C₅₃	—	4	1	no	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	—	4	4	no	—
54	C₄²×C₅₃	—	4	1	no	—
55	C₄²×C₅₃	SD₁₆	1	2	yes	—
56	C₄²×C₅₃	Q₁₆	1	2	yes	—
57	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅₃	#41	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	2	yes	—
60	C₄⋊Dic₅₃	—	4	8	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	#11719919	—	4	8	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂³×C₅₃	#120	1	1	yes	characteristic
66	Q₈×C₅₃	—	8	2	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#124	1	2	yes	—
76	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
81	C₂×Dic₅₃	—	4	8	no	—
82	C₂×Dic₅₃	—	4	4	no	—
83	C₈×C₅₃	—	8	1	no	—
84	#88322	—	8	2	no	—
85	#10365	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	#393	1	1	yes	characteristic
88	C₂²×C₅₃	#450	1	1	yes	characteristic
89	C₂²×C₅₃	#451	1	1	yes	characteristic
90	C₄×C₅₃	—	2	4	no	—
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	4	2	no	—
94	C₄×C₅₃	—	8	2	no	—
95	C₄×C₅₃	#475	1	1	yes	characteristic
96	C₄×C₅₃	#477	1	1	yes	characteristic
97	C₄×C₅₃	#487	1	1	yes	characteristic
98	C₄×C₅₃	#489	1	1	yes	characteristic
99	Dic₅₃	—	8	4	no	—
100	#2859	—	53	1	no	—
101	#1994	D₅₃	1	1	yes	characteristic
102	#1518	—	53	2	no	—
103	#1286	—	53	2	no	—
104	#1276	—	53	1	no	—
105	C₂×C₅₃	—	2	1	no	—
106	C₂×C₅₃	—	2	1	no	—
107	C₂×C₅₃	#2024	1	1	yes	characteristic
108	C₂×C₅₃	#2031	1	2	yes	—
109	#533	C₂×D₅₃	1	1	yes	characteristic
110	C₂×#125	—	106	1	no	—
111	#431	—	53	2	no	—
112	#423	C₂×D₅₃	1	1	yes	characteristic
113	#389	—	106	4	no	—
114	#384	—	106	1	no	—
115	#382	—	53	2	no	—
116	C₄³	C₂×D₅₃	1	1	yes	characteristic
117	#339	—	2	1	no	—
118	#339	—	2	1	no	—
119	#328	—	106	4	no	—
120	C₅₃	#10365	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	—	2	1	no	—
122	C₄⋊Q₈	Dic₁₀₆	1	2	yes	—
123	#125	—	2	1	no	—
124	#125	—	212	2	no	—
125	#125	—	212	2	no	—
126	C₂²⋊Q₈	—	2	2	no	—
127	C₄×Q₈	—	2	2	no	—
128	#117	—	2	1	no	—
129	C₂×C₄⋊C₄	—	106	4	no	—
130	C₂×C₄⋊C₄	—	106	4	no	—
131	C₂×C₄⋊C₄	—	106	2	no	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	106	2	no	—
138	C₂²⋊C₈	—	106	2	no	—
139	C₄×C₈	—	106	2	no	—
140	#95	—	53	4	no	—
141	#95	—	106	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	106	4	no	—
146	C₂²×C₄	D₄×D₅₃	1	2	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	M₁₆	—	4	1	no	—
149	M₁₆	—	4	1	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	212	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	212	8	no	—
156	C₄⋊C₄	—	212	8	no	—
157	C₄⋊C₄	—	212	2	no	—
158	C₄⋊C₄	—	212	2	no	—
159	C₂²⋊C₄	—	4	1	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	2	1	no	—
162	C₄²	—	4	4	no	—
163	C₄²	—	4	1	no	—
164	C₄²	#11719924	1	2	yes	—
165	C₄²	#11719925	1	2	yes	—
166	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
167	C₄²	#11719945	1	1	yes	characteristic
168	C₄²	#11719947	1	2	yes	—
169	C₂³	#13133185	1	1	yes	characteristic
170	Q₈	—	8	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	212	8	no	—
178	C₂×C₄	—	212	4	no	—
179	C₂×C₄	#13133194	1	2	yes	—
180	C₂×C₄	#13133196	1	1	yes	characteristic
181	C₂×C₄	#13133203	1	2	yes	—
182	C₂×C₄	#13133204	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
185	C₂×C₄	#13133242	1	1	yes	characteristic
186	C₂×C₄	#13133247	1	1	yes	characteristic
187	C₈	—	8	1	no	—
188	C₈	—	424	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#16810535	1	1	yes	characteristic
191	C₂²	#16810773	1	1	yes	characteristic
192	C₂²	#16810778	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	8	2	no	—
198	C₄	—	424	4	no	—
199	C₄	#16810908	1	1	yes	characteristic
200	C₄	#16810918	1	1	yes	characteristic
201	C₄	#16810973	1	1	yes	characteristic
202	C₄	#16810982	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#20275856	1	1	yes	characteristic
206	C₂	#20275887	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283078

C₂

yes

maximal, characteristic

#20273279

C₂

yes

maximal

#20272575

C₂

yes

maximal

#20272548

C₂

yes

maximal, characteristic

C₅₃×#1994

C₂

yes

maximal, characteristic

C₂×#13133209

—

#16810676

C₂²

yes

—

#16810521

—

#16810493

—

#16810489

C₂²

yes

—

#16810360

—

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#423

C₂²

yes

characteristic

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#339

—

C₅₃×#339

—

#13133209

—

#13133209

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

—

#125×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#10365

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

100

#2859

—

101

#1994

D₅₃

yes

characteristic

102

#1518

—

103

#1286

—

104

#1276

—

105

C₂×C₅₃

—

106

C₂×C₅₃

—

107

C₂×C₅₃

#2024

yes

characteristic

108

C₂×C₅₃

#2031

yes

—

109

#533

C₂×D₅₃

yes

characteristic

110

C₂×#125

—

106

—

111

#431

—

112

#423

C₂×D₅₃

yes

characteristic

113

#389

—

106

—

114

#384

—

106

—

115

#382

—

116

C₄³

C₂×D₅₃

yes

characteristic

117

#339

—

118

#339

—

119

#328

—

106

—

120

C₅₃

#10365

yes

characteristic, complemented, Sylow

121

C₂×M₁₆

—

122

C₄⋊Q₈

Dic₁₀₆

yes

—

123

#125

—

124

#125

—

212

—

125

#125

—

212

—

126

C₂²⋊Q₈

—

127

C₄×Q₈

—

128

#117

—

129

C₂×C₄⋊C₄

—

106

—

130

C₂×C₄⋊C₄

—

106

—

131

C₂×C₄⋊C₄

—

106

—

132

C₂×C₄²

—

133

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

134

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

135

C₂×C₄²

C₂²×D₅₃

yes

characteristic

136

C₄⋊C₈

—

137

C₄⋊C₈

—

106

—

138

C₂²⋊C₈

—

106

—

139

C₄×C₈

—

106

—

140

#95

—

141

#95

—

106

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

106

—

146

C₂²×C₄

D₄×D₅₃

yes

—

147

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

212

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

212

—

156

C₄⋊C₄

—

212

—

157

C₄⋊C₄

—

212

—

158

C₄⋊C₄

—

212

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

#11719924

yes

—

165

C₄²

#11719925

yes

—

166

C₄²

C₂×Dic₁₀₆

yes

characteristic

167

C₄²

#11719945

yes

characteristic

168

C₄²

#11719947

yes

—

169

C₂³

#13133185

yes

characteristic

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

212

—

178

C₂×C₄

—

212

—

179

C₂×C₄

#13133194

yes

—

180

C₂×C₄

#13133196

yes

characteristic

181

C₂×C₄

#13133203

yes

—

182

C₂×C₄

#13133204

yes

characteristic

183

C₂×C₄

C₂×#11719924

yes

characteristic

184

C₂×C₄

C₂×#11719925

yes

characteristic

185

C₂×C₄

#13133242

yes

characteristic

186

C₂×C₄

#13133247

yes

characteristic

187

C₈

—

188

C₈

—

424

—

189

C₂²

—

190

C₂²

#16810535

yes

characteristic

191

C₂²

#16810773

yes

characteristic

192

C₂²

#16810778

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

424

—

199

C₄

#16810908

yes

characteristic

200

C₄

#16810918

yes

characteristic

201

C₄

#16810973

yes

characteristic

202

C₄

#16810982

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

#20275856

yes

characteristic

206

C₂

#20275887

yes

—

	subgroup	G/H
center	C₂²	#16810535
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1994	C₂
socle	C₂²×C₅₃	#393
prime radical	C₂³×C₅₃	#120
hypercenter	#1994	D₅₃
hypocenter	C₅₃	#10365
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#16810535

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1994

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

N	H ≅ G/N	type
120. C₅₃	#10365	semidirect

H ≅ G/N

type

120. C₅₃

#10365

semidirect

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

subgroup	index
#20283078	2
#20273279	2
#20272575	2
#20272548	2
C₅₃×#1994	2
#10365	53

subgroup

index

C₅₃×#1994

Subgroups — #27287195

Conjugacy classes of subgroups

Complements

Maximal subgroups