Group #27539593 — subgroups

Subgroups — #27539593

Order 13568.

482 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 187 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	#20285227	C₂	1	2	yes	maximal
2	#20277000	C₂	1	4	yes	maximal, complemented
3	#20276999	C₂	1	4	yes	maximal, complemented
4	#20276499	C₂	1	1	yes	maximal, characteristic
5	#20276497	C₂	1	2	yes	maximal
6	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic
8	#16811176	—	2	2	no	—
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	#16811027	C₂²	1	4	yes	—
11	#16811024	C₂²	1	4	yes	—
12	#16811023	—	2	2	no	—
13	#16811020	—	2	2	no	—
14	#16810661	C₂²	1	2	yes	—
15	#16810659	—	2	2	no	—
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	#16810451	C₂²	1	2	yes	—
18	#16810450	C₂²	1	4	yes	—
19	#16810449	C₂²	1	2	yes	—
20	#16810448	C₂²	1	4	yes	—
21	#16810398	C₂²	1	2	yes	—
22	#16810398	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#427	C₂²	1	2	yes	—
25	C₅₃×#367	C₂²	1	2	yes	—
26	C₅₃×#366	C₂²	1	2	yes	—
27	C₂²×#88322	D₄	1	1	yes	characteristic
28	#13133216	—	2	4	no	—
29	#13133208	—	2	4	no	—
30	#13133194	—	4	4	no	—
31	C₄×Dic₁₀₆	—	4	4	no	—
32	#13133176	—	2	2	no	—
33	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
34	C₂×#11719908	Q₈	1	4	yes	complemented
35	C₂×#11719908	C₂³	1	2	yes	—
36	#13133151	C₂³	1	4	yes	—
37	#13133150	—	2	4	no	—
38	#13133149	—	2	2	no	—
39	#13133148	—	2	4	no	—
40	C₈×Dic₅₃	D₄	1	2	yes	—
41	C₈×Dic₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	#117×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	Q₈	1	4	yes	complemented
45	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
47	#107×C₅₃	C₂³	1	4	yes	—
48	#106×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	D₈	1	2	yes	—
51	C₂×Dic₁₀₆	—	4	4	no	—
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₂²⋊C₄×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₂²⋊Dic₅₃	—	4	2	no	—
60	C₄⋊Dic₅₃	—	2	4	no	—
61	C₄⋊Dic₅₃	—	4	4	no	—
62	#11719919	—	4	4	no	—
63	#11719919	—	4	4	no	—
64	C₄×Dic₅₃	—	4	2	no	—
65	C₄×Dic₅₃	D₈	1	2	yes	—
66	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
67	C₂×#88322	Q₁₆	1	4	yes	—
68	C₂×#88322	C₂×D₄	1	2	yes	—
69	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
70	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
71	#11719908	C₂×Q₈	1	4	yes	—
72	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	4	4	no	—
74	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₅₃	—	4	2	no	—
79	C₂×Dic₅₃	—	4	4	no	—
80	C₂×Dic₅₃	C₂×D₈	1	4	yes	—
81	Dic₁₀₆	—	8	8	no	—
82	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
83	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
85	#88322	C₂×Q₁₆	1	4	yes	—
86	#33728	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	2	2	no	—
88	C₂²×C₅₃	#500	1	2	yes	—
89	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
90	C₄×C₅₃	—	8	4	no	—
91	C₄×C₅₃	#500	1	2	yes	—
92	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
94	Dic₅₃	—	2	4	no	—
95	Dic₅₃	—	8	4	no	—
96	#3160	—	53	2	no	—
97	#2255	—	53	4	no	—
98	#2255	—	53	4	no	—
99	#2158	—	53	1	no	—
100	#2158	—	53	2	no	—
101	#2158	D₅₃	1	1	yes	characteristic
102	C₂×#346	—	53	1	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
105	C₂×C₅₃	#3414	1	2	yes	—
106	#533	—	106	2	no	—
107	C₂²×C₁₆	—	53	1	no	—
108	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
109	#500	—	53	4	no	—
110	#499	—	53	4	no	—
111	#499	—	106	2	no	—
112	#498	—	106	2	no	—
113	#427	—	53	2	no	—
114	#427	—	106	2	no	—
115	#427	C₂×D₅₃	1	2	yes	—
116	C₂×C₄×C₈	—	53	1	no	—
117	#367	—	53	2	no	—
118	#367	—	53	4	no	—
119	#367	C₂×D₅₃	1	2	yes	—
120	#366	—	53	2	no	—
121	#366	—	53	4	no	—
122	#366	C₂×D₅₃	1	2	yes	—
123	#346	—	53	2	no	—
124	#346	—	53	2	no	—
125	C₅₃	#33728	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	53	1	no	—
127	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₄⋊Q₈	—	106	4	no	—
129	#125	—	106	4	no	—
130	C₂²⋊Q₈	—	212	4	no	—
131	C₄×Q₈	—	212	4	no	—
132	#117	—	2	2	no	—
133	#117	—	106	2	no	—
134	C₂×C₄²	—	53	1	no	—
135	C₂×C₁₆	—	53	4	no	—
136	C₂×C₁₆	—	53	2	no	—
137	C₂×C₁₆	Dic₁₀₆	1	4	yes	complemented
138	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
139	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
140	#107	—	53	4	no	—
141	#107	—	106	4	no	—
142	#107	C₂²×D₅₃	1	4	yes	—
143	#106	—	2	2	no	—
144	#106	—	106	4	no	—
145	#106	—	106	2	no	—
146	C₄×C₈	—	53	2	no	—
147	C₄×C₈	—	53	2	no	—
148	C₂×Q₈	—	212	4	no	—
149	C₂²×C₄	—	53	2	no	—
150	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
151	C₂×C₈	—	53	4	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
154	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₅₃	1	2	yes	—
156	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	106	4	no	—
159	C₄⋊C₄	—	212	4	no	—
160	C₄⋊C₄	—	212	4	no	—
161	C₄⋊C₄	—	212	4	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	212	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	53	2	no	—
166	C₄²	—	53	2	no	—
167	C₄²	—	212	2	no	—
168	C₁₆	—	53	4	no	—
169	C₁₆	C₂×Dic₁₀₆	1	2	yes	—
170	C₁₆	C₂×Dic₁₀₆	1	2	yes	—
171	C₂³	D₈×D₅₃	1	1	yes	characteristic
172	Q₈	—	424	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	53	4	no	—
175	C₂×C₄	—	212	2	no	—
176	C₂×C₄	—	212	4	no	—
177	C₂×C₄	#13133216	1	2	yes	—
178	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
180	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
181	C₈	—	53	4	no	—
182	C₈	#13133216	1	2	yes	—
183	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
184	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
185	C₂²	—	2	2	no	—
186	C₂²	#16811027	1	2	yes	—
187	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	106	4	no	—
190	C₄	—	424	4	no	—
191	C₄	#16811027	1	2	yes	—
192	C₄	C₂×#13133216	1	1	yes	characteristic
193	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₂×#16811027	1	1	yes	characteristic
196	C₂	#20287848	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20285227

C₂

yes

maximal

#20277000

C₂

yes

maximal, complemented

#20276999

C₂

yes

maximal, complemented

#20276499

C₂

yes

maximal, characteristic

#20276497

C₂

yes

maximal

C₂×#16810398

C₂

yes

maximal, characteristic

C₅₃×#2158

C₂

yes

maximal, characteristic

#16811176

—

C₂²×#11719908

C₂²

yes

characteristic

#16811027

C₂²

yes

—

#16811024

C₂²

yes

—

#16811023

—

#16811020

—

#16810661

C₂²

yes

—

#16810659

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810451

C₂²

yes

—

#16810450

C₂²

yes

—

#16810449

C₂²

yes

—

#16810448

C₂²

yes

—

#16810398

C₂²

yes

—

#16810398

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#427

C₂²

yes

—

C₅₃×#367

C₂²

yes

—

C₅₃×#366

C₂²

yes

—

C₂²×#88322

D₄

yes

—

—

—

C₄×Dic₁₀₆

—

#13133176

—

C₂×C₄×Dic₅₃

D₄

yes

characteristic

C₂×#11719908

Q₈

yes

complemented

C₂×#11719908

C₂³

yes

—

#13133151

C₂³

yes

—

#13133150

—

#13133149

—

#13133148

—

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

C₂³

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

—

C₂×C₁₆×C₅₃

Q₈

yes

complemented

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

#107×C₅₃

C₂³

yes

—

#106×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

D₈

yes

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₈

yes

—

C₄×Dic₅₃

C₂×D₄

yes

—

C₂×#88322

Q₁₆

yes

—

C₂×#88322

C₂×D₄

yes

—

C₁₆×C₅₃

C₂×Q₈

yes

—

C₁₆×C₅₃

C₂×Q₈

yes

—

#11719908

C₂×Q₈

yes

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₈

yes

—

Dic₁₀₆

—

C₈×C₅₃

C₄⋊Q₈

yes

—

C₈×C₅₃

C₂²×Q₈

yes

characteristic

C₈×C₅₃

C₂²×Q₈

yes

characteristic

#88322

C₂×Q₁₆

yes

—

#33728

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#500

yes

—

C₂²×C₅₃

C₂²×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#500

yes

—

C₄×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₅₃

C₂²×Q₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

—

—

—

100

#2158

—

101

#2158

D₅₃

yes

characteristic

102

C₂×#346

—

103

C₂×C₅₃

—

104

C₂×C₅₃

C₂×#500

yes

characteristic

105

C₂×C₅₃

#3414

yes

—

106

#533

—

106

—

107

C₂²×C₁₆

—

108

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

109

#500

—

110

#499

—

111

#499

—

106

—

112

#498

—

106

—

113

#427

—

114

#427

—

106

—

115

#427

C₂×D₅₃

yes

—

116

C₂×C₄×C₈

—

117

#367

—

118

#367

—

119

#367

C₂×D₅₃

yes

—

120

#366

—

121

#366

—

122

#366

C₂×D₅₃

yes

—

123

#346

—

124

#346

—

125

C₅₃

#33728

yes

characteristic, complemented, Sylow

126

C₂²×C₈

—

127

C₂²×C₈

C₂²×D₅₃

yes

characteristic

128

C₄⋊Q₈

—

106

—

129

#125

—

106

—

130

C₂²⋊Q₈

—

212

—

131

C₄×Q₈

—

212

—

132

#117

—

133

#117

—

106

—

134

C₂×C₄²

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

Dic₁₀₆

yes

complemented

138

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

139

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

140

#107

—

141

#107

—

106

—

142

#107

C₂²×D₅₃

yes

—

143

#106

—

144

#106

—

106

—

145

#106

—

106

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₂×Q₈

—

212

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₅₃

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂×Dic₁₀₆

yes

—

154

C₂×C₈

D₄×D₅₃

yes

characteristic

155

C₂×C₈

Q₈×D₅₃

yes

—

156

C₂×C₈

C₂³×D₅₃

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

106

—

159

C₄⋊C₄

—

212

—

160

C₄⋊C₄

—

212

—

161

C₄⋊C₄

—

212

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

212

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

212

—

168

C₁₆

—

169

C₁₆

C₂×Dic₁₀₆

yes

—

170

C₁₆

C₂×Dic₁₀₆

yes

—

171

C₂³

D₈×D₅₃

yes

characteristic

172

Q₈

—

424

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

212

—

176

C₂×C₄

—

212

—

177

C₂×C₄

#13133216

yes

—

178

C₂×C₄

D₈×D₅₃

yes

characteristic

179

C₂×C₄

Q₁₆×D₅₃

yes

—

180

C₂×C₄

C₂×D₄×D₅₃

yes

characteristic

181

C₈

—

182

C₈

#13133216

yes

—

183

C₈

C₂×Q₈×D₅₃

yes

characteristic

184

C₈

C₂²×Dic₁₀₆

yes

characteristic

185

C₂²

—

186

C₂²

#16811027

yes

—

187

C₂²

C₂×D₈×D₅₃

yes

characteristic

188

C₄

—

189

C₄

—

106

—

190

C₄

—

424

—

191

C₄

#16811027

yes

—

192

C₄

C₂×#13133216

yes

characteristic

193

C₄

C₂×Q₁₆×D₅₃

yes

characteristic

194

C₂

—

195

C₂

C₂×#16811027

yes

characteristic

196

C₂

#20287848

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2158	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	#2158	D₅₃
hypocenter	C₅₃	#33728
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂×C₄

D₈×D₅₃

commutator, cosocle

C₂×C₈×C₅₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₃

Fitting

C₅₃×#2158

C₂

socle

C₂²×C₅₃

C₂²×D₈

prime radical

C₂³×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
2. #20277000	C₂	semidirect
3. #20276999	C₂	semidirect
34. C₂×#11719908	Q₈	semidirect
44. C₂×C₁₆×C₅₃	Q₈	semidirect
125. C₅₃	#33728	semidirect
137. C₂×C₁₆	Dic₁₀₆	semidirect

H ≅ G/N

type

2. #20277000

C₂

semidirect

3. #20276999

C₂

semidirect

34. C₂×#11719908

Q₈

semidirect

44. C₂×C₁₆×C₅₃

Q₈

semidirect

125. C₅₃

#33728

semidirect

137. C₂×C₁₆

Dic₁₀₆

semidirect

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

subgroup	index
#20285227	2
#20277000	2
#20276999	2
#20276499	2
#20276497	2
C₂×#16810398	2
C₅₃×#2158	2
#33728	53

subgroup

index

C₅₃×#2158

Subgroups — #27539593

Conjugacy classes of subgroups

Complements

Maximal subgroups