Group #27289123 — subgroups

Subgroups — #27289123

Order 13568.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20275695	C₂	1	1	yes	maximal, characteristic
2	#20274868	C₂	1	1	yes	maximal, characteristic
3	#20273487	C₂	1	1	yes	maximal, characteristic
4	#20273119	C₂	1	1	yes	maximal, characteristic
5	#20273117	C₂	1	1	yes	maximal, characteristic
6	#20272543	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1570	C₂	1	1	yes	maximal, characteristic
8	#16811180	C₂²	1	1	yes	characteristic
9	#16810676	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
11	C₂×#13133146	—	2	2	no	—
12	#16810469	C₂²	1	1	yes	characteristic
13	#16810384	—	2	2	no	—
14	#16810373	—	2	2	no	—
15	#16810355	—	2	2	no	—
16	#16810353	—	2	2	no	—
17	C₅₃×#431	C₂²	1	1	yes	characteristic
18	C₅₃×#423	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#334	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	#13133217	Q₈	1	1	yes	characteristic
23	#13133217	Q₈	1	1	yes	characteristic
24	#13133209	—	2	1	no	—
25	#13133196	—	2	1	no	—
26	#13133196	—	2	1	no	—
27	C₄×Dic₁₀₆	—	2	1	no	—
28	C₄×Dic₁₀₆	—	2	1	no	—
29	#13133176	—	2	1	no	—
30	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
31	C₄⋊#88322	—	2	1	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	—	2	1	no	—
34	#13133146	—	4	4	no	—
35	C₂²⋊#88322	—	2	1	no	—
36	C₄×#88322	D₄	1	2	yes	—
37	#13133124	—	2	1	no	—
38	#13133124	—	2	1	no	—
39	C₂×M₁₆×C₅₃	—	2	1	no	—
40	C₂²×C₈×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	1	no	—
43	C₄⋊C₈×C₅₃	—	2	1	no	—
44	C₄⋊C₈×C₅₃	—	2	1	no	—
45	C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₄⋊C₈×C₅₃	—	2	1	no	—
47	C₂²⋊C₈×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
52	C₂²×Dic₅₃	—	2	2	no	—
53	C₂×Dic₁₀₆	—	2	1	no	—
54	C₂×Dic₁₀₆	—	2	1	no	—
55	M₁₆×C₅₃	—	4	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	4	1	no	—
58	C₂×C₈×C₅₃	—	4	1	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₅₃	#41	1	1	yes	characteristic
64	C₂²⋊Dic₅₃	—	4	1	no	—
65	C₄⋊Dic₅₃	—	2	1	no	—
66	C₄⋊Dic₅₃	—	2	1	no	—
67	C₄⋊Dic₅₃	—	2	1	no	—
68	C₄⋊Dic₅₃	—	2	1	no	—
69	#11719919	—	4	1	no	—
70	#11719919	—	4	1	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₄×Dic₅₃	—	4	1	no	—
73	C₄×Dic₅₃	SD₁₆	1	4	yes	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	—	4	1	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂³×C₅₃	#123	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	4	1	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₅₃	#137	1	2	yes	—
88	C₂×Dic₅₃	—	2	4	no	—
89	C₂×Dic₅₃	—	4	2	no	—
90	C₂×Dic₅₃	—	4	2	no	—
91	C₂×Dic₅₃	—	4	1	no	—
92	C₂×Dic₅₃	—	4	1	no	—
93	Dic₁₀₆	—	4	2	no	—
94	Dic₁₀₆	—	4	2	no	—
95	C₈×C₅₃	—	4	2	no	—
96	C₈×C₅₃	—	8	1	no	—
97	C₈×C₅₃	—	8	1	no	—
98	#88322	—	4	2	no	—
99	#88322	—	8	1	no	—
100	#10670	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	#399	1	1	yes	characteristic
104	C₂²×C₅₃	#483	1	2	yes	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	#475	1	2	yes	—
110	C₄×C₅₃	#492	1	1	yes	characteristic
111	C₄×C₅₃	#497	1	1	yes	characteristic
112	Dic₅₃	—	4	4	no	—
113	Dic₅₃	—	8	1	no	—
114	Dic₅₃	—	8	1	no	—
115	#1995	—	53	1	no	—
116	#1843	—	53	1	no	—
117	#1570	—	53	1	no	—
118	#1570	D₅₃	1	1	yes	characteristic
119	#1481	—	53	1	no	—
120	#1481	—	53	1	no	—
121	#1274	—	53	1	no	—
122	C₂×C₅₃	—	2	2	no	—
123	C₂×C₅₃	#2096	1	1	yes	characteristic
124	C₂×C₅₃	#2098	1	1	yes	characteristic
125	C₂×C₅₃	#2098	1	1	yes	characteristic
126	#533	—	53	1	no	—
127	#431	—	53	1	no	—
128	#431	C₂×D₅₃	1	1	yes	characteristic
129	#423	C₂×D₅₃	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	53	1	no	—
131	C₂×C₄⋊C₈	—	106	2	no	—
132	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
133	#376	—	53	1	no	—
134	#339	—	106	2	no	—
135	#334	—	2	2	no	—
136	#334	—	106	2	no	—
137	Q₈⋊C₈	—	106	2	no	—
138	Q₈⋊C₈	—	106	2	no	—
139	C₅₃	#10670	1	1	yes	characteristic, complemented, Sylow
140	C₂×M₁₆	—	2	1	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	106	1	no	—
143	C₄⋊Q₈	—	53	1	no	—
144	C₄⋊Q₈	—	53	1	no	—
145	#125	—	106	1	no	—
146	C₂²⋊Q₈	—	106	1	no	—
147	C₂²⋊Q₈	—	106	1	no	—
148	C₄×Q₈	—	106	1	no	—
149	C₄×Q₈	—	106	1	no	—
150	#117	—	106	1	no	—
151	C₂×C₄²	—	53	2	no	—
152	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	106	1	no	—
159	C₄⋊C₈	—	106	1	no	—
160	C₄⋊C₈	—	106	1	no	—
161	C₄⋊C₈	—	212	4	no	—
162	C₂²⋊C₈	—	2	1	no	—
163	C₂²⋊C₈	—	106	1	no	—
164	C₄×C₈	—	53	2	no	—
165	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
166	#95	—	106	1	no	—
167	#95	—	106	1	no	—
168	C₂×Q₈	—	106	1	no	—
169	C₂×Q₈	—	106	1	no	—
170	C₂²×C₄	—	106	2	no	—
171	C₂²×C₄	#11719945	1	1	yes	characteristic
172	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
173	C₂²×C₄	#11719949	1	1	yes	characteristic
174	M₁₆	—	4	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	106	2	no	—
180	C₂×C₈	—	212	1	no	—
181	C₂×C₈	—	212	2	no	—
182	C₄⋊C₄	—	106	1	no	—
183	C₄⋊C₄	—	106	1	no	—
184	C₄⋊C₄	—	106	1	no	—
185	C₄⋊C₄	—	106	1	no	—
186	C₄⋊C₄	—	212	1	no	—
187	C₄⋊C₄	—	212	1	no	—
188	C₂²⋊C₄	—	212	1	no	—
189	C₄²	—	53	4	no	—
190	C₄²	—	106	2	no	—
191	C₄²	—	212	1	no	—
192	C₄²	#11719945	1	1	yes	characteristic
193	C₄²	D₄×D₅₃	1	1	yes	characteristic
194	C₄²	Q₈×D₅₃	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂³	#13133202	1	1	yes	characteristic
197	Q₈	—	212	2	no	—
198	Q₈	—	212	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	106	4	no	—
202	C₂×C₄	—	212	1	no	—
203	C₂×C₄	—	212	1	no	—
204	C₂×C₄	—	212	2	no	—
205	C₂×C₄	—	212	2	no	—
206	C₂×C₄	#13133193	1	1	yes	characteristic
207	C₂×C₄	#13133195	1	1	yes	characteristic
208	C₂×C₄	#13133195	1	1	yes	characteristic
209	C₂×C₄	#13133200	1	1	yes	characteristic
210	C₂×C₄	#13133202	1	1	yes	characteristic
211	C₂×C₄	#13133213	1	1	yes	characteristic
212	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
213	C₂×C₄	#13133246	1	2	yes	—
214	C₈	—	4	2	no	—
215	C₈	—	8	1	no	—
216	C₈	—	8	1	no	—
217	C₈	—	424	1	no	—
218	C₈	—	212	2	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#16810558	1	1	yes	characteristic
222	C₂²	#16810955	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	424	1	no	—
228	C₄	—	424	1	no	—
229	C₄	—	212	4	no	—
230	C₄	#16810907	1	2	yes	—
231	C₄	#16810994	1	1	yes	characteristic
232	C₄	#16811014	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	#20276255	1	1	yes	characteristic
235	C₂	#20276267	1	1	yes	characteristic
236	C₂	#20276271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275695

C₂

yes

maximal, characteristic

#20274868

C₂

yes

maximal, characteristic

#20273487

C₂

yes

maximal, characteristic

#20273119

C₂

yes

maximal, characteristic

#20273117

C₂

yes

maximal, characteristic

#20272543

C₂

yes

maximal, characteristic

C₅₃×#1570

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

characteristic

#16810676

C₂²

yes

characteristic

C₂×C₄⋊#88322

C₂²

yes

characteristic

C₂×#13133146

—

#16810469

C₂²

yes

—

—

—

—

C₅₃×#431

C₂²

yes

characteristic

C₅₃×#423

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₂²×#88322

—

#13133217

Q₈

yes

characteristic

#13133217

Q₈

yes

—

—

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

#13133176

—

C₂×C₄×Dic₅₃

D₄

yes

—

—

—

—

—

—

D₄

yes

—

#13133124

—

#13133124

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

C₂²×C₄×C₅₃

yes

C₂²×C₄×C₅₃

yes

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

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

SD₁₆

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

100

#10670

—

maximal, Sylow

101

C₂²×C₅₃

—

102

C₂²×C₅₃

—

103

C₂²×C₅₃

#399

yes

characteristic

104

C₂²×C₅₃

#483

yes

—

105

C₄×C₅₃

—

106

C₄×C₅₃

—

107

C₄×C₅₃

—

108

C₄×C₅₃

—

109

C₄×C₅₃

#475

yes

—

110

C₄×C₅₃

#492

yes

characteristic

111

C₄×C₅₃

#497

yes

characteristic

112

Dic₅₃

—

113

Dic₅₃

—

114

Dic₅₃

—

115

#1995

—

116

#1843

—

117

#1570

—

118

#1570

D₅₃

yes

characteristic

119

#1481

—

120

#1481

—

121

#1274

—

122

C₂×C₅₃

—

123

C₂×C₅₃

#2096

yes

characteristic

124

C₂×C₅₃

#2098

yes

characteristic

125

C₂×C₅₃

#2098

yes

characteristic

126

#533

—

127

#431

—

128

#431

C₂×D₅₃

yes

characteristic

129

#423

C₂×D₅₃

yes

characteristic

130

C₂×C₄⋊C₈

—

131

C₂×C₄⋊C₈

—

106

—

132

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

133

#376

—

134

#339

—

106

—

135

#334

—

136

#334

—

106

—

137

Q₈⋊C₈

—

106

—

138

Q₈⋊C₈

—

106

—

139

C₅₃

#10670

yes

characteristic, complemented, Sylow

140

C₂×M₁₆

—

141

C₂²×C₈

—

142

C₂²×C₈

—

106

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

#125

—

106

—

146

C₂²⋊Q₈

—

106

—

147

C₂²⋊Q₈

—

106

—

148

C₄×Q₈

—

106

—

149

C₄×Q₈

—

106

—

150

#117

—

106

—

151

C₂×C₄²

—

152

C₂×C₄²

C₂²×D₅₃

yes

characteristic

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

C₄⋊C₈

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

106

—

159

C₄⋊C₈

—

106

—

160

C₄⋊C₈

—

106

—

161

C₄⋊C₈

—

212

—

162

C₂²⋊C₈

—

163

C₂²⋊C₈

—

106

—

164

C₄×C₈

—

165

C₄×C₈

C₂²⋊D₅₃

yes

—

166

#95

—

106

—

167

#95

—

106

—

168

C₂×Q₈

—

106

—

169

C₂×Q₈

—

106

—

170

C₂²×C₄

—

106

—

171

C₂²×C₄

#11719945

yes

characteristic

172

C₂²×C₄

D₄×D₅₃

yes

characteristic

173

C₂²×C₄

#11719949

yes

characteristic

174

M₁₆

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

179

C₂×C₈

—

106

—

180

C₂×C₈

—

212

—

181

C₂×C₈

—

212

—

182

C₄⋊C₄

—

106

—

183

C₄⋊C₄

—

106

—

184

C₄⋊C₄

—

106

—

185

C₄⋊C₄

—

106

—

186

C₄⋊C₄

—

212

—

187

C₄⋊C₄

—

212

—

188

C₂²⋊C₄

—

212

—

189

C₄²

—

190

C₄²

—

106

—

191

C₄²

—

212

—

192

C₄²

#11719945

yes

characteristic

193

C₄²

D₄×D₅₃

yes

characteristic

194

C₄²

Q₈×D₅₃

yes

characteristic

195

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

196

C₂³

#13133202

yes

characteristic

197

Q₈

—

212

—

198

Q₈

—

212

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

106

—

202

C₂×C₄

—

212

—

203

C₂×C₄

—

212

—

204

C₂×C₄

—

212

—

205

C₂×C₄

—

212

—

206

C₂×C₄

#13133193

yes

characteristic

207

C₂×C₄

#13133195

yes

characteristic

208

C₂×C₄

#13133195

yes

characteristic

209

C₂×C₄

#13133200

yes

characteristic

210

C₂×C₄

#13133202

yes

characteristic

211

C₂×C₄

#13133213

yes

characteristic

212

C₂×C₄

SD₁₆×D₅₃

yes

—

213

C₂×C₄

#13133246

yes

—

214

C₈

—

215

C₈

—

216

C₈

—

217

C₈

—

424

—

218

C₈

—

212

—

219

C₂²

—

220

C₂²

—

221

C₂²

#16810558

yes

characteristic

222

C₂²

#16810955

yes

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

424

—

228

C₄

—

424

—

229

C₄

—

212

—

230

C₄

#16810907

yes

—

231

C₄

#16810994

yes

characteristic

232

C₄

#16811014

yes

characteristic

233

C₂

—

234

C₂

#20276255

yes

characteristic

235

C₂

#20276267

yes

characteristic

236

C₂

#20276271

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1570	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂³×C₅₃	#123
hypercenter	#1570	D₅₃
hypocenter	C₅₃	#10670
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

subgroup

G/H

center

C₂×C₄

#13133213

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1570

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

N	H ≅ G/N	type
139. C₅₃	#10670	semidirect

H ≅ G/N

type

139. C₅₃

#10670

semidirect

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

subgroup	index
#20275695	2
#20274868	2
#20273487	2
#20273119	2
#20273117	2
#20272543	2
C₅₃×#1570	2
#10670	53

subgroup

index

C₅₃×#1570

Subgroups — #27289123

Conjugacy classes of subgroups

Complements

Maximal subgroups