Group #27284220 — subgroups

Subgroups — #27284220

Order 13568.

430 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 159 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	#20283078	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272599	C₂	1	4	yes	maximal
4	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133209	—	2	1	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	—	2	1	no	—
8	#16810521	—	2	2	no	—
9	#16810521	C₄	1	4	yes	—
10	#16810493	C₂²	1	2	yes	—
11	#16810489	—	2	1	no	—
12	#16810378	—	2	1	no	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#336	—	2	2	no	—
16	C₂²×#88322	—	2	2	no	—
17	#13133209	—	4	2	no	—
18	#13133209	—	4	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
22	C₂×#11719919	—	2	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	Q₈	1	4	yes	—
25	C₄×#88322	—	4	2	no	—
26	#13133124	—	2	4	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	4	no	—
42	C₄²×C₅₃	—	2	4	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	4	2	no	—
45	C₄²×C₅₃	D₈	1	2	yes	—
46	C₄²×C₅₃	SD₁₆	1	2	yes	—
47	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
48	C₄²×C₅₃	#41	1	2	yes	—
49	C₄⋊Dic₅₃	—	4	2	no	—
50	C₄⋊Dic₅₃	—	4	2	no	—
51	C₄⋊Dic₅₃	—	4	4	no	—
52	C₄⋊Dic₅₃	—	4	4	no	—
53	#11719919	—	4	8	no	—
54	C₂×#88322	—	2	4	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	#102	1	4	yes	—
65	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
66	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#135	1	4	yes	—
72	C₂×C₄×C₅₃	#137	1	2	yes	—
73	C₂×Dic₅₃	—	4	4	no	—
74	C₂×Dic₅₃	—	4	4	no	—
75	C₂×Dic₅₃	—	4	4	no	—
76	C₈×C₅₃	—	8	2	no	—
77	#88322	—	4	4	no	—
78	#9899	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	#391	1	1	yes	characteristic
80	C₂²×C₅₃	#413	1	2	yes	—
81	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
82	C₂²×C₅₃	#416	1	2	yes	—
83	C₂²×C₅₃	#419	1	1	yes	characteristic
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	—	4	2	no	—
87	C₄×C₅₃	#474	1	2	yes	—
88	C₄×C₅₃	#475	1	2	yes	—
89	C₄×C₅₃	#488	1	2	yes	—
90	C₄×C₅₃	#499	1	2	yes	—
91	Dic₅₃	—	8	4	no	—
92	#2859	—	53	1	no	—
93	#1831	D₅₃	1	1	yes	characteristic
94	C₄×C₄⋊C₈	—	53	1	no	—
95	#1296	—	53	4	no	—
96	C₂×C₅₃	#1540	1	2	yes	—
97	C₂×C₅₃	#1554	1	2	yes	—
98	C₂×C₅₃	#1988	1	1	yes	characteristic
99	C₂×C₅₃	#1991	1	1	yes	characteristic
100	C₂×C₅₃	#1998	1	1	yes	characteristic
101	C₂×#125	—	106	1	no	—
102	C₂×C₄⋊C₈	—	53	2	no	—
103	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
104	C₂×C₄×C₈	—	106	1	no	—
105	#389	—	53	4	no	—
106	#389	—	106	2	no	—
107	#384	—	53	2	no	—
108	#382	—	106	1	no	—
109	C₄³	C₂×D₅₃	1	1	yes	characteristic
110	#336	—	2	2	no	—
111	#336	—	106	1	no	—
112	C₅₃	#9899	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	106	2	no	—
115	#125	—	212	2	no	—
116	#125	—	212	2	no	—
117	C₂×C₄⋊C₄	—	53	4	no	—
118	C₂×C₄⋊C₄	—	106	2	no	—
119	C₂×C₄⋊C₄	—	106	2	no	—
120	C₂×C₄⋊C₄	—	106	4	no	—
121	C₂×C₄²	C₄×D₅₃	1	2	yes	—
122	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
123	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
124	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
125	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
126	C₄⋊C₈	—	2	4	no	—
127	C₄⋊C₈	—	53	4	no	—
128	C₄⋊C₈	—	106	2	no	—
129	C₄×C₈	—	212	2	no	—
130	#95	—	106	4	no	—
131	C₂²×C₄	—	106	4	no	—
132	C₂²×C₄	#11719921	1	2	yes	—
133	C₂²×C₄	#11719921	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	—	106	4	no	—
140	C₂×C₈	—	212	2	no	—
141	C₂×C₈	—	212	2	no	—
142	C₄⋊C₄	—	212	4	no	—
143	C₄⋊C₄	—	212	8	no	—
144	C₄⋊C₄	—	212	2	no	—
145	C₄⋊C₄	—	212	2	no	—
146	C₄⋊C₄	—	212	4	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	#11719913	1	2	yes	—
151	C₄²	D₄₂₄	1	2	yes	—
152	C₄²	#11719947	1	2	yes	—
153	C₄²	Q₈×D₅₃	1	2	yes	—
154	C₂³	C₂×#11719921	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	212	4	no	—
162	C₂×C₄	—	212	4	no	—
163	C₂×C₄	—	212	4	no	—
164	C₂×C₄	#13133139	1	4	yes	—
165	C₂×C₄	#13133182	1	2	yes	—
166	C₂×C₄	#13133197	1	2	yes	—
167	C₂×C₄	#13133204	1	1	yes	characteristic
168	C₂×C₄	#13133215	1	1	yes	characteristic
169	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
170	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
171	C₂×C₄	#13133237	1	4	yes	—
172	C₂×C₄	#13133250	1	2	yes	—
173	C₈	—	8	2	no	—
174	C₈	—	212	4	no	—
175	C₂²	#16810528	1	1	yes	characteristic
176	C₂²	#16810605	1	2	yes	—
177	C₂²	C₂×#13133139	1	1	yes	characteristic
178	C₂²	#16810617	1	2	yes	—
179	C₂²	#16810633	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	424	4	no	—
184	C₄	#16810904	1	2	yes	—
185	C₄	#16810909	1	2	yes	—
186	C₄	#16810977	1	2	yes	—
187	C₄	#16811022	1	2	yes	—
188	C₂	#20273362	1	2	yes	—
189	C₂	#20273426	1	2	yes	—
190	C₂	#20275666	1	1	yes	characteristic
191	C₂	#20275679	1	1	yes	characteristic
192	C₂	#20275707	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283078

C₂

yes

maximal, characteristic

C₄×C₄⋊#88322

C₂

yes

maximal, characteristic

#20272599

C₂

yes

maximal

C₅₃×#1831

C₂

yes

maximal, characteristic

C₂×#13133209

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄×#88322

—

#16810521

—

#16810521

C₄

yes

—

#16810493

C₂²

yes

—

#16810489

—

#16810378

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂²×#88322

—

#13133209

—

#13133209

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#11719919

—

C₄⋊#88322

—

C₄⋊#88322

Q₈

yes

—

C₄×#88322

—

#13133124

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×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×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

—

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

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₄×Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#102

yes

characteristic

C₂²×C₅₃

#416

yes

—

C₂²×C₅₃

#419

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#474

yes

—

C₄×C₅₃

#475

yes

—

C₄×C₅₃

#488

yes

—

C₄×C₅₃

#499

yes

—

Dic₅₃

—

#2859

—

#1831

D₅₃

yes

characteristic

C₄×C₄⋊C₈

—

#1296

—

C₂×C₅₃

#1540

yes

—

C₂×C₅₃

#1554

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

100

C₂×C₅₃

#1998

yes

characteristic

101

C₂×#125

—

106

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

104

C₂×C₄×C₈

—

106

—

105

#389

—

106

#389

—

106

—

107

#384

—

108

#382

—

106

—

109

C₄³

C₂×D₅₃

yes

characteristic

110

#336

—

111

#336

—

106

—

112

C₅₃

#9899

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

114

C₂²×C₈

—

106

—

115

#125

—

212

—

116

#125

—

212

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

106

—

119

C₂×C₄⋊C₄

—

106

—

120

C₂×C₄⋊C₄

—

106

—

121

C₂×C₄²

C₄×D₅₃

yes

—

122

C₂×C₄²

D₂₁₂

yes

characteristic

123

C₂×C₄²

D₂₁₂

yes

characteristic

124

C₂×C₄²

C₂²⋊D₅₃

yes

—

125

C₂×C₄²

C₂²×D₅₃

yes

characteristic

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

106

—

129

C₄×C₈

—

212

—

130

#95

—

106

—

131

C₂²×C₄

—

106

—

132

C₂²×C₄

#11719921

yes

—

133

C₂²×C₄

#11719921

yes

—

134

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

135

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

136

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

106

—

140

C₂×C₈

—

212

—

141

C₂×C₈

—

212

—

142

C₄⋊C₄

—

212

—

143

C₄⋊C₄

—

212

—

144

C₄⋊C₄

—

212

—

145

C₄⋊C₄

—

212

—

146

C₄⋊C₄

—

212

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

#11719913

yes

—

151

C₄²

D₄₂₄

yes

—

152

C₄²

#11719947

yes

—

153

C₄²

Q₈×D₅₃

yes

—

154

C₂³

C₂×#11719921

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

212

—

162

C₂×C₄

—

212

—

163

C₂×C₄

—

212

—

164

C₂×C₄

#13133139

yes

—

165

C₂×C₄

#13133182

yes

—

166

C₂×C₄

#13133197

yes

—

167

C₂×C₄

#13133204

yes

characteristic

168

C₂×C₄

#13133215

yes

characteristic

169

C₂×C₄

C₂×D₄₂₄

yes

characteristic

170

C₂×C₄

C₂×#11719913

yes

characteristic

171

C₂×C₄

#13133237

yes

—

172

C₂×C₄

#13133250

yes

—

173

C₈

—

174

C₈

—

212

—

175

C₂²

#16810528

yes

characteristic

176

C₂²

#16810605

yes

—

177

C₂²

C₂×#13133139

yes

characteristic

178

C₂²

#16810617

yes

—

179

C₂²

#16810633

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

424

—

184

C₄

#16810904

yes

—

185

C₄

#16810909

yes

—

186

C₄

#16810977

yes

—

187

C₄

#16811022

yes

—

188

C₂

#20273362

yes

—

189

C₂

#20273426

yes

—

190

C₂

#20275666

yes

characteristic

191

C₂

#20275679

yes

characteristic

192

C₂

#20275707

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#9899
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×#11719921

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1831

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 ◁ #1831

N	H ≅ G/N	type
112. C₅₃	#9899	semidirect

H ≅ G/N

type

112. C₅₃

#9899

semidirect

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

subgroup	index
#20283078	2
C₄×C₄⋊#88322	2
#20272599	2
C₅₃×#1831	2
#9899	53

subgroup

index

#20283078

C₄×C₄⋊#88322

#20272599

C₅₃×#1831

#9899

Subgroups — #27284220

Conjugacy classes of subgroups

Complements

Maximal subgroups