Group #27216948 — subgroups

Subgroups — C₅₃×#9436 #27216948

Order 13568.

400 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 126 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×#3139	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2986	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1561	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1549	C₂	1	2	yes	maximal
5	C₅₃×#1299	C₂	1	2	yes	maximal
6	C₅₃×#533	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#491	C₄	1	4	yes	—
9	C₅₃×#487	—	2	2	no	—
10	C₅₃×#446	—	2	2	no	—
11	C₈×Q₈×C₅₃	—	2	2	no	—
12	C₅₃×#433	—	2	2	no	—
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₅₃×#423	C₂²	1	2	yes	—
15	C₅₃×#419	—	2	2	no	—
16	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#332	—	2	2	no	—
18	C₅₃×#332	—	2	2	no	—
19	C₅₃×#332	—	2	2	no	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂×M₁₆×C₅₃	—	2	1	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₄⋊Q₈×C₅₃	—	2	1	no	—
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
27	#125×C₅₃	C₂×C₄	1	2	yes	—
28	C₂²⋊Q₈×C₅₃	—	2	2	no	—
29	C₄×Q₈×C₅₃	—	2	2	no	—
30	#117×C₅₃	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	4	no	—
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
37	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
38	#103×C₅₃	—	2	8	no	—
39	C₂²⋊C₈×C₅₃	—	2	2	no	—
40	#97×C₅₃	—	2	2	no	—
41	#97×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	—	2	1	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	—	2	2	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	4	4	no	—
48	C₂×Q₈×C₅₃	—	4	1	no	—
49	C₂²×C₄×C₅₃	—	4	1	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	2	yes	—
52	M₁₆×C₅₃	—	4	2	no	—
53	M₁₆×C₅₃	—	8	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	2	no	—
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
63	C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	2	yes	—
68	C₂³×C₅₃	#123	1	1	yes	characteristic
69	Q₈×C₅₃	—	4	4	no	—
70	Q₈×C₅₃	—	4	2	no	—
71	Q₈×C₅₃	—	4	2	no	—
72	Q₈×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	8	2	no	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#121	1	2	yes	—
83	C₈×C₅₃	—	4	2	no	—
84	C₈×C₅₃	—	4	2	no	—
85	C₈×C₅₃	—	4	2	no	—
86	C₈×C₅₃	—	8	2	no	—
87	#9436	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₅₃	—	2	1	no	—
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	#386	1	1	yes	characteristic
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	4	4	no	—
94	C₄×C₅₃	—	8	2	no	—
95	C₄×C₅₃	#454	1	2	yes	—
96	C₄×C₅₃	#471	1	2	yes	—
97	C₄×C₅₃	#473	1	2	yes	—
98	#3139	C₂×C₅₃	1	1	yes	characteristic
99	#2986	C₂×C₅₃	1	1	yes	characteristic
100	#1561	C₂×C₅₃	1	1	yes	characteristic
101	#1549	C₂×C₅₃	1	2	yes	—
102	#1299	C₂×C₅₃	1	2	yes	—
103	C₂×C₅₃	—	4	1	no	—
104	C₂×C₅₃	#1888	1	1	yes	characteristic
105	C₂×C₅₃	#1908	1	2	yes	—
106	#533	C₂²×C₅₃	1	1	yes	characteristic
107	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
108	#491	C₄×C₅₃	1	4	yes	—
109	#487	—	2	2	no	—
110	#446	—	2	2	no	—
111	C₈×Q₈	—	2	2	no	—
112	#433	—	2	2	no	—
113	#431	C₂²×C₅₃	1	2	yes	—
114	#423	C₂²×C₅₃	1	2	yes	—
115	#419	—	2	2	no	—
116	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
117	#332	—	2	2	no	—
118	#332	—	2	2	no	—
119	#332	—	2	2	no	—
120	C₅₃	#9436	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²×Q₈	—	2	1	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	1	no	—
126	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
127	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
128	#125	C₂×C₄×C₅₃	1	2	yes	—
129	C₂²⋊Q₈	—	2	2	no	—
130	C₄×Q₈	—	2	2	no	—
131	#117	D₄×C₅₃	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
134	C₄⋊C₈	—	2	4	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
138	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
139	#103	—	2	8	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	#97	—	2	2	no	—
142	#97	C₂×C₄×C₅₃	1	2	yes	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	—	2	2	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	4	4	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂²×C₄	—	4	1	no	—
151	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
152	C₂²×C₄	#41×C₅₃	1	2	yes	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	8	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
167	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₅₃	1	2	yes	—
169	C₂³	#123×C₅₃	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	2	no	—
172	Q₈	—	4	2	no	—
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
182	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
183	C₂×C₄	#121×C₅₃	1	2	yes	—
184	C₈	—	4	2	no	—
185	C₈	—	4	2	no	—
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	C₅₃×#386	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	8	2	no	—
195	C₄	C₅₃×#454	1	2	yes	—
196	C₄	C₅₃×#471	1	2	yes	—
197	C₄	C₅₃×#473	1	2	yes	—
198	C₂	—	4	1	no	—
199	C₂	C₅₃×#1888	1	1	yes	characteristic
200	C₂	C₅₃×#1908	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3139

C₂

yes

maximal, characteristic

C₅₃×#2986

C₂

yes

maximal, characteristic

C₅₃×#1561

C₂

yes

maximal, characteristic

C₅₃×#1549

C₂

yes

maximal

C₅₃×#1299

C₂

yes

maximal

C₅₃×#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×#491

C₄

yes

—

C₅₃×#487

—

C₅₃×#446

—

C₈×Q₈×C₅₃

—

C₅₃×#433

—

C₅₃×#431

C₂²

yes

—

C₅₃×#423

C₂²

yes

—

C₅₃×#419

—

C₄×M₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#332

—

C₅₃×#332

—

C₅₃×#332

—

C₂²×Q₈×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

D₄

yes

—

#125×C₅₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

C₄⋊C₈×C₅₃

D₄

yes

—

#103×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

#97×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×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

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#9436

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#386

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#454

yes

—

C₄×C₅₃

#471

yes

—

C₄×C₅₃

#473

yes

—

#3139

C₂×C₅₃

yes

characteristic

#2986

C₂×C₅₃

yes

characteristic

100

#1561

C₂×C₅₃

yes

characteristic

101

#1549

C₂×C₅₃

yes

—

102

#1299

C₂×C₅₃

yes

—

103

C₂×C₅₃

—

104

C₂×C₅₃

#1888

yes

characteristic

105

C₂×C₅₃

#1908

yes

—

106

#533

C₂²×C₅₃

yes

characteristic

107

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

characteristic

108

#491

C₄×C₅₃

yes

—

109

#487

—

110

#446

—

111

C₈×Q₈

—

112

#433

—

113

#431

C₂²×C₅₃

yes

—

114

#423

C₂²×C₅₃

yes

—

115

#419

—

116

C₄×M₁₆

C₂²×C₅₃

yes

characteristic

117

#332

—

118

#332

—

119

#332

—

120

C₅₃

#9436

yes

characteristic, direct product factor, Sylow

121

C₂²×Q₈

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

127

C₄⋊Q₈

D₄×C₅₃

yes

—

128

#125

C₂×C₄×C₅₃

yes

—

129

C₂²⋊Q₈

—

130

C₄×Q₈

—

131

#117

D₄×C₅₃

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂³×C₅₃

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

D₄×C₅₃

yes

—

138

C₄⋊C₈

D₄×C₅₃

yes

—

139

#103

—

140

C₂²⋊C₈

—

141

#97

—

142

#97

C₂×C₄×C₅₃

yes

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂²×C₄

—

151

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

152

C₂²×C₄

#41×C₅₃

yes

—

153

M₁₆

—

154

M₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

C₂²⋊C₄×C₅₃

yes

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

C₂²×C₄×C₅₃

yes

characteristic

167

C₄²

C₂×D₄×C₅₃

yes

characteristic

168

C₄²

C₂×D₄×C₅₃

yes

—

169

C₂³

#123×C₅₃

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

181

C₂×C₄

C₄×D₄×C₅₃

yes

—

182

C₂×C₄

#120×C₅₃

yes

characteristic

183

C₂×C₄

#121×C₅₃

yes

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

188

C₂²

—

189

C₂²

—

190

C₂²

C₅₃×#386

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

C₅₃×#454

yes

—

196

C₄

C₅₃×#471

yes

—

197

C₄

C₅₃×#473

yes

—

198

C₂

—

199

C₂

C₅₃×#1888

yes

characteristic

200

C₂

C₅₃×#1908

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#386

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂³×C₅₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
87. #9436	C₅₃	direct (N × H)
120. C₅₃	#9436	direct (N × H)

H ≅ G/N

type

87. #9436

C₅₃

direct (N × H)

120. C₅₃

#9436

direct (N × H)

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

subgroup	index
C₅₃×#3139	2
C₅₃×#2986	2
C₅₃×#1561	2
C₅₃×#1549	2
C₅₃×#1299	2
#9436	53

subgroup

index

C₅₃×#3139

C₅₃×#2986

C₅₃×#1561

C₅₃×#1549

C₅₃×#1299

#9436

Subgroups — C₅₃×#9436 #27216948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9436 #27216948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9436 #27216948