Group #27284052 — subgroups

Subgroups — #27284052

Order 13568.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20274853	C₂	1	1	yes	maximal, characteristic
2	#20273170	C₂	1	2	yes	maximal
3	#20273170	C₂	1	2	yes	maximal
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
6	#16810647	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810388	—	2	2	no	—
10	Q₈⋊#88322	—	2	4	no	—
11	Q₈⋊#88322	—	2	4	no	—
12	#16810345	—	2	2	no	—
13	C₅₃×#533	C₂²	1	2	yes	—
14	C₅₃×#500	C₄	1	1	yes	characteristic
15	C₅₃×#500	C₄	1	1	yes	characteristic
16	C₅₃×#499	—	2	2	no	—
17	C₅₃×#498	C₄	1	2	yes	—
18	C₅₃×#427	—	2	2	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	C₂²×#88322	—	2	1	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	D₄	1	2	yes	—
28	C₄×#88322	D₄	1	2	yes	—
29	C₂²×C₈×C₅₃	D₄	1	2	yes	—
30	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
32	#125×C₅₃	—	2	2	no	—
33	C₂²⋊Q₈×C₅₃	—	2	2	no	—
34	C₂²⋊Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	C₄×Q₈×C₅₃	—	2	2	no	—
37	#117×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#107×C₅₃	—	2	2	no	—
40	#107×C₅₃	—	2	2	no	—
41	#106×C₅₃	—	2	4	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂×Q₈×C₅₃	—	2	2	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₂²⋊C₄×C₅₃	—	4	2	no	—
60	C₄²×C₅₃	—	4	2	no	—
61	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	#11719917	—	4	4	no	—
66	C₂×#88322	—	2	4	no	—
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	—	2	2	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂³×C₅₃	#124	1	1	yes	characteristic
72	Q₈×C₅₃	—	4	4	no	—
73	Q₈×C₅₃	—	4	4	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#137	1	4	yes	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	2	2	no	—
87	#88322	—	2	4	no	—
88	#88322	—	4	2	no	—
89	#88322	—	8	2	no	—
90	#9859	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	#390	1	1	yes	characteristic
94	C₂²×C₅₃	#491	1	2	yes	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	8	2	no	—
98	C₄×C₅₃	—	8	2	no	—
99	C₄×C₅₃	#419	1	2	yes	—
100	C₄×C₅₃	#443	1	2	yes	—
101	C₄×C₅₃	#444	1	2	yes	—
102	C₄×C₅₃	#497	1	2	yes	—
103	#3160	D₅₃	1	1	yes	characteristic
104	#1837	—	53	1	no	—
105	#1493	—	53	2	no	—
106	#1493	—	53	2	no	—
107	C₂×#322	—	53	1	no	—
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	#1960	1	1	yes	characteristic
110	C₂×C₅₃	#1974	1	1	yes	characteristic
111	C₂×C₅₃	#1975	1	1	yes	characteristic
112	#533	C₂×D₅₃	1	2	yes	—
113	#500	Dic₅₃	1	1	yes	characteristic
114	#500	Dic₅₃	1	1	yes	characteristic
115	#499	—	2	2	no	—
116	#498	Dic₅₃	1	2	yes	—
117	#427	—	2	2	no	—
118	#423	—	53	2	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
122	#341	—	106	2	no	—
123	Q₈⋊C₈	—	106	4	no	—
124	Q₈⋊C₈	—	106	4	no	—
125	#322	—	106	2	no	—
126	C₅₃	#9859	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	106	2	no	—
128	C₂²×C₈	—	106	1	no	—
129	C₂²×C₈	—	106	1	no	—
130	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
131	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
132	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
133	#125	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₄×Q₈	—	2	2	no	—
138	#117	—	2	2	no	—
139	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
140	#107	—	2	2	no	—
141	#107	—	2	2	no	—
142	#106	—	2	4	no	—
143	C₄⋊C₈	—	106	2	no	—
144	C₄⋊C₈	—	106	2	no	—
145	C₄×C₈	—	53	2	no	—
146	C₄×C₈	—	53	2	no	—
147	C₄×C₈	—	106	1	no	—
148	C₄×C₈	—	106	1	no	—
149	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
150	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂²×C₄	#11719947	1	1	yes	characteristic
155	C₂²×C₄	#11719947	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
157	M₁₆	—	212	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	106	4	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	2	no	—
162	C₂×C₈	—	212	2	no	—
163	C₂×C₈	—	212	2	no	—
164	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
165	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	D₄×D₅₃	1	1	yes	characteristic
175	C₄²	D₄×D₅₃	1	1	yes	characteristic
176	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
177	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂³	#13133204	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	4	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
186	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
188	C₂×C₄	#13133188	1	1	yes	characteristic
189	C₂×C₄	#13133188	1	1	yes	characteristic
190	C₂×C₄	#13133213	1	1	yes	characteristic
191	C₂×C₄	#13133248	1	4	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	106	4	no	—
195	C₈	—	424	2	no	—
196	C₈	—	212	2	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#16810526	1	1	yes	characteristic
200	C₂²	#16810991	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	2	no	—
204	C₄	—	8	2	no	—
205	C₄	#16810632	1	2	yes	—
206	C₄	#16810740	1	2	yes	—
207	C₄	#16810745	1	2	yes	—
208	C₄	#16811015	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	#20275536	1	1	yes	characteristic
211	C₂	#20275604	1	1	yes	characteristic
212	C₂	#20275609	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274853

C₂

yes

maximal, characteristic

#20273170

C₂

yes

maximal

#20273170

C₂

yes

maximal

C₂×#16810345

C₂

yes

maximal, characteristic

C₅₃×#3160

C₂

yes

maximal, characteristic

#16810647

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

—

—

—

—

C₅₃×#533

C₂²

yes

—

C₅₃×#500

C₄

yes

characteristic

C₅₃×#500

C₄

yes

characteristic

C₅₃×#499

—

C₅₃×#498

C₄

yes

—

C₅₃×#427

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

C₂²×#88322

—

C₂²×#88322

—

—

—

—

—

D₄

yes

—

C₄×#88322

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

#125×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

#107×C₅₃

—

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

C₂²×C₄×C₅₃

yes

C₂²×C₄×C₅₃

yes

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

—

—

—

—

—

C₂³×C₅₃

#124

yes

characteristic

Q₈×C₅₃

—

Q₈×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

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#491

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#419

yes

—

100

C₄×C₅₃

#443

yes

—

101

C₄×C₅₃

#444

yes

—

102

C₄×C₅₃

#497

yes

—

103

#3160

D₅₃

yes

characteristic

104

#1837

—

105

#1493

—

106

#1493

—

107

C₂×#322

—

108

C₂×C₅₃

—

109

C₂×C₅₃

#1960

yes

characteristic

110

C₂×C₅₃

#1974

yes

characteristic

111

C₂×C₅₃

#1975

yes

characteristic

112

#533

C₂×D₅₃

yes

—

113

#500

Dic₅₃

yes

characteristic

114

#500

Dic₅₃

yes

characteristic

115

#499

—

116

#498

Dic₅₃

yes

—

117

#427

—

118

#423

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

122

#341

—

106

—

123

Q₈⋊C₈

—

106

—

124

Q₈⋊C₈

—

106

—

125

#322

—

106

—

126

C₅₃

#9859

yes

characteristic, complemented, Sylow

127

C₂×M₁₆

—

106

—

128

C₂²×C₈

—

106

—

129

C₂²×C₈

—

106

—

130

C₂²×C₈

C₂²⋊D₅₃

yes

—

131

C₄⋊Q₈

C₂×Dic₅₃

yes

—

132

C₄⋊Q₈

C₂×Dic₅₃

yes

—

133

#125

—

134

C₂²⋊Q₈

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

C₄×Q₈

—

138

#117

—

139

C₂×C₄²

C₂²×D₅₃

yes

characteristic

140

#107

—

141

#107

—

142

#106

—

143

C₄⋊C₈

—

106

—

144

C₄⋊C₈

—

106

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

106

—

148

C₄×C₈

—

106

—

149

C₄×C₈

C₂×Dic₅₃

yes

characteristic

150

C₄×C₈

C₂×Dic₅₃

yes

characteristic

151

C₄×C₈

C₂²⋊D₅₃

yes

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂²×C₄

#11719947

yes

characteristic

155

C₂²×C₄

#11719947

yes

characteristic

156

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

157

M₁₆

—

212

—

158

C₂×C₈

—

159

C₂×C₈

—

106

—

160

C₂×C₈

—

106

—

161

C₂×C₈

—

106

—

162

C₂×C₈

—

212

—

163

C₂×C₈

—

212

—

164

C₂×C₈

C₂²⋊Dic₅₃

yes

—

165

C₂×C₈

C₂²⋊Dic₅₃

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

D₄×D₅₃

yes

characteristic

175

C₄²

D₄×D₅₃

yes

characteristic

176

C₄²

C₂²×Dic₅₃

yes

characteristic

177

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

178

C₂³

#13133204

yes

characteristic

179

Q₈

—

180

Q₈

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

186

C₂×C₄

D₄×Dic₅₃

yes

characteristic

187

C₂×C₄

D₄×Dic₅₃

yes

characteristic

188

C₂×C₄

#13133188

yes

characteristic

189

C₂×C₄

#13133188

yes

characteristic

190

C₂×C₄

#13133213

yes

characteristic

191

C₂×C₄

#13133248

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

106

—

195

C₈

—

424

—

196

C₈

—

212

—

197

C₂²

—

198

C₂²

—

199

C₂²

#16810526

yes

characteristic

200

C₂²

#16810991

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

#16810632

yes

—

206

C₄

#16810740

yes

—

207

C₄

#16810745

yes

—

208

C₄

#16811015

yes

—

209

C₂

—

210

C₂

#20275536

yes

characteristic

211

C₂

#20275604

yes

characteristic

212

C₂

#20275609

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	#124
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#9859
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#13133213

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	#9859	semidirect

H ≅ G/N

type

126. C₅₃

#9859

semidirect

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

subgroup	index
#20274853	2
#20273170	2
C₂×#16810345	2
C₅₃×#3160	2
#9859	53

subgroup

index

C₅₃×#3160

Subgroups — #27284052

Conjugacy classes of subgroups

Complements

Maximal subgroups