Group #27283897 — subgroups

Subgroups — #27283897

Order 13568.

494 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	#20275675	C₂	1	2	yes	maximal
3	#20275516	C₂	1	2	yes	maximal
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	2	yes	—
7	#16811028	—	2	2	no	—
8	#16811028	C₄	1	4	yes	complemented
9	C₂×#13133151	—	2	2	no	—
10	C₂×#13133140	—	2	4	no	—
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810385	—	2	4	no	—
14	#16810345	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#384	C₂²	1	2	yes	—
17	C₅₃×#340	—	2	2	no	—
18	C₂²×Dic₁₀₆	—	2	2	no	—
19	C₂²×#88322	D₄	1	2	yes	—
20	C₂²×#88322	D₄	1	2	yes	—
21	#13133217	—	2	2	no	—
22	#13133217	—	2	2	no	—
23	#13133217	—	2	2	no	—
24	#13133217	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	#13133151	—	2	8	no	—
28	#13133140	—	4	8	no	—
29	C₄×#88322	—	2	1	no	—
30	C₄×#88322	—	2	1	no	—
31	C₄×#88322	—	2	1	no	—
32	C₄×#88322	—	2	1	no	—
33	C₂²×C₈×C₅₃	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	—	4	2	no	—
45	C₂×Dic₁₀₆	—	2	4	no	—
46	C₂×Dic₁₀₆	—	4	4	no	—
47	C₂×Dic₁₀₆	—	4	4	no	—
48	C₂×Dic₁₀₆	—	4	2	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
51	C₄⋊C₄×C₅₃	—	4	4	no	—
52	C₄⋊C₄×C₅₃	—	4	4	no	—
53	C₄⋊C₄×C₅₃	—	4	4	no	—
54	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	#41	1	1	yes	characteristic
57	C₄²×C₅₃	#41	1	1	yes	characteristic
58	C₄⋊Dic₅₃	—	2	4	no	—
59	C₄⋊Dic₅₃	—	2	4	no	—
60	C₄⋊Dic₅₃	—	4	2	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	4	no	—
66	C₂×#88322	D₈	1	4	yes	—
67	C₂³×C₅₃	#127	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	8	2	no	—
69	C₂×C₄×C₅₃	—	8	2	no	—
70	C₂×C₄×C₅₃	—	8	2	no	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
78	C₂×C₄×C₅₃	#135	1	2	yes	—
79	C₂×C₄×C₅₃	#137	1	4	yes	—
80	C₂×C₄×C₅₃	#137	1	4	yes	—
81	C₂×Dic₅₃	—	4	4	no	—
82	C₂×Dic₅₃	—	8	2	no	—
83	C₂×Dic₅₃	—	8	2	no	—
84	Dic₁₀₆	—	4	8	no	—
85	Dic₁₀₆	—	8	4	no	—
86	C₈×C₅₃	—	2	4	no	—
87	#88322	—	2	4	no	—
88	#88322	—	4	2	no	—
89	#9831	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	#390	1	1	yes	characteristic
91	C₂²×C₅₃	#493	1	1	yes	characteristic
92	C₂²×C₅₃	#495	1	1	yes	characteristic
93	C₂²×C₅₃	#497	1	2	yes	—
94	C₂²×C₅₃	#497	1	2	yes	—
95	C₄×C₅₃	—	16	2	no	—
96	C₄×C₅₃	#419	1	2	yes	—
97	C₄×C₅₃	C₄×D₈	1	2	yes	—
98	C₄×C₅₃	#441	1	2	yes	—
99	C₄×C₅₃	#466	1	2	yes	—
100	C₄×C₅₃	#470	1	2	yes	—
101	C₄×C₅₃	#491	1	2	yes	—
102	Dic₅₃	—	8	4	no	—
103	C₂×#500	—	53	1	no	—
104	#1990	—	53	2	no	—
105	#1956	—	53	2	no	—
106	#1834	D₅₃	1	1	yes	characteristic
107	C₂×#322	—	53	1	no	—
108	C₂×C₅₃	#1718	1	2	yes	—
109	C₂×C₅₃	#1722	1	2	yes	—
110	C₂×C₅₃	#1960	1	1	yes	characteristic
111	C₂×C₅₃	#1970	1	1	yes	characteristic
112	C₂×C₅₃	#1971	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	53	2	no	—
114	#500	—	53	4	no	—
115	#500	—	106	2	no	—
116	C₂×#107	—	106	2	no	—
117	C₂×#103	—	106	4	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
121	#384	C₂×D₅₃	1	2	yes	—
122	#340	—	2	2	no	—
123	#340	—	106	4	no	—
124	#322	—	106	2	no	—
125	C₅₃	#9831	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	106	2	no	—
127	C₂²×C₈	—	53	2	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
130	C₄⋊Q₈	—	53	4	no	—
131	C₄⋊Q₈	—	106	2	no	—
132	C₄⋊Q₈	—	106	2	no	—
133	C₄⋊Q₈	—	106	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	106	2	no	—
138	C₂×C₄⋊C₄	—	106	2	no	—
139	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
140	#107	—	106	8	no	—
141	#103	—	212	8	no	—
142	C₄×C₈	—	106	1	no	—
143	C₄×C₈	—	106	1	no	—
144	C₄×C₈	—	106	1	no	—
145	C₄×C₈	—	106	1	no	—
146	C₄×C₈	C₄×D₅₃	1	2	yes	—
147	C₄×C₈	D₂₁₂	1	2	yes	—
148	C₂×Q₈	—	106	4	no	—
149	C₂×Q₈	—	212	4	no	—
150	C₂×Q₈	—	212	4	no	—
151	C₂×Q₈	—	212	2	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	212	2	no	—
154	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
155	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	53	4	no	—
159	C₂×C₈	—	106	2	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	2	no	—
162	C₂×C₈	—	106	4	no	—
163	C₂×C₈	#11719921	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	106	4	no	—
168	C₄⋊C₄	—	106	4	no	—
169	C₄⋊C₄	—	212	2	no	—
170	C₄⋊C₄	—	212	2	no	—
171	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
172	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
173	C₄²	#11719949	1	1	yes	characteristic
174	C₄²	#11719949	1	1	yes	characteristic
175	C₂³	#13133213	1	1	yes	characteristic
176	Q₈	—	424	4	no	—
177	Q₈	—	212	8	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	424	2	no	—
182	C₂×C₄	—	424	2	no	—
183	C₂×C₄	—	212	4	no	—
184	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
185	C₂×C₄	#13133179	1	1	yes	characteristic
186	C₂×C₄	#13133179	1	1	yes	characteristic
187	C₂×C₄	#13133192	1	1	yes	characteristic
188	C₂×C₄	#13133192	1	1	yes	characteristic
189	C₂×C₄	#13133206	1	1	yes	characteristic
190	C₂×C₄	D₈×D₅₃	1	2	yes	—
191	C₂×C₄	#13133240	1	2	yes	—
192	C₂×C₄	#13133246	1	4	yes	—
193	C₂×C₄	#13133247	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	106	4	no	—
196	C₈	—	212	2	no	—
197	C₂²	#16810524	1	1	yes	characteristic
198	C₂²	#16810998	1	1	yes	characteristic
199	C₂²	#16811008	1	1	yes	characteristic
200	C₂²	#16811014	1	2	yes	—
201	C₂²	#16811018	1	2	yes	—
202	C₄	—	16	2	no	—
203	C₄	—	424	4	no	—
204	C₄	#16810631	1	2	yes	—
205	C₄	#16810705	1	2	yes	—
206	C₄	#16810729	1	2	yes	—
207	C₄	#16810860	1	2	yes	—
208	C₄	#16810882	1	2	yes	—
209	C₄	#16810993	1	2	yes	—
210	C₂	#20274364	1	2	yes	—
211	C₂	#20274388	1	2	yes	—
212	C₂	#20275534	1	1	yes	characteristic
213	C₂	#20275579	1	1	yes	characteristic
214	C₂	#20275584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811028

C₂

yes

maximal, characteristic

#20275675

C₂

yes

maximal

#20275516

C₂

yes

maximal

C₂×#16810345

C₂

yes

maximal, characteristic

C₅₃×#1834

C₂

yes

maximal, characteristic

C₂×#13133217

C₂²

yes

—

#16811028

—

#16811028

C₄

yes

complemented

C₂×#13133151

—

C₂×#13133140

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

characteristic

#16810385

—

#16810345

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

—

C₅₃×#340

—

C₂²×Dic₁₀₆

—

C₂²×#88322

D₄

yes

—

C₂²×#88322

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

—

D₈

yes

—

C₂³×C₅₃

#127

yes

characteristic

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

C₂×D₈

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#497

yes

—

C₄×C₅₃

—

C₄×C₅₃

#419

yes

—

C₄×C₅₃

C₄×D₈

yes

—

C₄×C₅₃

#441

yes

—

C₄×C₅₃

#466

yes

—

100

C₄×C₅₃

#470

yes

—

101

C₄×C₅₃

#491

yes

—

102

Dic₅₃

—

103

C₂×#500

—

104

#1990

—

105

#1956

—

106

#1834

D₅₃

yes

characteristic

107

C₂×#322

—

108

C₂×C₅₃

#1718

yes

—

109

C₂×C₅₃

#1722

yes

—

110

C₂×C₅₃

#1960

yes

characteristic

111

C₂×C₅₃

#1970

yes

characteristic

112

C₂×C₅₃

#1971

yes

characteristic

113

C₂×C₄⋊Q₈

—

114

#500

—

115

#500

—

106

—

116

C₂×#107

—

106

—

117

C₂×#103

—

106

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

121

#384

C₂×D₅₃

yes

—

122

#340

—

123

#340

—

106

—

124

#322

—

106

—

125

C₅₃

#9831

yes

characteristic, complemented, Sylow

126

C₂²×Q₈

—

106

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

C₂²⋊D₅₃

yes

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

106

—

132

C₄⋊Q₈

—

106

—

133

C₄⋊Q₈

—

106

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

106

—

138

C₂×C₄⋊C₄

—

106

—

139

C₂×C₄²

C₂²×D₅₃

yes

characteristic

140

#107

—

106

—

141

#103

—

212

—

142

C₄×C₈

—

106

—

143

C₄×C₈

—

106

—

144

C₄×C₈

—

106

—

145

C₄×C₈

—

106

—

146

C₄×C₈

C₄×D₅₃

yes

—

147

C₄×C₈

D₂₁₂

yes

—

148

C₂×Q₈

—

106

—

149

C₂×Q₈

—

212

—

150

C₂×Q₈

—

212

—

151

C₂×Q₈

—

212

—

152

C₂²×C₄

—

153

C₂²×C₄

—

212

—

154

C₂²×C₄

D₄×D₅₃

yes

characteristic

155

C₂²×C₄

D₄×D₅₃

yes

characteristic

156

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

106

—

160

C₂×C₈

—

106

—

161

C₂×C₈

—

106

—

162

C₂×C₈

—

106

—

163

C₂×C₈

#11719921

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

106

—

168

C₄⋊C₄

—

106

—

169

C₄⋊C₄

—

212

—

170

C₄⋊C₄

—

212

—

171

C₄²

C₂×C₄×D₅₃

yes

characteristic

172

C₄²

C₂×D₂₁₂

yes

characteristic

173

C₄²

#11719949

yes

characteristic

174

C₄²

#11719949

yes

characteristic

175

C₂³

#13133213

yes

characteristic

176

Q₈

—

424

—

177

Q₈

—

212

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

424

—

182

C₂×C₄

—

424

—

183

C₂×C₄

—

212

—

184

C₂×C₄

C₂×#11719921

yes

characteristic

185

C₂×C₄

#13133179

yes

characteristic

186

C₂×C₄

#13133179

yes

characteristic

187

C₂×C₄

#13133192

yes

characteristic

188

C₂×C₄

#13133192

yes

characteristic

189

C₂×C₄

#13133206

yes

characteristic

190

C₂×C₄

D₈×D₅₃

yes

—

191

C₂×C₄

#13133240

yes

—

192

C₂×C₄

#13133246

yes

—

193

C₂×C₄

#13133247

yes

—

194

C₈

—

195

C₈

—

106

—

196

C₈

—

212

—

197

C₂²

#16810524

yes

characteristic

198

C₂²

#16810998

yes

characteristic

199

C₂²

#16811008

yes

characteristic

200

C₂²

#16811014

yes

—

201

C₂²

#16811018

yes

—

202

C₄

—

203

C₄

—

424

—

204

C₄

#16810631

yes

—

205

C₄

#16810705

yes

—

206

C₄

#16810729

yes

—

207

C₄

#16810860

yes

—

208

C₄

#16810882

yes

—

209

C₄

#16810993

yes

—

210

C₂

#20274364

yes

—

211

C₂

#20274388

yes

—

212

C₂

#20275534

yes

characteristic

213

C₂

#20275579

yes

characteristic

214

C₂

#20275584

yes

characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1834	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1834	D₅₃
hypocenter	C₅₃	#9831
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1834

C₂

socle, prime radical

C₂³×C₅₃

#127

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
8. #16811028	C₄	semidirect
125. C₅₃	#9831	semidirect

H ≅ G/N

type

8. #16811028

C₄

semidirect

125. C₅₃

#9831

semidirect

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

subgroup	index
C₂×#16811028	2
#20275675	2
#20275516	2
C₂×#16810345	2
C₅₃×#1834	2
#9831	53

subgroup

index

C₅₃×#1834

Subgroups — #27283897

Conjugacy classes of subgroups

Complements

Maximal subgroups