Group #27288225 — subgroups

Subgroups — #27288225

Order 13568.

382 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20286060	C₂	1	1	yes	maximal, characteristic
2	#20275692	C₂	1	2	yes	maximal
3	#20273403	C₂	1	2	yes	maximal
4	#20272590	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	2	yes	—
7	#16810919	—	2	2	no	—
8	#16810864	—	2	2	no	—
9	#16810834	—	2	2	no	—
10	#16810647	C₂²	1	2	yes	—
11	#16810383	—	2	2	no	—
12	#16810383	—	2	2	no	—
13	#16810371	—	2	2	no	—
14	#16810370	—	2	2	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	2	no	—
19	C₂×Dic₂₁₂	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	#13133217	D₄	1	2	yes	—
22	#13133217	D₄	1	2	yes	—
23	#13133209	—	2	2	no	—
24	#13133196	—	2	2	no	—
25	#13133196	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	2	no	—
27	C₄×Dic₁₀₆	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	#13133151	—	2	2	no	—
30	C₄⋊#88322	—	2	2	no	—
31	C₄⋊#88322	—	2	2	no	—
32	#13133142	—	2	4	no	—
33	C₂²×C₈×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	—	2	1	no	—
36	C₂×C₄²×C₅₃	—	2	1	no	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂×Dic₁₀₆	—	2	2	no	—
48	C₂×Dic₁₀₆	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	—	4	1	no	—
53	C₄²×C₅₃	—	2	1	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	4	1	no	—
56	C₄²×C₅₃	—	4	1	no	—
57	C₄²×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	Q₁₆	1	2	yes	complemented
59	C₄²×C₅₃	C₂×D₄	1	2	yes	—
60	C₄²×C₅₃	#41	1	2	yes	—
61	C₂²⋊Dic₅₃	—	4	2	no	—
62	C₄⋊Dic₅₃	—	2	2	no	—
63	C₄⋊Dic₅₃	—	2	2	no	—
64	C₄⋊Dic₅₃	—	2	2	no	—
65	C₄⋊Dic₅₃	—	2	2	no	—
66	#11719919	—	4	2	no	—
67	#11719919	—	4	2	no	—
68	C₄×Dic₅₃	—	4	2	no	—
69	#11719917	—	4	2	no	—
70	#11719917	—	4	2	no	—
71	C₂×#88322	—	4	2	no	—
72	Dic₂₁₂	—	8	2	no	—
73	C₂³×C₅₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	4	1	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	#120	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#121	1	2	yes	—
85	C₂×C₄×C₅₃	#123	1	2	yes	—
86	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#137	1	2	yes	—
90	C₂×Dic₅₃	—	4	2	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	Dic₁₀₆	—	4	2	no	—
93	Dic₁₀₆	—	4	2	no	—
94	Dic₁₀₆	—	8	2	no	—
95	C₈×C₅₃	—	4	2	no	—
96	C₈×C₅₃	—	8	1	no	—
97	#88322	—	8	2	no	—
98	#10523	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	#394	1	1	yes	characteristic
102	C₂²×C₅₃	#467	1	1	yes	characteristic
103	C₂²×C₅₃	#470	1	1	yes	characteristic
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	4	2	no	—
107	C₄×C₅₃	—	4	2	no	—
108	C₄×C₅₃	#454	1	1	yes	characteristic
109	C₄×C₅₃	#455	1	1	yes	characteristic
110	C₄×C₅₃	#462	1	2	yes	—
111	C₄×C₅₃	#464	1	2	yes	—
112	Dic₅₃	—	8	2	no	—
113	Dic₅₃	—	8	2	no	—
114	#3249	—	53	1	no	—
115	#1994	—	53	2	no	—
116	#1831	D₅₃	1	1	yes	characteristic
117	#1550	—	53	2	no	—
118	#1292	—	53	1	no	—
119	C₂×C₅₃	—	2	1	no	—
120	C₂×C₅₃	—	2	1	no	—
121	C₂×C₅₃	#2040	1	1	yes	characteristic
122	C₂×C₅₃	#2050	1	1	yes	characteristic
123	C₂×C₅₃	#2056	1	1	yes	characteristic
124	#533	—	53	2	no	—
125	#477	—	106	2	no	—
126	#467	—	106	2	no	—
127	#462	—	106	2	no	—
128	#423	—	53	2	no	—
129	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
130	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
131	C₄³	C₂×D₅₃	1	1	yes	characteristic
132	#339	—	106	2	no	—
133	#339	—	106	2	no	—
134	#336	—	2	2	no	—
135	#333	—	106	2	no	—
136	#332	—	106	2	no	—
137	C₅₃	#10523	1	1	yes	characteristic, complemented, Sylow
138	C₂×Q₁₆	—	106	2	no	—
139	C₂×M₁₆	—	106	2	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
142	C₄⋊Q₈	—	53	2	no	—
143	C₄⋊Q₈	—	53	2	no	—
144	#125	—	106	2	no	—
145	C₂²⋊Q₈	—	106	2	no	—
146	C₂²⋊Q₈	—	106	2	no	—
147	C₄×Q₈	—	106	2	no	—
148	C₄×Q₈	—	106	2	no	—
149	#117	—	106	2	no	—
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
153	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
155	#107	—	106	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	106	2	no	—
159	C₄⋊C₈	—	106	2	no	—
160	#103	—	106	4	no	—
161	C₂×Q₈	—	106	2	no	—
162	C₂×Q₈	—	106	2	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
166	C₂²×C₄	#11719945	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
168	Q₁₆	—	424	2	no	—
169	M₁₆	—	212	2	no	—
170	M₁₆	—	212	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	212	2	no	—
176	C₄⋊C₄	—	106	2	no	—
177	C₄⋊C₄	—	106	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	106	2	no	—
180	C₄⋊C₄	—	212	2	no	—
181	C₄⋊C₄	—	212	2	no	—
182	C₂²⋊C₄	—	212	2	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	1	no	—
186	C₄²	—	4	1	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	212	2	no	—
189	C₄²	Dic₂₁₂	1	2	yes	complemented
190	C₄²	D₄×D₅₃	1	2	yes	—
191	C₄²	#11719949	1	2	yes	—
192	C₂³	#13133191	1	1	yes	characteristic
193	Q₈	—	424	2	no	—
194	Q₈	—	212	2	no	—
195	Q₈	—	212	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	212	2	no	—
206	C₂×C₄	—	212	2	no	—
207	C₂×C₄	#13133185	1	1	yes	characteristic
208	C₂×C₄	#13133192	1	2	yes	—
209	C₂×C₄	#13133202	1	2	yes	—
210	C₂×C₄	#13133206	1	1	yes	characteristic
211	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
212	C₂×C₄	#13133239	1	1	yes	characteristic
213	C₂×C₄	#13133250	1	2	yes	—
214	C₈	—	4	2	no	—
215	C₈	—	8	1	no	—
216	C₈	—	424	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#16810540	1	1	yes	characteristic
220	C₂²	#16810864	1	1	yes	characteristic
221	C₂²	#16810881	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	424	2	no	—
227	C₄	—	424	2	no	—
228	C₄	#16810795	1	1	yes	characteristic
229	C₄	#16810800	1	1	yes	characteristic
230	C₄	#16810834	1	2	yes	—
231	C₄	#16810846	1	2	yes	—
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	#20275944	1	1	yes	characteristic
235	C₂	#20276002	1	1	yes	characteristic
236	C₂	#20276038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20286060

C₂

yes

maximal, characteristic

#20275692

C₂

yes

maximal

#20273403

C₂

yes

maximal

#20272590

C₂

yes

maximal, characteristic

C₅₃×#1831

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂×Dic₂₁₂

—

C₂×#11719917

—

#13133217

D₄

yes

—

#13133217

D₄

yes

—

#13133209

—

#13133196

—

#13133196

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

—

—

—

—

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

Q₁₆

yes

complemented

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

Dic₂₁₂

—

C₂³×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#10523

—

maximal, Sylow

C₂²×C₅₃

—

100

C₂²×C₅₃

—

101

C₂²×C₅₃

#394

yes

characteristic

102

C₂²×C₅₃

#467

yes

characteristic

103

C₂²×C₅₃

#470

yes

characteristic

104

C₄×C₅₃

—

105

C₄×C₅₃

—

106

C₄×C₅₃

—

107

C₄×C₅₃

—

108

C₄×C₅₃

#454

yes

characteristic

109

C₄×C₅₃

#455

yes

characteristic

110

C₄×C₅₃

#462

yes

—

111

C₄×C₅₃

#464

yes

—

112

Dic₅₃

—

113

Dic₅₃

—

114

#3249

—

115

#1994

—

116

#1831

D₅₃

yes

characteristic

117

#1550

—

118

#1292

—

119

C₂×C₅₃

—

120

C₂×C₅₃

—

121

C₂×C₅₃

#2040

yes

characteristic

122

C₂×C₅₃

#2050

yes

characteristic

123

C₂×C₅₃

#2056

yes

characteristic

124

#533

—

125

#477

—

106

—

126

#467

—

106

—

127

#462

—

106

—

128

#423

—

129

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

130

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

131

C₄³

C₂×D₅₃

yes

characteristic

132

#339

—

106

—

133

#339

—

106

—

134

#336

—

135

#333

—

106

—

136

#332

—

106

—

137

C₅₃

#10523

yes

characteristic, complemented, Sylow

138

C₂×Q₁₆

—

106

—

139

C₂×M₁₆

—

106

—

140

C₂²×C₈

—

141

C₂²×C₈

C₂²⋊D₅₃

yes

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

#125

—

106

—

145

C₂²⋊Q₈

—

106

—

146

C₂²⋊Q₈

—

106

—

147

C₄×Q₈

—

106

—

148

C₄×Q₈

—

106

—

149

#117

—

106

—

150

C₂×C₄²

—

151

C₂×C₄²

—

152

C₂×C₄²

D₂₁₂

yes

characteristic

153

C₂×C₄²

D₂₁₂

yes

characteristic

154

C₂×C₄²

C₂²×D₅₃

yes

characteristic

155

#107

—

106

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

106

—

159

C₄⋊C₈

—

106

—

160

#103

—

106

—

161

C₂×Q₈

—

106

—

162

C₂×Q₈

—

106

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

166

C₂²×C₄

#11719945

yes

characteristic

167

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

168

Q₁₆

—

424

—

169

M₁₆

—

212

—

170

M₁₆

—

212

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

212

—

176

C₄⋊C₄

—

106

—

177

C₄⋊C₄

—

106

—

178

C₄⋊C₄

—

106

—

179

C₄⋊C₄

—

106

—

180

C₄⋊C₄

—

212

—

181

C₄⋊C₄

—

212

—

182

C₂²⋊C₄

—

212

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

—

188

C₄²

—

212

—

189

C₄²

Dic₂₁₂

yes

complemented

190

C₄²

D₄×D₅₃

yes

—

191

C₄²

#11719949

yes

—

192

C₂³

#13133191

yes

characteristic

193

Q₈

—

424

—

194

Q₈

—

212

—

195

Q₈

—

212

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

212

—

206

C₂×C₄

—

212

—

207

C₂×C₄

#13133185

yes

characteristic

208

C₂×C₄

#13133192

yes

—

209

C₂×C₄

#13133202

yes

—

210

C₂×C₄

#13133206

yes

characteristic

211

C₂×C₄

C₂×Dic₂₁₂

yes

characteristic

212

C₂×C₄

#13133239

yes

characteristic

213

C₂×C₄

#13133250

yes

—

214

C₈

—

215

C₈

—

216

C₈

—

424

—

217

C₂²

—

218

C₂²

—

219

C₂²

#16810540

yes

characteristic

220

C₂²

#16810864

yes

characteristic

221

C₂²

#16810881

yes

characteristic

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

424

—

227

C₄

—

424

—

228

C₄

#16810795

yes

characteristic

229

C₄

#16810800

yes

characteristic

230

C₄

#16810834

yes

—

231

C₄

#16810846

yes

—

232

C₂

—

233

C₂

—

234

C₂

#20275944

yes

characteristic

235

C₂

#20276002

yes

characteristic

236

C₂

#20276038

yes

characteristic

	subgroup	G/H
center	C₂²	#16810540
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle	C₂²×C₅₃	#394
prime radical	C₂³×C₅₃	#121
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#10523
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1831

subgroup

G/H

center

C₂²

#16810540

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1831

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1831

N	H ≅ G/N	type
58. C₄²×C₅₃	Q₁₆	semidirect
137. C₅₃	#10523	semidirect
189. C₄²	Dic₂₁₂	semidirect

H ≅ G/N

type

58. C₄²×C₅₃

Q₁₆

semidirect

137. C₅₃

#10523

semidirect

189. C₄²

Dic₂₁₂

semidirect

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

subgroup	index
#20286060	2
#20275692	2
#20273403	2
#20272590	2
C₅₃×#1831	2
#10523	53

subgroup

index

C₅₃×#1831

Subgroups — #27288225

Conjugacy classes of subgroups

Complements

Maximal subgroups