Group #27289117 — subgroups

Subgroups — #27289117

Order 13568.

296 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20275697	C₂	1	1	yes	maximal, characteristic
2	#20274834	C₂	1	1	yes	maximal, characteristic
3	#20273489	C₂	1	1	yes	maximal, characteristic
4	#20273486	C₂	1	1	yes	maximal, characteristic
5	#20273118	C₂	1	1	yes	maximal, characteristic
6	#20272543	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1481	C₂	1	1	yes	maximal, characteristic
8	#16810676	C₂²	1	1	yes	characteristic
9	#16810647	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
11	#16810469	C₂²	1	1	yes	characteristic
12	#16810382	—	2	2	no	—
13	#16810377	—	2	2	no	—
14	#16810375	—	2	2	no	—
15	#16810374	—	2	2	no	—
16	Q₈⋊#88322	—	2	2	no	—
17	C₅₃×#533	C₂²	1	1	yes	characteristic
18	C₅₃×#431	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×Q₈⋊C₈	—	2	2	no	—
21	C₂×#11719917	—	2	1	no	—
22	C₂²×#88322	—	2	1	no	—
23	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
24	C₄⋊#88322	—	2	1	no	—
25	C₄⋊#88322	—	2	1	no	—
26	C₄⋊#88322	—	2	1	no	—
27	C₄⋊#88322	—	2	1	no	—
28	C₄⋊#88322	—	2	1	no	—
29	C₂²⋊#88322	—	2	1	no	—
30	C₄×#88322	D₄	1	2	yes	—
31	#13133124	—	2	1	no	—
32	#13133124	—	2	1	no	—
33	C₂²×C₈×C₅₃	—	2	1	no	—
34	C₄⋊Q₈×C₅₃	Q₈	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₃	Q₈	1	1	yes	characteristic
36	#125×C₅₃	—	2	1	no	—
37	C₂²⋊Q₈×C₅₃	—	2	1	no	—
38	C₂²⋊Q₈×C₅₃	—	2	1	no	—
39	C₄×Q₈×C₅₃	—	2	1	no	—
40	C₄×Q₈×C₅₃	—	2	1	no	—
41	#117×C₅₃	—	2	1	no	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	1	no	—
44	C₄⋊C₈×C₅₃	—	2	1	no	—
45	C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₂²⋊C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₂×Q₈×C₅₃	—	2	1	no	—
49	C₂×Q₈×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂²×Dic₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	4	1	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₄⋊C₄×C₅₃	—	4	1	no	—
62	C₄⋊C₄×C₅₃	—	4	1	no	—
63	C₂²⋊C₄×C₅₃	—	4	1	no	—
64	C₄²×C₅₃	—	4	1	no	—
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₅₃	#41	1	1	yes	characteristic
69	C₄×Dic₅₃	—	2	2	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	#11719917	—	4	2	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	4	1	no	—
74	C₂×#88322	—	4	1	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂³×C₅₃	#123	1	1	yes	characteristic
77	Q₈×C₅₃	—	4	2	no	—
78	Q₈×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	—	4	1	no	—
81	C₂×C₄×C₅₃	—	4	1	no	—
82	C₂×C₄×C₅₃	—	4	1	no	—
83	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#135	1	2	yes	—
90	C₂×C₄×C₅₃	#137	1	2	yes	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	—	4	2	no	—
94	C₈×C₅₃	—	4	2	no	—
95	C₈×C₅₃	—	8	1	no	—
96	#88322	—	4	2	no	—
97	#88322	—	8	1	no	—
98	#88322	—	8	1	no	—
99	#10669	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	#399	1	1	yes	characteristic
103	C₂²×C₅₃	#485	1	2	yes	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	8	1	no	—
109	C₄×C₅₃	—	8	1	no	—
110	C₄×C₅₃	#479	1	2	yes	—
111	C₄×C₅₃	#495	1	1	yes	characteristic
112	C₄×C₅₃	#497	1	1	yes	characteristic
113	Dic₅₃	—	8	2	no	—
114	#1995	—	53	1	no	—
115	#1832	—	53	1	no	—
116	#1571	—	53	1	no	—
117	#1570	—	53	1	no	—
118	#1481	—	53	1	no	—
119	#1481	D₅₃	1	1	yes	characteristic
120	#1274	—	53	1	no	—
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	#2097	1	1	yes	characteristic
123	C₂×C₅₃	#2098	1	1	yes	characteristic
124	C₂×C₅₃	#2098	1	1	yes	characteristic
125	#533	C₂×D₅₃	1	1	yes	characteristic
126	#431	—	53	1	no	—
127	#431	C₂×D₅₃	1	1	yes	characteristic
128	#423	—	53	1	no	—
129	C₂×C₄⋊C₈	—	53	1	no	—
130	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
131	#376	—	53	1	no	—
132	#339	—	106	2	no	—
133	#336	—	106	2	no	—
134	#335	—	106	2	no	—
135	#334	—	106	2	no	—
136	Q₈⋊C₈	—	2	2	no	—
137	Q₈⋊C₈	—	106	2	no	—
138	C₅₃	#10669	1	1	yes	characteristic, complemented, Sylow
139	C₂×M₁₆	—	106	1	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	106	1	no	—
142	C₄⋊Q₈	Dic₁₀₆	1	1	yes	characteristic
143	C₄⋊Q₈	Dic₁₀₆	1	1	yes	characteristic
144	#125	—	2	1	no	—
145	C₂²⋊Q₈	—	2	1	no	—
146	C₂²⋊Q₈	—	2	1	no	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	—	2	1	no	—
149	#117	—	2	1	no	—
150	C₂×C₄²	—	53	2	no	—
151	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	106	1	no	—
156	C₄⋊C₈	—	106	1	no	—
157	C₄⋊C₈	—	106	1	no	—
158	C₄⋊C₈	—	106	1	no	—
159	C₄⋊C₈	—	106	1	no	—
160	C₂²⋊C₈	—	2	1	no	—
161	C₂²⋊C₈	—	106	1	no	—
162	C₄×C₈	—	53	2	no	—
163	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
164	#95	—	106	1	no	—
165	#95	—	106	1	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	2	1	no	—
168	C₂²×C₄	—	106	2	no	—
169	C₂²×C₄	#11719945	1	1	yes	characteristic
170	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
171	C₂²×C₄	#11719947	1	1	yes	characteristic
172	M₁₆	—	212	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	106	2	no	—
177	C₂×C₈	—	212	1	no	—
178	C₂×C₈	—	212	1	no	—
179	C₂×C₈	—	212	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₂²⋊C₄	—	4	1	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	106	2	no	—
189	C₄²	—	106	2	no	—
190	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
191	C₄²	D₄×D₅₃	1	1	yes	characteristic
192	C₄²	#11719947	1	1	yes	characteristic
193	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂³	#13133199	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	4	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	—	212	2	no	—
203	C₂×C₄	—	212	2	no	—
204	C₂×C₄	#13133194	1	1	yes	characteristic
205	C₂×C₄	#13133194	1	1	yes	characteristic
206	C₂×C₄	#13133196	1	1	yes	characteristic
207	C₂×C₄	#13133198	1	1	yes	characteristic
208	C₂×C₄	#13133199	1	1	yes	characteristic
209	C₂×C₄	#13133213	1	1	yes	characteristic
210	C₂×C₄	#13133238	1	2	yes	—
211	C₂×C₄	#13133249	1	2	yes	—
212	C₈	—	4	2	no	—
213	C₈	—	8	1	no	—
214	C₈	—	424	1	no	—
215	C₈	—	424	1	no	—
216	C₈	—	212	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#16810559	1	1	yes	characteristic
220	C₂²	#16810962	1	2	yes	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	8	1	no	—
226	C₄	—	8	1	no	—
227	C₄	—	424	2	no	—
228	C₄	#16810928	1	2	yes	—
229	C₄	#16811004	1	1	yes	characteristic
230	C₄	#16811016	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	#20276265	1	1	yes	characteristic
233	C₂	#20276268	1	1	yes	characteristic
234	C₂	#20276270	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275697

C₂

yes

maximal, characteristic

#20274834

C₂

yes

maximal, characteristic

#20273489

C₂

yes

maximal, characteristic

#20273486

C₂

yes

maximal, characteristic

#20273118

C₂

yes

maximal, characteristic

#20272543

C₂

yes

maximal, characteristic

C₅₃×#1481

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

characteristic

#16810647

C₂²

yes

characteristic

C₂×C₄⋊#88322

C₂²

yes

characteristic

#16810469

C₂²

yes

—

—

—

—

—

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#431

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₅₃×Q₈⋊C₈

—

C₂×#11719917

—

C₂²×#88322

—

C₂×C₄×Dic₅₃

D₄

yes

—

—

—

—

—

—

—

D₄

yes

—

#13133124

—

#13133124

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

characteristic

C₄⋊Q₈×C₅₃

Q₈

yes

characteristic

#125×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

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

C₂²×Dic₅₃

—

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×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

100

C₂²×C₅₃

—

101

C₂²×C₅₃

—

102

C₂²×C₅₃

#399

yes

characteristic

103

C₂²×C₅₃

#485

yes

—

104

C₄×C₅₃

—

105

C₄×C₅₃

—

106

C₄×C₅₃

—

107

C₄×C₅₃

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

C₄×C₅₃

#479

yes

—

111

C₄×C₅₃

#495

yes

characteristic

112

C₄×C₅₃

#497

yes

characteristic

113

Dic₅₃

—

114

#1995

—

115

#1832

—

116

#1571

—

117

#1570

—

118

#1481

—

119

#1481

D₅₃

yes

characteristic

120

#1274

—

121

C₂×C₅₃

—

122

C₂×C₅₃

#2097

yes

characteristic

123

C₂×C₅₃

#2098

yes

characteristic

124

C₂×C₅₃

#2098

yes

characteristic

125

#533

C₂×D₅₃

yes

characteristic

126

#431

—

127

#431

C₂×D₅₃

yes

characteristic

128

#423

—

129

C₂×C₄⋊C₈

—

130

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

131

#376

—

132

#339

—

106

—

133

#336

—

106

—

134

#335

—

106

—

135

#334

—

106

—

136

Q₈⋊C₈

—

137

Q₈⋊C₈

—

106

—

138

C₅₃

#10669

yes

characteristic, complemented, Sylow

139

C₂×M₁₆

—

106

—

140

C₂²×C₈

—

141

C₂²×C₈

—

106

—

142

C₄⋊Q₈

Dic₁₀₆

yes

characteristic

143

C₄⋊Q₈

Dic₁₀₆

yes

characteristic

144

#125

—

145

C₂²⋊Q₈

—

146

C₂²⋊Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

#117

—

150

C₂×C₄²

—

151

C₂×C₄²

C₂²×D₅₃

yes

characteristic

152

C₄⋊C₈

—

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

C₄⋊C₈

—

106

—

156

C₄⋊C₈

—

106

—

157

C₄⋊C₈

—

106

—

158

C₄⋊C₈

—

106

—

159

C₄⋊C₈

—

106

—

160

C₂²⋊C₈

—

161

C₂²⋊C₈

—

106

—

162

C₄×C₈

—

163

C₄×C₈

C₂²⋊D₅₃

yes

—

164

#95

—

106

—

165

#95

—

106

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂²×C₄

—

106

—

169

C₂²×C₄

#11719945

yes

characteristic

170

C₂²×C₄

D₄×D₅₃

yes

characteristic

171

C₂²×C₄

#11719947

yes

characteristic

172

M₁₆

—

212

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

106

—

177

C₂×C₈

—

212

—

178

C₂×C₈

—

212

—

179

C₂×C₈

—

212

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₂²⋊C₄

—

187

C₄²

—

188

C₄²

—

106

—

189

C₄²

—

106

—

190

C₄²

C₂×Dic₁₀₆

yes

characteristic

191

C₄²

D₄×D₅₃

yes

characteristic

192

C₄²

#11719947

yes

characteristic

193

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

194

C₂³

#13133199

yes

characteristic

195

Q₈

—

196

Q₈

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

212

—

202

C₂×C₄

—

212

—

203

C₂×C₄

—

212

—

204

C₂×C₄

#13133194

yes

characteristic

205

C₂×C₄

#13133194

yes

characteristic

206

C₂×C₄

#13133196

yes

characteristic

207

C₂×C₄

#13133198

yes

characteristic

208

C₂×C₄

#13133199

yes

characteristic

209

C₂×C₄

#13133213

yes

characteristic

210

C₂×C₄

#13133238

yes

—

211

C₂×C₄

#13133249

yes

—

212

C₈

—

213

C₈

—

214

C₈

—

424

—

215

C₈

—

424

—

216

C₈

—

212

—

217

C₂²

—

218

C₂²

—

219

C₂²

#16810559

yes

characteristic

220

C₂²

#16810962

yes

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

424

—

228

C₄

#16810928

yes

—

229

C₄

#16811004

yes

characteristic

230

C₄

#16811016

yes

characteristic

231

C₂

—

232

C₂

#20276265

yes

characteristic

233

C₂

#20276268

yes

characteristic

234

C₂

#20276270

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1481	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂³×C₅₃	#123
hypercenter	#1481	D₅₃
hypocenter	C₅₃	#10669
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1481

subgroup

G/H

center

C₂×C₄

#13133213

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1481

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1481

N	H ≅ G/N	type
138. C₅₃	#10669	semidirect

H ≅ G/N

type

138. C₅₃

#10669

semidirect

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

subgroup	index
#20275697	2
#20274834	2
#20273489	2
#20273486	2
#20273118	2
#20272543	2
C₅₃×#1481	2
#10669	53

subgroup

index

C₅₃×#1481

Subgroups — #27289117

Conjugacy classes of subgroups

Complements

Maximal subgroups