Group #27279409 — subgroups

Subgroups — #27279409

Order 13568.

404 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 111 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	#20283817	C₂	1	1	yes	maximal, characteristic
2	#20273310	C₂	1	1	yes	maximal, characteristic
3	#20273136	C₂	1	2	yes	maximal
4	#20272552	C₂	1	2	yes	maximal
5	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
6	#16810754	—	2	2	no	—
7	Q₈×#88322	—	2	2	no	—
8	#16810684	—	2	2	no	—
9	#16810676	C₂²	1	2	yes	—
10	C₄×#11719917	C₂²	1	2	yes	—
11	#16810365	—	2	2	no	—
12	Q₈⋊#88322	—	2	2	no	—
13	Q₈⋊#88322	—	2	2	no	—
14	C₅₃×#533	C₂²	1	1	yes	characteristic
15	C₅₃×#533	C₂²	1	1	yes	characteristic
16	C₅₃×#500	C₄	1	2	yes	—
17	C₅₃×#499	—	2	2	no	—
18	C₅₃×#498	C₄	1	2	yes	—
19	C₅₃×#427	—	2	2	no	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#11719917	—	2	2	no	—
22	C₄⋊#88322	—	2	4	no	—
23	C₄⋊#88322	D₄	1	2	yes	—
24	C₄⋊#88322	D₄	1	2	yes	—
25	C₂²⋊#88322	—	2	2	no	—
26	#13133128	—	2	2	no	—
27	#13133128	—	2	2	no	—
28	C₄×#88322	—	2	2	no	—
29	C₄×#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	2	1	no	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
34	#125×C₅₃	—	2	1	no	—
35	#125×C₅₃	D₄	1	2	yes	—
36	C₂²⋊Q₈×C₅₃	—	2	2	no	—
37	C₂²⋊Q₈×C₅₃	—	2	2	no	—
38	C₄×Q₈×C₅₃	—	2	2	no	—
39	C₄×Q₈×C₅₃	—	2	2	no	—
40	#117×C₅₃	—	2	1	no	—
41	#117×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#107×C₅₃	—	2	4	no	—
44	#106×C₅₃	—	2	4	no	—
45	C₄×C₈×C₅₃	—	2	1	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	2	yes	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
61	C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂²⋊C₄×C₅₃	—	4	1	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	4	1	no	—
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	2	yes	—
68	#11719917	—	2	4	no	—
69	#11719917	—	4	2	no	—
70	C₂×#88322	—	2	2	no	—
71	C₂×#88322	—	2	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₂³×C₅₃	#123	1	1	yes	characteristic
74	Q₈×C₅₃	—	4	2	no	—
75	Q₈×C₅₃	—	4	2	no	—
76	Q₈×C₅₃	—	4	2	no	—
77	Q₈×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#121	1	2	yes	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	—	2	2	no	—
89	#88322	—	4	4	no	—
90	#88322	—	4	2	no	—
91	#88322	—	4	2	no	—
92	#9068	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	#386	1	1	yes	characteristic
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	4	4	no	—
99	C₄×C₅₃	—	8	2	no	—
100	C₄×C₅₃	#443	1	2	yes	—
101	C₄×C₅₃	#444	1	2	yes	—
102	C₄×C₅₃	#456	1	2	yes	—
103	#3160	D₅₃	1	1	yes	characteristic
104	#2986	—	53	1	no	—
105	#1526	—	53	1	no	—
106	#1485	—	53	2	no	—
107	#1278	—	53	2	no	—
108	C₂×C₅₃	—	4	1	no	—
109	C₂×C₅₃	#1876	1	2	yes	—
110	C₂×C₅₃	#1886	1	1	yes	characteristic
111	#533	C₂×D₅₃	1	1	yes	characteristic
112	#533	C₂×D₅₃	1	1	yes	characteristic
113	#500	Dic₅₃	1	2	yes	—
114	#499	—	2	2	no	—
115	#498	Dic₅₃	1	2	yes	—
116	#446	—	106	2	no	—
117	C₈×Q₈	—	106	2	no	—
118	#433	—	106	2	no	—
119	#431	—	53	2	no	—
120	#427	—	2	2	no	—
121	C₄×M₁₆	—	53	2	no	—
122	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
123	#330	—	106	2	no	—
124	Q₈⋊C₈	—	106	2	no	—
125	Q₈⋊C₈	—	106	2	no	—
126	C₅₃	#9068	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	106	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
130	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
131	#125	—	2	1	no	—
132	#125	C₂²⋊D₅₃	1	2	yes	—
133	C₂²⋊Q₈	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₄×Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	#117	—	2	1	no	—
138	#117	C₂²⋊D₅₃	1	2	yes	—
139	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
140	#107	—	2	4	no	—
141	#106	—	2	4	no	—
142	C₄⋊C₈	—	53	2	no	—
143	C₄⋊C₈	—	53	2	no	—
144	C₄⋊C₈	—	106	4	no	—
145	C₂²⋊C₈	—	106	2	no	—
146	#97	—	106	2	no	—
147	#97	—	106	2	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	—	106	2	no	—
150	C₄×C₈	—	106	2	no	—
151	C₄×C₈	—	106	2	no	—
152	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
153	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂²×C₄	#11719947	1	2	yes	—
157	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
158	M₁₆	—	106	4	no	—
159	M₁₆	—	212	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	—	106	2	no	—
165	C₂×C₈	—	212	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	C₂²⋊Dic₅₃	1	4	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	D₄×D₅₃	1	2	yes	—
177	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
178	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂³	#13133198	1	1	yes	characteristic
180	Q₈	—	4	2	no	—
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	4	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
191	C₂×C₄	#13133185	1	1	yes	characteristic
192	C₂×C₄	#13133189	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	212	4	no	—
196	C₈	—	212	2	no	—
197	C₈	—	212	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#16810503	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	4	no	—
204	C₄	—	8	2	no	—
205	C₄	#16810740	1	2	yes	—
206	C₄	#16810745	1	2	yes	—
207	C₄	#16810805	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	#20275044	1	2	yes	—
210	C₂	#20275109	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283817

C₂

yes

maximal, characteristic

#20273310

C₂

yes

maximal, characteristic

#20273136

C₂

yes

maximal

#20272552

C₂

yes

maximal

C₅₃×#3160

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×#11719917

C₂²

yes

—

#16810365

—

Q₈⋊#88322

—

Q₈⋊#88322

—

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#500

C₄

yes

—

C₅₃×#499

—

C₅₃×#498

C₄

yes

—

C₅₃×#427

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

D₄

yes

—

C₄⋊#88322

D₄

yes

—

—

—

—

—

—

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

#125×C₅₃

—

#125×C₅₃

D₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

#117×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

#106×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

—

—

—

—

—

C₂³×C₅₃

#123

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

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#386

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

100

C₄×C₅₃

#443

yes

—

101

C₄×C₅₃

#444

yes

—

102

C₄×C₅₃

#456

yes

—

103

#3160

D₅₃

yes

characteristic

104

#2986

—

105

#1526

—

106

#1485

—

107

#1278

—

108

C₂×C₅₃

—

109

C₂×C₅₃

#1876

yes

—

110

C₂×C₅₃

#1886

yes

characteristic

111

#533

C₂×D₅₃

yes

characteristic

112

#533

C₂×D₅₃

yes

characteristic

113

#500

Dic₅₃

yes

—

114

#499

—

115

#498

Dic₅₃

yes

—

116

#446

—

106

—

117

C₈×Q₈

—

106

—

118

#433

—

106

—

119

#431

—

120

#427

—

121

C₄×M₁₆

—

122

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

123

#330

—

106

—

124

Q₈⋊C₈

—

106

—

125

Q₈⋊C₈

—

106

—

126

C₅₃

#9068

yes

characteristic, complemented, Sylow

127

C₂×M₁₆

—

106

—

128

C₂²×C₈

—

129

C₄⋊Q₈

C₂×Dic₅₃

yes

—

130

C₄⋊Q₈

C₂×Dic₅₃

yes

—

131

#125

—

132

#125

C₂²⋊D₅₃

yes

—

133

C₂²⋊Q₈

—

134

C₂²⋊Q₈

—

135

C₄×Q₈

—

136

C₄×Q₈

—

137

#117

—

138

#117

C₂²⋊D₅₃

yes

—

139

C₂×C₄²

C₂²×D₅₃

yes

characteristic

140

#107

—

141

#106

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

106

—

145

C₂²⋊C₈

—

106

—

146

#97

—

106

—

147

#97

—

106

—

148

C₄×C₈

—

149

C₄×C₈

—

106

—

150

C₄×C₈

—

106

—

151

C₄×C₈

—

106

—

152

C₄×C₈

C₂×Dic₅₃

yes

characteristic

153

C₄×C₈

C₂×Dic₅₃

yes

characteristic

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂²×C₄

#11719947

yes

—

157

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

158

M₁₆

—

106

—

159

M₁₆

—

212

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

106

—

164

C₂×C₈

—

106

—

165

C₂×C₈

—

212

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

C₂²⋊Dic₅₃

yes

—

172

C₂²⋊C₄

—

173

C₂²⋊C₄

—

174

C₄²

—

175

C₄²

—

176

C₄²

D₄×D₅₃

yes

—

177

C₄²

C₂²×Dic₅₃

yes

characteristic

178

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

179

C₂³

#13133198

yes

characteristic

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

190

C₂×C₄

D₄×Dic₅₃

yes

—

191

C₂×C₄

#13133185

yes

characteristic

192

C₂×C₄

#13133189

yes

—

193

C₈

—

194

C₈

—

195

C₈

—

212

—

196

C₈

—

212

—

197

C₈

—

212

—

198

C₂²

—

199

C₂²

—

200

C₂²

#16810503

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

#16810740

yes

—

206

C₄

#16810745

yes

—

207

C₄

#16810805

yes

—

208

C₂

—

209

C₂

#20275044

yes

—

210

C₂

#20275109

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	#123
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#9068
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#13133185

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#3160

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
126. C₅₃	#9068	semidirect

H ≅ G/N

type

126. C₅₃

#9068

semidirect

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

subgroup	index
#20283817	2
#20273310	2
#20273136	2
#20272552	2
C₅₃×#3160	2
#9068	53

subgroup

index

C₅₃×#3160

Subgroups — #27279409

Conjugacy classes of subgroups

Complements

Maximal subgroups