Group #27284017 — subgroups

Subgroups — #27284017

Order 13568.

422 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20275691	C₂	1	2	yes	maximal
3	#20274460	C₂	1	1	yes	maximal, characteristic
4	#20273169	C₂	1	2	yes	maximal
5	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	2	yes	—
7	#16811028	C₄	1	1	yes	characteristic, complemented
8	#16811003	C₄	1	1	yes	characteristic, complemented
9	#16810974	C₄	1	2	yes	complemented
10	#16810864	—	2	2	no	—
11	C₄×Dic₂₁₂	—	2	2	no	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810384	—	2	4	no	—
15	#16810377	—	2	2	no	—
16	#16810353	—	2	4	no	—
17	C₅₃×#423	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#334	—	2	2	no	—
20	C₂×Dic₂₁₂	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	#13133217	C₂×C₄	1	2	yes	—
23	#13133217	C₂×C₄	1	2	yes	—
24	#13133209	—	2	2	no	—
25	#13133196	—	2	2	no	—
26	#13133196	—	2	2	no	—
27	C₄×Dic₁₀₆	—	2	2	no	—
28	C₄×Dic₁₀₆	—	2	2	no	—
29	#13133176	—	2	2	no	—
30	C₂×C₄×Dic₅₃	—	2	1	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
33	#13133151	—	2	2	no	—
34	#13133142	—	2	4	no	—
35	C₄×#88322	—	2	1	no	—
36	C₄×#88322	D₄	1	2	yes	—
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	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	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	2	2	no	—
49	C₂×Dic₁₀₆	—	2	2	no	—
50	C₂×Dic₁₀₆	—	2	2	no	—
51	M₁₆×C₅₃	—	4	4	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	1	yes	characteristic
60	C₂²⋊Dic₅₃	—	4	2	no	—
61	C₄⋊Dic₅₃	—	2	2	no	—
62	C₄⋊Dic₅₃	—	2	2	no	—
63	C₄⋊Dic₅₃	—	2	2	no	—
64	C₄⋊Dic₅₃	—	2	2	no	—
65	#11719919	—	4	2	no	—
66	#11719919	—	4	2	no	—
67	C₄×Dic₅₃	—	2	4	no	—
68	C₄×Dic₅₃	—	2	4	no	—
69	C₄×Dic₅₃	—	4	2	no	—
70	C₄×Dic₅₃	—	4	2	no	—
71	C₂×#88322	—	2	2	no	—
72	C₂×#88322	—	4	2	no	—
73	Dic₂₁₂	—	4	4	no	—
74	C₂³×C₅₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#137	1	4	yes	—
84	C₂×Dic₅₃	—	2	4	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	C₂×Dic₅₃	—	4	2	no	—
87	C₂×Dic₅₃	—	4	2	no	—
88	C₂×Dic₅₃	—	4	2	no	—
89	Dic₁₀₆	—	4	4	no	—
90	Dic₁₀₆	—	4	4	no	—
91	C₈×C₅₃	—	2	2	no	—
92	C₈×C₅₃	—	2	2	no	—
93	C₈×C₅₃	—	8	2	no	—
94	#88322	—	4	2	no	—
95	#9853	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	#390	1	1	yes	characteristic
99	C₂²×C₅₃	#470	1	2	yes	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	#421	1	2	yes	—
103	C₄×C₅₃	#441	1	2	yes	—
104	C₄×C₅₃	#443	1	2	yes	—
105	C₄×C₅₃	#497	1	2	yes	—
106	Dic₅₃	—	4	4	no	—
107	Dic₅₃	—	8	2	no	—
108	Dic₅₃	—	8	2	no	—
109	#3170	—	53	1	no	—
110	#1994	—	53	2	no	—
111	#1744	—	53	1	no	—
112	#1566	D₅₃	1	1	yes	characteristic
113	#1493	—	53	2	no	—
114	C₂×C₅₃	—	2	2	no	—
115	C₂×C₅₃	#1963	1	1	yes	characteristic
116	C₂×C₅₃	#1971	1	1	yes	characteristic
117	C₂×C₅₃	#1974	1	1	yes	characteristic
118	#533	—	53	2	no	—
119	#500	—	53	1	no	—
120	#494	—	53	1	no	—
121	#487	—	53	2	no	—
122	#467	—	106	2	no	—
123	C₄×Q₁₆	—	106	2	no	—
124	#423	C₂×D₅₃	1	2	yes	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₄³	—	53	1	no	—
128	#339	—	106	4	no	—
129	#336	—	106	2	no	—
130	#334	—	2	2	no	—
131	Q₈⋊C₈	—	106	4	no	—
132	C₅₃	#9853	1	1	yes	characteristic, complemented, Sylow
133	C₂×Q₁₆	—	106	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂²×C₈	—	106	1	no	—
136	C₂²×C₈	D₂₁₂	1	2	yes	—
137	C₄⋊Q₈	—	53	2	no	—
138	C₄⋊Q₈	—	53	2	no	—
139	#125	—	106	2	no	—
140	C₂²⋊Q₈	—	106	2	no	—
141	C₂²⋊Q₈	—	106	2	no	—
142	C₄×Q₈	—	106	2	no	—
143	C₄×Q₈	—	106	2	no	—
144	#117	—	106	2	no	—
145	C₂×C₄²	—	53	2	no	—
146	C₂×C₄²	—	106	1	no	—
147	C₂×C₄²	—	106	1	no	—
148	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
149	#107	—	106	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	#103	—	106	4	no	—
153	C₄×C₈	—	53	2	no	—
154	C₄×C₈	—	106	1	no	—
155	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
156	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
158	C₂×Q₈	—	106	2	no	—
159	C₂×Q₈	—	106	2	no	—
160	C₂²×C₄	—	106	2	no	—
161	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
162	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
163	C₂²×C₄	#11719949	1	1	yes	characteristic
164	Q₁₆	—	212	4	no	—
165	M₁₆	—	4	4	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	#11719921	1	2	yes	—
171	C₂×C₈	#11719921	1	2	yes	—
172	C₄⋊C₄	—	106	2	no	—
173	C₄⋊C₄	—	106	2	no	—
174	C₄⋊C₄	—	106	2	no	—
175	C₄⋊C₄	—	106	2	no	—
176	C₄⋊C₄	—	212	2	no	—
177	C₄⋊C₄	—	212	2	no	—
178	C₂²⋊C₄	—	212	2	no	—
179	C₄²	—	106	4	no	—
180	C₄²	—	106	4	no	—
181	C₄²	—	212	2	no	—
182	C₄²	—	212	2	no	—
183	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
184	C₄²	D₄×D₅₃	1	1	yes	characteristic
185	C₄²	#11719949	1	1	yes	characteristic
186	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂³	#13133192	1	1	yes	characteristic
188	Q₈	—	212	4	no	—
189	Q₈	—	212	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	106	4	no	—
193	C₂×C₄	—	212	2	no	—
194	C₂×C₄	—	212	2	no	—
195	C₂×C₄	—	212	2	no	—
196	C₂×C₄	—	212	2	no	—
197	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
198	C₂×C₄	#13133179	1	1	yes	characteristic
199	C₂×C₄	#13133179	1	1	yes	characteristic
200	C₂×C₄	#13133192	1	1	yes	characteristic
201	C₂×C₄	#13133206	1	1	yes	characteristic
202	C₂×C₄	#13133213	1	1	yes	characteristic
203	C₂×C₄	#13133246	1	4	yes	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	8	2	no	—
207	C₈	—	212	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#16810524	1	1	yes	characteristic
211	C₂²	#16810882	1	2	yes	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	424	2	no	—
215	C₄	—	424	2	no	—
216	C₄	—	212	4	no	—
217	C₄	#16810639	1	2	yes	—
218	C₄	#16810729	1	2	yes	—
219	C₄	#16810739	1	2	yes	—
220	C₄	#16811014	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	#20275546	1	1	yes	characteristic
223	C₂	#20275584	1	1	yes	characteristic
224	C₂	#20275601	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20285301

C₂

yes

maximal, characteristic

#20275691

C₂

yes

maximal

#20274460

C₂

yes

maximal, characteristic

#20273169

C₂

yes

maximal

C₅₃×#1566

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

—

#16811028

C₄

yes

characteristic, complemented

#16811003

C₄

yes

characteristic, complemented

#16810974

C₄

yes

complemented

#16810864

—

C₄×Dic₂₁₂

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₄²×Dic₅₃

C₂²

yes

—

—

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₂×Dic₂₁₂

—

C₂²×#88322

—

#13133217

C₂×C₄

yes

—

#13133217

C₂×C₄

yes

—

#13133209

—

#13133196

—

#13133196

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

#13133176

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

D₄

yes

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

Dic₂₁₂

—

C₂³×C₅₃

#121

yes

characteristic

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

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#9853

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#470

yes

—

100

C₄×C₅₃

—

101

C₄×C₅₃

—

102

C₄×C₅₃

#421

yes

—

103

C₄×C₅₃

#441

yes

—

104

C₄×C₅₃

#443

yes

—

105

C₄×C₅₃

#497

yes

—

106

Dic₅₃

—

107

Dic₅₃

—

108

Dic₅₃

—

109

#3170

—

110

#1994

—

111

#1744

—

112

#1566

D₅₃

yes

characteristic

113

#1493

—

114

C₂×C₅₃

—

115

C₂×C₅₃

#1963

yes

characteristic

116

C₂×C₅₃

#1971

yes

characteristic

117

C₂×C₅₃

#1974

yes

characteristic

118

#533

—

119

#500

—

120

#494

—

121

#487

—

122

#467

—

106

—

123

C₄×Q₁₆

—

106

—

124

#423

C₂×D₅₃

yes

—

125

C₂×C₄×C₈

—

126

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

127

C₄³

—

128

#339

—

106

—

129

#336

—

106

—

130

#334

—

131

Q₈⋊C₈

—

106

—

132

C₅₃

#9853

yes

characteristic, complemented, Sylow

133

C₂×Q₁₆

—

106

—

134

C₂×M₁₆

—

135

C₂²×C₈

—

106

—

136

C₂²×C₈

D₂₁₂

yes

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

#125

—

106

—

140

C₂²⋊Q₈

—

106

—

141

C₂²⋊Q₈

—

106

—

142

C₄×Q₈

—

106

—

143

C₄×Q₈

—

106

—

144

#117

—

106

—

145

C₂×C₄²

—

146

C₂×C₄²

—

106

—

147

C₂×C₄²

—

106

—

148

C₂×C₄²

C₂²×D₅₃

yes

characteristic

149

#107

—

106

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

#103

—

106

—

153

C₄×C₈

—

154

C₄×C₈

—

106

—

155

C₄×C₈

C₄×D₅₃

yes

characteristic

156

C₄×C₈

C₄×D₅₃

yes

characteristic

157

C₄×C₈

C₂²⋊D₅₃

yes

—

158

C₂×Q₈

—

106

—

159

C₂×Q₈

—

106

—

160

C₂²×C₄

—

106

—

161

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

162

C₂²×C₄

D₄×D₅₃

yes

characteristic

163

C₂²×C₄

#11719949

yes

characteristic

164

Q₁₆

—

212

—

165

M₁₆

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

106

—

169

C₂×C₈

—

212

—

170

C₂×C₈

#11719921

yes

—

171

C₂×C₈

#11719921

yes

—

172

C₄⋊C₄

—

106

—

173

C₄⋊C₄

—

106

—

174

C₄⋊C₄

—

106

—

175

C₄⋊C₄

—

106

—

176

C₄⋊C₄

—

212

—

177

C₄⋊C₄

—

212

—

178

C₂²⋊C₄

—

212

—

179

C₄²

—

106

—

180

C₄²

—

106

—

181

C₄²

—

212

—

182

C₄²

—

212

—

183

C₄²

C₂×C₄×D₅₃

yes

characteristic

184

C₄²

D₄×D₅₃

yes

characteristic

185

C₄²

#11719949

yes

characteristic

186

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

187

C₂³

#13133192

yes

characteristic

188

Q₈

—

212

—

189

Q₈

—

212

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

106

—

193

C₂×C₄

—

212

—

194

C₂×C₄

—

212

—

195

C₂×C₄

—

212

—

196

C₂×C₄

—

212

—

197

C₂×C₄

C₂×#11719921

yes

characteristic

198

C₂×C₄

#13133179

yes

characteristic

199

C₂×C₄

#13133179

yes

characteristic

200

C₂×C₄

#13133192

yes

characteristic

201

C₂×C₄

#13133206

yes

characteristic

202

C₂×C₄

#13133213

yes

characteristic

203

C₂×C₄

#13133246

yes

—

204

C₈

—

205

C₈

—

206

C₈

—

207

C₈

—

212

—

208

C₂²

—

209

C₂²

—

210

C₂²

#16810524

yes

characteristic

211

C₂²

#16810882

yes

—

212

C₄

—

213

C₄

—

214

C₄

—

424

—

215

C₄

—

424

—

216

C₄

—

212

—

217

C₄

#16810639

yes

—

218

C₄

#16810729

yes

—

219

C₄

#16810739

yes

—

220

C₄

#16811014

yes

—

221

C₂

—

222

C₂

#20275546

yes

characteristic

223

C₂

#20275584

yes

characteristic

224

C₂

#20275601

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1566	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1566	D₅₃
hypocenter	C₅₃	#9853
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1566

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
7. #16811028	C₄	semidirect
8. #16811003	C₄	semidirect
9. #16810974	C₄	semidirect
132. C₅₃	#9853	semidirect

H ≅ G/N

type

7. #16811028

C₄

semidirect

8. #16811003

C₄

semidirect

9. #16810974

C₄

semidirect

132. C₅₃

#9853

semidirect

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

subgroup	index
#20285301	2
#20275691	2
#20274460	2
#20273169	2
C₅₃×#1566	2
#9853	53

subgroup

index

C₅₃×#1566

Subgroups — #27284017

Conjugacy classes of subgroups

Complements

Maximal subgroups