Group #27277110 — subgroups

Subgroups — #27277110

Order 13568.

404 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20273471	C₂	1	2	yes	maximal, complemented
2	#20273450	C₂	1	2	yes	maximal
3	#20272628	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
5	#16810647	C₂²	1	4	yes	—
6	#16810374	—	2	2	no	—
7	#16810374	—	2	2	no	—
8	#16810372	—	2	2	no	—
9	#16810367	—	2	2	no	—
10	C₂×#131×C₅₃	C₄	1	2	yes	—
11	C₅₃×#532	C₂²	1	1	yes	characteristic
12	C₅₃×#436	—	2	2	no	—
13	C₈×D₄×C₅₃	—	2	2	no	—
14	C₅₃×#423	C₂²	1	1	yes	characteristic
15	C₅₃×#408	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719917	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
21	#131×C₅₃	—	4	4	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
28	#127×C₅₃	D₄	1	1	yes	characteristic
29	#124×C₅₃	Q₈	1	2	yes	—
30	#121×C₅₃	—	2	2	no	—
31	C₄×D₄×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₂²⋊C₈×C₅₃	—	2	4	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	#41×C₅₃	—	4	2	no	—
38	#41×C₅₃	—	4	2	no	—
39	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂×D₄×C₅₃	—	2	2	no	—
41	C₂×D₄×C₅₃	—	2	2	no	—
42	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	#41	1	2	yes	—
46	M₁₆×C₅₃	—	4	4	no	—
47	M₁₆×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	4	1	no	—
51	C₂×C₈×C₅₃	—	4	4	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂²⋊C₄×C₅₃	—	2	4	no	—
57	C₄²×C₅₃	C₂×D₄	1	2	yes	—
58	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	#11719917	—	8	4	no	—
61	C₂×#88322	—	4	4	no	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	#117	1	1	yes	characteristic
64	Q₈×C₅₃	—	4	2	no	—
65	D₄×C₅₃	—	4	2	no	—
66	D₄×C₅₃	—	4	4	no	—
67	D₄×C₅₃	—	4	4	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#121	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₅₃	—	2	1	no	—
79	C₈×C₅₃	—	2	2	no	—
80	C₈×C₅₃	—	4	2	no	—
81	C₈×C₅₃	#100	1	2	yes	—
82	#88322	—	8	4	no	—
83	#8636	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	4	2	no	—
87	C₂²×C₅₃	—	4	4	no	—
88	C₂²×C₅₃	#380	1	1	yes	characteristic
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	4	2	no	—
92	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
93	C₄×C₅₃	#413	1	1	yes	characteristic
94	C₄×C₅₃	#471	1	4	yes	—
95	#2931	D₅₃	1	1	yes	characteristic
96	#1566	—	53	2	no	—
97	#1560	—	53	2	no	—
98	#1309	—	53	2	no	—
99	C₂×C₅₃	—	4	1	no	—
100	C₂×C₅₃	—	8	2	no	—
101	C₂×C₅₃	#1800	1	1	yes	characteristic
102	C₂×C₅₃	#1815	1	2	yes	—
103	C₂×#131	Dic₅₃	1	2	yes	—
104	#532	C₂×D₅₃	1	1	yes	characteristic
105	#436	—	2	2	no	—
106	C₈×D₄	—	2	2	no	—
107	#423	—	53	4	no	—
108	#423	C₂×D₅₃	1	1	yes	characteristic
109	#408	Dic₅₃	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
111	#334	—	106	2	no	—
112	#334	—	106	2	no	—
113	#333	—	106	2	no	—
114	#331	—	106	2	no	—
115	C₅₃	#8636	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	C₂×Dic₅₃	1	2	yes	—
117	#131	—	4	4	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	106	4	no	—
120	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
124	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
125	#127	D₂₁₂	1	1	yes	characteristic
126	#124	Dic₁₀₆	1	2	yes	—
127	#121	—	2	2	no	—
128	C₄×D₄	—	2	2	no	—
129	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	106	4	no	—
132	C₄⋊C₈	—	106	4	no	—
133	C₂²⋊C₈	—	2	4	no	—
134	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
135	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
136	#41	—	4	2	no	—
137	#41	—	4	2	no	—
138	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₄⋊Dic₅₃	1	2	yes	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	#11719945	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
145	M₁₆	—	4	4	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	424	4	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	212	4	no	—
153	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
155	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
159	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
160	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
161	C₂³	—	2	2	no	—
162	C₂³	#13133176	1	1	yes	characteristic
163	Q₈	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	4	4	no	—
166	D₄	—	4	4	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
175	C₂×C₄	#13133191	1	2	yes	—
176	C₂×C₄	#13133196	1	2	yes	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	424	4	no	—
181	C₈	#13133134	1	2	yes	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	4	no	—
186	C₂²	#16810483	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	2	no	—
190	C₄	C₂×#13133134	1	1	yes	characteristic
191	C₄	#16810607	1	1	yes	characteristic
192	C₄	#16810886	1	4	yes	—
193	C₂	—	4	1	no	—
194	C₂	—	8	2	no	—
195	C₂	#20274707	1	1	yes	characteristic
196	C₂	#20274774	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20273471

C₂

yes

maximal, complemented

#20273450

C₂

yes

maximal

#20272628

C₂

yes

maximal, complemented

C₅₃×#2931

C₂

yes

maximal, characteristic

#16810647

C₂²

yes

—

—

—

—

—

C₂×#131×C₅₃

C₄

yes

—

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#436

—

C₈×D₄×C₅₃

—

C₅₃×#423

C₂²

yes

characteristic

C₅₃×#408

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

C₂×#41×C₅₃

C₂×C₄

yes

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₅₃

D₄

yes

characteristic

#127×C₅₃

D₄

yes

characteristic

#124×C₅₃

Q₈

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₄⋊C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

#117

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#100

yes

—

#88322

—

#8636

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#380

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2931

D₅₃

yes

—

—

—

C₂×C₅₃

—

100

C₂×C₅₃

—

101

C₂×C₅₃

#1800

yes

characteristic

102

C₂×C₅₃

#1815

yes

—

103

C₂×#131

Dic₅₃

yes

—

104

#532

C₂×D₅₃

yes

characteristic

105

#436

—

106

C₈×D₄

—

107

#423

—

108

#423

C₂×D₅₃

yes

characteristic

109

#408

Dic₅₃

yes

—

110

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

111

#334

—

106

—

112

#334

—

106

—

113

#333

—

106

—

114

#331

—

106

—

115

C₅₃

#8636

yes

characteristic, complemented, Sylow

116

C₂×#41

C₂×Dic₅₃

yes

—

117

#131

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

106

—

120

C₂×M₁₆

C₂×Dic₅₃

yes

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

123

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

124

C₄⋊Q₈

D₂₁₂

yes

characteristic

125

#127

D₂₁₂

yes

characteristic

126

#124

Dic₁₀₆

yes

—

127

#121

—

128

C₄×D₄

—

129

C₂×C₄²

C₂²×D₅₃

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

106

—

132

C₄⋊C₈

—

106

—

133

C₂²⋊C₈

—

134

C₄×C₈

C₂²⋊D₅₃

yes

—

135

C₄×C₈

C₂²⋊D₅₃

yes

—

136

#41

—

137

#41

—

138

C₂×Q₈

C₄⋊Dic₅₃

yes

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

C₄⋊Dic₅₃

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

#11719945

yes

—

144

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

424

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

212

—

153

C₂×C₈

C₂²⋊Dic₅₃

yes

—

154

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

155

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

C₂×Dic₁₀₆

yes

characteristic

159

C₄²

C₂×D₂₁₂

yes

characteristic

160

C₄²

C₂×C₂²⋊D₅₃

yes

—

161

C₂³

—

162

C₂³

#13133176

yes

characteristic

163

Q₈

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

174

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

175

C₂×C₄

#13133191

yes

—

176

C₂×C₄

#13133196

yes

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

424

—

181

C₈

#13133134

yes

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

#16810483

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

C₂×#13133134

yes

characteristic

191

C₄

#16810607

yes

characteristic

192

C₄

#16810886

yes

—

193

C₂

—

194

C₂

—

195

C₂

#20274707

yes

characteristic

196

C₂

#20274774

yes

—

	subgroup	G/H
center	C₂²	#16810483
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2931	C₂
socle	C₂²×C₅₃	#380
prime radical	C₅₃×#532	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2931	D₅₃
hypocenter	C₅₃	#8636
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

subgroup

G/H

center

C₂²

#16810483

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2931

C₂

socle

C₂²×C₅₃

#380

prime radical

C₅₃×#532

C₂²

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

N	H ≅ G/N	type
1. #20273471	C₂	semidirect
3. #20272628	C₂	semidirect
115. C₅₃	#8636	semidirect

H ≅ G/N

type

1. #20273471

C₂

semidirect

3. #20272628

C₂

semidirect

115. C₅₃

#8636

semidirect

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

subgroup	index
#20273471	2
#20273450	2
#20272628	2
C₅₃×#2931	2
#8636	53

subgroup

index

C₅₃×#2931

Subgroups — #27277110

Conjugacy classes of subgroups

Complements

Maximal subgroups