Group #27283186 — subgroups

Subgroups — #27283186

Order 13568.

538 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 171 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	#20283113	C₂	1	1	yes	maximal, characteristic
2	#20274832	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810370	C₂	1	4	yes	maximal
4	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	—	2	1	no	—
6	C₂×#13133217	—	2	1	no	—
7	C₂×#13133217	C₂²	1	2	yes	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	#16810493	—	2	2	no	—
10	#16810493	C₄	1	4	yes	—
11	#16810378	—	2	2	no	—
12	#16810370	—	2	8	no	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×#97×C₅₃	—	2	1	no	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	1	no	—
17	C₂²×Dic₁₀₆	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	#13133217	—	2	4	no	—
20	#13133217	—	2	4	no	—
21	#13133217	—	4	4	no	—
22	#13133217	—	4	2	no	—
23	#13133217	—	4	2	no	—
24	C₂×C₄⋊Dic₅₃	—	2	4	no	—
25	C₂×C₄⋊Dic₅₃	—	2	4	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
28	C₄⋊#88322	—	2	4	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
37	#97×C₅₃	—	4	2	no	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	2	4	no	—
44	C₂×Dic₁₀₆	—	4	4	no	—
45	C₂×Dic₁₀₆	—	4	2	no	—
46	C₂×Dic₁₀₆	—	4	2	no	—
47	C₂×Dic₁₀₆	—	4	4	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	—	2	4	no	—
52	C₄²×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	D₈	1	2	yes	—
55	C₄²×C₅₃	SD₁₆	1	2	yes	—
56	C₄²×C₅₃	C₂×D₄	1	2	yes	—
57	C₄²×C₅₃	#41	1	2	yes	—
58	C₄⋊Dic₅₃	—	2	8	no	—
59	C₄⋊Dic₅₃	—	4	8	no	—
60	C₄⋊Dic₅₃	—	4	8	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	4	no	—
65	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	#101	1	4	yes	—
72	C₂×C₄×C₅₃	#102	1	4	yes	—
73	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₅₃	#121	1	2	yes	—
75	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₅₃	#137	1	4	yes	—
81	C₂×Dic₅₃	—	4	4	no	—
82	C₂×Dic₅₃	—	4	4	no	—
83	C₂×Dic₅₃	—	4	4	no	—
84	Dic₁₀₆	—	8	4	no	—
85	Dic₁₀₆	—	8	4	no	—
86	C₈×C₅₃	—	4	4	no	—
87	#88322	—	8	2	no	—
88	#9705	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#390	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×#101	1	2	yes	—
91	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
92	C₂²×C₅₃	#418	1	1	yes	characteristic
93	C₂²×C₅₃	#419	1	2	yes	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	#459	1	2	yes	—
98	C₄×C₅₃	#460	1	2	yes	—
99	C₄×C₅₃	#491	1	2	yes	—
100	C₄×C₅₃	#497	1	2	yes	—
101	Dic₅₃	—	8	4	no	—
102	#2868	—	53	1	no	—
103	#1831	—	53	1	no	—
104	#1767	D₅₃	1	1	yes	characteristic
105	C₂×#332	—	53	4	no	—
106	C₂×C₅₃	#1546	1	2	yes	—
107	C₂×C₅₃	#1556	1	2	yes	—
108	C₂×C₅₃	#1960	1	1	yes	characteristic
109	C₂×C₅₃	#1966	1	1	yes	characteristic
110	C₂×C₅₃	#1982	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	—	53	2	no	—
112	C₂×C₄⋊Q₈	—	106	1	no	—
113	C₂×C₄⋊Q₈	—	106	1	no	—
114	C₂×C₄⋊C₈	—	53	2	no	—
115	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
116	C₂×#97	—	2	1	no	—
117	#384	—	53	4	no	—
118	#384	—	106	2	no	—
119	C₄³	C₂×D₅₃	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#336	—	106	2	no	—
122	#332	—	106	8	no	—
123	C₅₃	#9705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	106	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	106	2	no	—
127	C₄⋊Q₈	—	106	4	no	—
128	C₄⋊Q₈	—	106	4	no	—
129	C₄⋊Q₈	—	212	4	no	—
130	C₄⋊Q₈	—	212	2	no	—
131	C₄⋊Q₈	—	212	2	no	—
132	C₂×C₄⋊C₄	—	53	4	no	—
133	C₂×C₄⋊C₄	—	106	2	no	—
134	C₂×C₄⋊C₄	—	106	4	no	—
135	C₂×C₄⋊C₄	—	106	4	no	—
136	C₂×C₄²	C₄×D₅₃	1	2	yes	—
137	C₂×C₄²	D₂₁₂	1	2	yes	—
138	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	106	4	no	—
143	C₄⋊C₈	D₂₁₂	1	4	yes	—
144	#97	—	4	2	no	—
145	C₂×Q₈	—	212	4	no	—
146	C₂×Q₈	—	212	2	no	—
147	C₂×Q₈	—	212	2	no	—
148	C₂×Q₈	—	212	4	no	—
149	C₂²×C₄	—	106	4	no	—
150	C₂²×C₄	#11719921	1	2	yes	—
151	C₂²×C₄	#11719921	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	212	2	no	—
159	C₂×C₈	—	212	4	no	—
160	C₄⋊C₄	—	106	8	no	—
161	C₄⋊C₄	—	212	8	no	—
162	C₄⋊C₄	—	212	8	no	—
163	C₄⋊C₄	—	212	2	no	—
164	C₄⋊C₄	—	212	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	#11719922	1	2	yes	—
169	C₄²	#11719924	1	2	yes	—
170	C₄²	C₂×D₂₁₂	1	2	yes	—
171	C₄²	#11719945	1	2	yes	—
172	C₂³	C₂×#11719921	1	1	yes	characteristic
173	Q₈	—	424	4	no	—
174	Q₈	—	424	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	212	4	no	—
181	C₂×C₄	—	212	4	no	—
182	C₂×C₄	—	212	4	no	—
183	C₂×C₄	#13133135	1	4	yes	—
184	C₂×C₄	#13133137	1	4	yes	—
185	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
186	C₂×C₄	#13133191	1	2	yes	—
187	C₂×C₄	#13133205	1	1	yes	characteristic
188	C₂×C₄	#13133214	1	1	yes	characteristic
189	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
190	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
191	C₂×C₄	#13133242	1	2	yes	—
192	C₂×C₄	#13133250	1	4	yes	—
193	C₈	—	4	4	no	—
194	C₈	—	424	2	no	—
195	C₂²	#16810525	1	1	yes	characteristic
196	C₂²	C₂×#13133135	1	2	yes	—
197	C₂²	C₂×#13133137	1	1	yes	characteristic
198	C₂²	#16810627	1	1	yes	characteristic
199	C₂²	#16810633	1	2	yes	—
200	C₄	—	2	4	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	4	no	—
203	C₄	—	424	4	no	—
204	C₄	#16810818	1	2	yes	—
205	C₄	#16810823	1	2	yes	—
206	C₄	#16810992	1	2	yes	—
207	C₄	#16811017	1	2	yes	—
208	C₂	#20273388	1	2	yes	—
209	C₂	#20273433	1	2	yes	—
210	C₂	#20275535	1	1	yes	characteristic
211	C₂	#20275559	1	1	yes	characteristic
212	C₂	#20275642	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283113

C₂

yes

maximal, characteristic

#20274832

C₂

yes

maximal, characteristic

C₂×#16810370

C₂

yes

maximal

C₅₃×#1767

C₂

yes

maximal, characteristic

C₂×#13133217

—

C₂×#13133217

—

C₂×#13133217

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

#16810493

—

#16810493

C₄

yes

—

#16810378

—

#16810370

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×#97×C₅₃

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂²×Dic₁₀₆

—

C₂²×#88322

—

—

—

—

—

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

#97×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

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#101

yes

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

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

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#459

yes

—

C₄×C₅₃

#460

yes

—

C₄×C₅₃

#491

yes

—

100

C₄×C₅₃

#497

yes

—

101

Dic₅₃

—

102

#2868

—

103

#1831

—

104

#1767

D₅₃

yes

characteristic

105

C₂×#332

—

106

C₂×C₅₃

#1546

yes

—

107

C₂×C₅₃

#1556

yes

—

108

C₂×C₅₃

#1960

yes

characteristic

109

C₂×C₅₃

#1966

yes

characteristic

110

C₂×C₅₃

#1982

yes

characteristic

111

C₂×C₄⋊Q₈

—

112

C₂×C₄⋊Q₈

—

106

—

113

C₂×C₄⋊Q₈

—

106

—

114

C₂×C₄⋊C₈

—

115

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

116

C₂×#97

—

117

#384

—

118

#384

—

106

—

119

C₄³

C₂×D₅₃

yes

characteristic

120

#336

—

121

#336

—

106

—

122

#332

—

106

—

123

C₅₃

#9705

yes

characteristic, complemented, Sylow

124

C₂²×Q₈

—

106

—

125

C₂²×C₈

—

126

C₂²×C₈

—

106

—

127

C₄⋊Q₈

—

106

—

128

C₄⋊Q₈

—

106

—

129

C₄⋊Q₈

—

212

—

130

C₄⋊Q₈

—

212

—

131

C₄⋊Q₈

—

212

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

106

—

134

C₂×C₄⋊C₄

—

106

—

135

C₂×C₄⋊C₄

—

106

—

136

C₂×C₄²

C₄×D₅₃

yes

—

137

C₂×C₄²

D₂₁₂

yes

—

138

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

139

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

140

C₂×C₄²

C₂²×D₅₃

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

106

—

143

C₄⋊C₈

D₂₁₂

yes

—

144

#97

—

145

C₂×Q₈

—

212

—

146

C₂×Q₈

—

212

—

147

C₂×Q₈

—

212

—

148

C₂×Q₈

—

212

—

149

C₂²×C₄

—

106

—

150

C₂²×C₄

#11719921

yes

—

151

C₂²×C₄

#11719921

yes

—

152

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

153

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

212

—

159

C₂×C₈

—

212

—

160

C₄⋊C₄

—

106

—

161

C₄⋊C₄

—

212

—

162

C₄⋊C₄

—

212

—

163

C₄⋊C₄

—

212

—

164

C₄⋊C₄

—

212

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

#11719922

yes

—

169

C₄²

#11719924

yes

—

170

C₄²

C₂×D₂₁₂

yes

—

171

C₄²

#11719945

yes

—

172

C₂³

C₂×#11719921

yes

characteristic

173

Q₈

—

424

—

174

Q₈

—

424

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

212

—

181

C₂×C₄

—

212

—

182

C₂×C₄

—

212

—

183

C₂×C₄

#13133135

yes

—

184

C₂×C₄

#13133137

yes

—

185

C₂×C₄

C₄×D₂₁₂

yes

—

186

C₂×C₄

#13133191

yes

—

187

C₂×C₄

#13133205

yes

characteristic

188

C₂×C₄

#13133214

yes

characteristic

189

C₂×C₄

C₂×#11719922

yes

characteristic

190

C₂×C₄

C₂×#11719924

yes

characteristic

191

C₂×C₄

#13133242

yes

—

192

C₂×C₄

#13133250

yes

—

193

C₈

—

194

C₈

—

424

—

195

C₂²

#16810525

yes

characteristic

196

C₂²

C₂×#13133135

yes

—

197

C₂²

C₂×#13133137

yes

characteristic

198

C₂²

#16810627

yes

characteristic

199

C₂²

#16810633

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

424

—

204

C₄

#16810818

yes

—

205

C₄

#16810823

yes

—

206

C₄

#16810992

yes

—

207

C₄

#16811017

yes

—

208

C₂

#20273388

yes

—

209

C₂

#20273433

yes

—

210

C₂

#20275535

yes

characteristic

211

C₂

#20275559

yes

characteristic

212

C₂

#20275642

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1767	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1767	D₅₃
hypocenter	C₅₃	#9705
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×#11719921

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1767

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 ◁ #1767

N	H ≅ G/N	type
123. C₅₃	#9705	semidirect

H ≅ G/N

type

123. C₅₃

#9705

semidirect

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

subgroup	index
#20283113	2
#20274832	2
C₂×#16810370	2
C₅₃×#1767	2
#9705	53

subgroup

index

C₅₃×#1767

Subgroups — #27283186

Conjugacy classes of subgroups

Complements

Maximal subgroups