Group #27218036 — subgroups

Subgroups — C₅₃×#10524 #27218036

Order 13568.

386 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₅₃×#3253	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1557	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1530	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1307	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#532	C₂²	1	2	yes	—
7	C₅₃×#476	—	2	2	no	—
8	C₅₃×#469	—	2	2	no	—
9	C₅₃×#459	—	2	2	no	—
10	C₅₃×#432	C₂²	1	2	yes	—
11	C₂×#108×C₅₃	—	2	2	no	—
12	C₅₃×#423	C₂²	1	1	yes	characteristic
13	C₅₃×#423	C₂²	1	1	yes	characteristic
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#332	—	2	2	no	—
16	C₅₃×#331	—	2	2	no	—
17	C₅₃×#329	—	2	4	no	—
18	C₂×#41×C₅₃	—	2	2	no	—
19	C₂×D₈×C₅₃	—	2	2	no	—
20	C₂×M₁₆×C₅₃	—	2	1	no	—
21	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
22	C₂²×C₈×C₅₃	—	2	1	no	—
23	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
24	#127×C₅₃	D₄	1	2	yes	—
25	#124×C₅₃	—	2	2	no	—
26	#121×C₅₃	—	2	2	no	—
27	#121×C₅₃	—	2	2	no	—
28	C₄×D₄×C₅₃	—	2	2	no	—
29	C₄×D₄×C₅₃	—	2	2	no	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#108×C₅₃	—	4	4	no	—
32	#106×C₅₃	—	2	2	no	—
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	#102×C₅₃	—	2	4	no	—
37	C₂²⋊C₈×C₅₃	—	2	2	no	—
38	#97×C₅₃	—	2	2	no	—
39	#97×C₅₃	—	2	1	no	—
40	#97×C₅₃	D₄	1	2	yes	—
41	#41×C₅₃	—	8	4	no	—
42	C₂×Q₈×C₅₃	—	2	2	no	—
43	C₂×D₄×C₅₃	—	2	2	no	—
44	C₂×D₄×C₅₃	—	2	2	no	—
45	C₂×D₄×C₅₃	—	2	2	no	—
46	C₂×D₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	2	yes	—
50	D₈×C₅₃	—	8	2	no	—
51	M₁₆×C₅₃	—	4	2	no	—
52	M₁₆×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	4	1	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	4	2	no	—
62	C₂²⋊C₄×C₅₃	—	4	2	no	—
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅₃	#41	1	1	yes	characteristic
66	C₂³×C₅₃	—	4	2	no	—
67	C₂³×C₅₃	#124	1	1	yes	characteristic
68	Q₈×C₅₃	—	8	2	no	—
69	D₄×C₅₃	—	4	2	no	—
70	D₄×C₅₃	—	4	2	no	—
71	D₄×C₅₃	—	8	2	no	—
72	D₄×C₅₃	—	8	2	no	—
73	D₄×C₅₃	—	8	4	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	8	4	no	—
79	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#121	1	2	yes	—
82	C₂×C₄×C₅₃	#123	1	2	yes	—
83	C₈×C₅₃	—	2	4	no	—
84	C₈×C₅₃	—	4	2	no	—
85	C₈×C₅₃	—	8	2	no	—
86	C₈×C₅₃	—	8	1	no	—
87	#10524	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	2	1	no	—
90	C₂²×C₅₃	—	8	2	no	—
91	C₂²×C₅₃	—	8	2	no	—
92	C₂²×C₅₃	—	8	2	no	—
93	C₂²×C₅₃	#394	1	1	yes	characteristic
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	8	2	no	—
98	C₄×C₅₃	#453	1	1	yes	characteristic
99	C₄×C₅₃	#456	1	1	yes	characteristic
100	#3253	C₂×C₅₃	1	1	yes	characteristic
101	#1849	C₂×C₅₃	1	1	yes	characteristic, complemented
102	#1557	C₂×C₅₃	1	2	yes	complemented
103	#1530	C₂×C₅₃	1	2	yes	complemented
104	#1307	C₂×C₅₃	1	1	yes	characteristic, complemented
105	C₂×C₅₃	—	4	1	no	—
106	C₂×C₅₃	—	16	2	no	—
107	C₂×C₅₃	#2039	1	1	yes	characteristic
108	C₂×C₅₃	#2048	1	2	yes	—
109	#532	C₂²×C₅₃	1	2	yes	—
110	#476	—	2	2	no	—
111	#469	—	2	2	no	—
112	#459	—	2	2	no	—
113	#432	C₂²×C₅₃	1	2	yes	—
114	C₂×#108	—	2	2	no	—
115	#423	C₂²×C₅₃	1	1	yes	characteristic
116	#423	C₂²×C₅₃	1	1	yes	characteristic
117	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
118	#332	—	2	2	no	—
119	#331	—	2	2	no	—
120	#329	—	2	4	no	—
121	C₅₃	#10524	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×#41	—	2	2	no	—
123	C₂×D₈	—	2	2	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
126	C₂²×C₈	—	2	1	no	—
127	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
128	#127	D₄×C₅₃	1	2	yes	—
129	#124	—	2	2	no	—
130	#121	—	2	2	no	—
131	#121	—	2	2	no	—
132	C₄×D₄	—	2	2	no	—
133	C₄×D₄	—	2	2	no	—
134	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
135	#108	—	4	4	no	—
136	#106	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	#102	—	2	4	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	#97	—	2	2	no	—
143	#97	—	2	1	no	—
144	#97	D₄×C₅₃	1	2	yes	—
145	#41	—	8	4	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	4	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	#41×C₅₃	1	2	yes	—
154	D₈	—	8	2	no	—
155	M₁₆	—	4	2	no	—
156	M₁₆	—	4	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	4	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	4	2	no	—
166	C₂²⋊C₄	—	4	2	no	—
167	C₄²	C₂×D₄×C₅₃	1	2	yes	—
168	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
169	C₄²	#41×C₅₃	1	1	yes	characteristic
170	C₂³	—	4	2	no	—
171	C₂³	#124×C₅₃	1	1	yes	characteristic
172	Q₈	—	8	2	no	—
173	D₄	—	4	2	no	—
174	D₄	—	4	2	no	—
175	D₄	—	8	2	no	—
176	D₄	—	8	2	no	—
177	D₄	—	8	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	8	4	no	—
183	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
184	C₂×C₄	#121×C₅₃	1	2	yes	—
185	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
186	C₂×C₄	#123×C₅₃	1	2	yes	—
187	C₈	—	2	4	no	—
188	C₈	—	4	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	8	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	8	2	no	—
194	C₂²	—	8	2	no	—
195	C₂²	—	8	2	no	—
196	C₂²	C₅₃×#394	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	8	2	no	—
201	C₄	C₅₃×#453	1	1	yes	characteristic
202	C₄	C₅₃×#456	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	16	2	no	—
205	C₂	C₅₃×#2039	1	1	yes	characteristic
206	C₂	C₅₃×#2048	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3253

C₂

yes

maximal, characteristic

C₅₃×#1849

C₂

yes

maximal, characteristic, complemented

C₅₃×#1557

C₂

yes

maximal, complemented

C₅₃×#1530

C₂

yes

maximal, complemented

C₅₃×#1307

C₂

yes

maximal, characteristic, complemented

C₅₃×#532

C₂²

yes

—

C₅₃×#476

—

C₅₃×#469

—

C₅₃×#459

—

C₅₃×#432

C₂²

yes

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

characteristic

C₅₃×#423

C₂²

yes

characteristic

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#332

—

C₅₃×#331

—

C₅₃×#329

—

C₂×#41×C₅₃

—

C₂×D₈×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

—

#127×C₅₃

D₄

yes

—

#124×C₅₃

—

#121×C₅₃

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

#106×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#102×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

#97×C₅₃

—

#97×C₅₃

D₄

yes

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

D₈×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

#124

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×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

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#123

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#10524

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#394

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

100

C₂×C₅₃

yes

characteristic

101

#1849

C₂×C₅₃

yes

characteristic, complemented

102

#1557

C₂×C₅₃

yes

complemented

103

#1530

C₂×C₅₃

yes

complemented

104

#1307

C₂×C₅₃

yes

characteristic, complemented

105

C₂×C₅₃

—

106

C₂×C₅₃

—

107

C₂×C₅₃

#2039

yes

characteristic

108

C₂×C₅₃

#2048

yes

—

109

#532

C₂²×C₅₃

yes

—

110

#476

—

111

#469

—

112

#459

—

113

#432

C₂²×C₅₃

yes

—

114

C₂×#108

—

115

#423

C₂²×C₅₃

yes

characteristic

116

#423

C₂²×C₅₃

yes

characteristic

117

C₂×#97

C₂²×C₅₃

yes

characteristic

118

#332

—

119

#331

—

120

#329

—

121

C₅₃

#10524

yes

characteristic, direct product factor, Sylow

122

C₂×#41

—

123

C₂×D₈

—

124

C₂×M₁₆

—

125

C₂×M₁₆

D₄×C₅₃

yes

—

126

C₂²×C₈

—

127

C₄⋊Q₈

D₄×C₅₃

yes

—

128

#127

D₄×C₅₃

yes

—

129

#124

—

130

#121

—

131

#121

—

132

C₄×D₄

—

133

C₄×D₄

—

134

C₂×C₄²

C₂³×C₅₃

yes

characteristic

135

#108

—

136

#106

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

#102

—

141

C₂²⋊C₈

—

142

#97

—

143

#97

—

144

#97

D₄×C₅₃

yes

—

145

#41

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

153

C₂²×C₄

#41×C₅₃

yes

—

154

D₈

—

155

M₁₆

—

156

M₁₆

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₂²⋊C₄

—

166

C₂²⋊C₄

—

167

C₄²

C₂×D₄×C₅₃

yes

—

168

C₄²

C₂×D₄×C₅₃

yes

characteristic

169

C₄²

#41×C₅₃

yes

characteristic

170

C₂³

—

171

C₂³

#124×C₅₃

yes

characteristic

172

Q₈

—

173

D₄

—

174

D₄

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

#120×C₅₃

yes

characteristic

184

C₂×C₄

#121×C₅₃

yes

—

185

C₂×C₄

#121×C₅₃

yes

characteristic

186

C₂×C₄

#123×C₅₃

yes

—

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

C₅₃×#394

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

C₅₃×#453

yes

characteristic

202

C₄

C₅₃×#456

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

C₅₃×#2039

yes

characteristic

206

C₂

C₅₃×#2048

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₅₃×#3253

C₂

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#1849	C₂	semidirect
3. C₅₃×#1557	C₂	semidirect
4. C₅₃×#1530	C₂	semidirect
5. C₅₃×#1307	C₂	semidirect
87. #10524	C₅₃	direct (N × H)
101. #1849	C₂×C₅₃	semidirect
102. #1557	C₂×C₅₃	semidirect
103. #1530	C₂×C₅₃	semidirect
104. #1307	C₂×C₅₃	semidirect
121. C₅₃	#10524	direct (N × H)

H ≅ G/N

type

2. C₅₃×#1849

C₂

semidirect

3. C₅₃×#1557

C₂

semidirect

4. C₅₃×#1530

C₂

semidirect

5. C₅₃×#1307

C₂

semidirect

87. #10524

C₅₃

direct (N × H)

101. #1849

C₂×C₅₃

semidirect

102. #1557

C₂×C₅₃

semidirect

103. #1530

C₂×C₅₃

semidirect

104. #1307

C₂×C₅₃

semidirect

121. C₅₃

#10524

direct (N × H)

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

subgroup	index
C₅₃×#3253	2
C₅₃×#1849	2
C₅₃×#1557	2
C₅₃×#1530	2
C₅₃×#1307	2
#10524	53

subgroup

index

C₅₃×#3253

C₅₃×#1849

C₅₃×#1557

C₅₃×#1530

C₅₃×#1307

#10524

Subgroups — C₅₃×#10524 #27218036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10524 #27218036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10524 #27218036