Group #27283174 — subgroups

Subgroups — #27283174

Order 13568.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	#20274832	C₂	1	1	yes	maximal, characteristic
2	#20274568	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810372	C₂	1	4	yes	maximal
4	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	2	yes	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133128	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	—	2	2	no	—
10	#16810372	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₅₃	C₄	1	4	yes	—
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#384	—	2	1	no	—
14	C₅₃×#384	—	2	2	no	—
15	C₅₃×#384	—	2	1	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₄⋊#88322	—	2	4	no	—
21	C₄⋊#88322	D₄	1	4	yes	—
22	#13133128	—	4	2	no	—
23	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₄⋊Q₈×C₅₃	—	2	4	no	—
25	C₄⋊Q₈×C₅₃	—	2	4	no	—
26	C₄⋊Q₈×C₅₃	—	4	8	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×Q₈×C₅₃	—	2	8	no	—
37	C₂×Q₈×C₅₃	—	4	8	no	—
38	C₂²×C₄×C₅₃	—	2	4	no	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	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	C₄⋊C₄×C₅₃	—	4	8	no	—
45	C₄⋊C₄×C₅₃	—	4	8	no	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄²×C₅₃	—	2	4	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	4	no	—
52	C₄²×C₅₃	SD₁₆	1	2	yes	—
53	C₄²×C₅₃	Q₁₆	1	2	yes	—
54	C₄²×C₅₃	C₂×D₄	1	2	yes	—
55	C₄²×C₅₃	#41	1	2	yes	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂×#88322	—	4	4	no	—
61	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	Q₈×C₅₃	—	8	8	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	4	8	no	—
70	C₂×C₄×C₅₃	#100	1	4	yes	—
71	C₂×C₄×C₅₃	#103	1	4	yes	—
72	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₅₃	#121	1	2	yes	—
74	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#137	1	2	yes	—
79	C₂×C₄×C₅₃	#137	1	4	yes	—
80	#88322	—	4	4	no	—
81	#88322	—	8	2	no	—
82	#9702	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	#390	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×#100	1	2	yes	—
85	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
86	C₂²×C₅₃	#419	1	2	yes	—
87	C₂²×C₅₃	#419	1	1	yes	characteristic
88	C₄×C₅₃	—	2	4	no	—
89	C₄×C₅₃	—	2	4	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	8	4	no	—
92	C₄×C₅₃	#461	1	2	yes	—
93	C₄×C₅₃	#462	1	2	yes	—
94	C₄×C₅₃	#491	1	2	yes	—
95	C₄×C₅₃	#497	1	2	yes	—
96	#2868	D₅₃	1	1	yes	characteristic
97	#1831	—	53	1	no	—
98	#1767	—	53	1	no	—
99	C₂×#333	—	53	4	no	—
100	C₂×C₅₃	#1548	1	2	yes	—
101	C₂×C₅₃	#1558	1	2	yes	—
102	C₂×C₅₃	#1960	1	1	yes	characteristic
103	C₂×C₅₃	#1965	1	1	yes	characteristic
104	C₂×C₅₃	#1981	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	Dic₅₃	1	4	yes	—
106	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
107	C₂×C₄⋊C₈	—	53	2	no	—
108	C₂×C₄⋊C₈	—	53	2	no	—
109	C₂×#97	—	106	1	no	—
110	#384	—	2	1	no	—
111	#384	—	2	2	no	—
112	#384	—	2	1	no	—
113	C₄³	C₂×D₅₃	1	1	yes	characteristic
114	#336	—	106	2	no	—
115	#336	—	106	1	no	—
116	#333	—	106	8	no	—
117	C₅₃	#9702	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	C₂×Dic₅₃	1	4	yes	—
119	C₂²×C₈	—	106	2	no	—
120	C₂²×C₈	—	106	2	no	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	4	8	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
129	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
133	C₄⋊C₈	—	53	4	no	—
134	C₄⋊C₈	—	106	4	no	—
135	C₄⋊C₈	—	106	2	no	—
136	#97	—	212	2	no	—
137	C₂×Q₈	—	2	8	no	—
138	C₂×Q₈	—	4	8	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
141	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
145	C₂×C₈	—	106	4	no	—
146	C₂×C₈	—	212	2	no	—
147	C₂×C₈	—	212	4	no	—
148	C₂×C₈	—	212	2	no	—
149	C₂×C₈	—	212	2	no	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	#11719924	1	2	yes	—
159	C₄²	#11719925	1	2	yes	—
160	C₄²	D₄×D₅₃	1	2	yes	—
161	C₄²	#11719947	1	2	yes	—
162	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
163	Q₈	—	8	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	8	no	—
171	C₂×C₄	#13133134	1	4	yes	—
172	C₂×C₄	#13133141	1	4	yes	—
173	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
174	C₂×C₄	#13133188	1	2	yes	—
175	C₂×C₄	#13133204	1	1	yes	characteristic
176	C₂×C₄	#13133213	1	1	yes	characteristic
177	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
178	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
179	C₂×C₄	#13133248	1	4	yes	—
180	C₂×C₄	#13133248	1	2	yes	—
181	C₈	—	424	2	no	—
182	C₈	—	212	4	no	—
183	C₂²	#16810526	1	1	yes	characteristic
184	C₂²	C₂×#13133134	1	2	yes	—
185	C₂²	C₂×#13133141	1	1	yes	characteristic
186	C₂²	#16810632	1	1	yes	characteristic
187	C₂²	#16810632	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	4	no	—
191	C₄	—	8	4	no	—
192	C₄	#16810825	1	2	yes	—
193	C₄	#16810830	1	2	yes	—
194	C₄	#16810991	1	2	yes	—
195	C₄	#16811015	1	2	yes	—
196	C₂	#20273396	1	2	yes	—
197	C₂	#20273441	1	2	yes	—
198	C₂	#20275536	1	1	yes	characteristic
199	C₂	#20275556	1	1	yes	characteristic
200	C₂	#20275639	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal, characteristic

#20274568

C₂

yes

maximal, characteristic

C₂×#16810372

C₂

yes

maximal

C₅₃×#2868

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₅₃

C₄

yes

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#384

—

C₅₃×#384

—

C₅₃×#384

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄⋊#88322

D₄

yes

—

#13133128

—

C₂²×Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×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

#100

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

#137

yes

—

C₂×C₄×C₅₃

#137

yes

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#103

yes

characteristic

C₂²×C₅₃

#419

yes

—

C₂²×C₅₃

#419

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#461

yes

—

C₄×C₅₃

#462

yes

—

C₄×C₅₃

#491

yes

—

C₄×C₅₃

#497

yes

—

#2868

D₅₃

yes

—

—

—

100

C₂×C₅₃

#1548

yes

—

101

C₂×C₅₃

#1558

yes

—

102

C₂×C₅₃

#1960

yes

characteristic

103

C₂×C₅₃

#1965

yes

characteristic

104

C₂×C₅₃

#1981

yes

characteristic

105

C₂×C₄⋊Q₈

Dic₅₃

yes

—

106

C₂×C₄⋊Q₈

C₂×D₅₃

yes

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

109

C₂×#97

—

106

—

110

#384

—

111

#384

—

112

#384

—

113

C₄³

C₂×D₅₃

yes

characteristic

114

#336

—

106

—

115

#336

—

106

—

116

#333

—

106

—

117

C₅₃

#9702

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

C₂×Dic₅₃

yes

—

119

C₂²×C₈

—

106

—

120

C₂²×C₈

—

106

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂×Dic₅₃

yes

—

129

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

130

C₂×C₄²

C₂²⋊D₅₃

yes

—

131

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

132

C₂×C₄²

C₂²×D₅₃

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

106

—

135

C₄⋊C₈

—

106

—

136

#97

—

212

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

141

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

142

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

143

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

144

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

145

C₂×C₈

—

106

—

146

C₂×C₈

—

212

—

147

C₂×C₈

—

212

—

148

C₂×C₈

—

212

—

149

C₂×C₈

—

212

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#11719924

yes

—

159

C₄²

#11719925

yes

—

160

C₄²

D₄×D₅₃

yes

—

161

C₄²

#11719947

yes

—

162

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

163

Q₈

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

#13133134

yes

—

172

C₂×C₄

#13133141

yes

—

173

C₂×C₄

D₄×Dic₅₃

yes

—

174

C₂×C₄

#13133188

yes

—

175

C₂×C₄

#13133204

yes

characteristic

176

C₂×C₄

#13133213

yes

characteristic

177

C₂×C₄

C₂×#11719924

yes

characteristic

178

C₂×C₄

C₂×#11719925

yes

characteristic

179

C₂×C₄

#13133248

yes

—

180

C₂×C₄

#13133248

yes

—

181

C₈

—

424

—

182

C₈

—

212

—

183

C₂²

#16810526

yes

characteristic

184

C₂²

C₂×#13133134

yes

—

185

C₂²

C₂×#13133141

yes

characteristic

186

C₂²

#16810632

yes

characteristic

187

C₂²

#16810632

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#16810825

yes

—

193

C₄

#16810830

yes

—

194

C₄

#16810991

yes

—

195

C₄

#16811015

yes

—

196

C₂

#20273396

yes

—

197

C₂

#20273441

yes

—

198

C₂

#20275536

yes

characteristic

199

C₂

#20275556

yes

characteristic

200

C₂

#20275639

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2868	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2868	D₅₃
hypocenter	C₅₃	#9702
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2868

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

N	H ≅ G/N	type
117. C₅₃	#9702	semidirect

H ≅ G/N

type

117. C₅₃

#9702

semidirect

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

subgroup	index
#20274832	2
#20274568	2
C₂×#16810372	2
C₅₃×#2868	2
#9702	53

subgroup

index

C₅₃×#2868

Subgroups — #27283174

Conjugacy classes of subgroups

Complements

Maximal subgroups