Group #27275097 — subgroups

Subgroups — #27275097

Order 13568.

452 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20283427	C₂	1	1	yes	maximal, characteristic
2	#20273377	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#88322	C₂	1	2	yes	maximal, complemented
4	#20272605	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1525	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133224	C₄	1	2	yes	—
7	#16810701	—	2	2	no	—
8	D₄×#88322	—	2	2	no	—
9	#16810676	C₂²	1	2	yes	—
10	#16810647	C₂²	1	1	yes	characteristic
11	#16810590	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810372	C₄	1	2	yes	—
14	#16810367	C₄	1	2	yes	—
15	#16810344	—	2	2	no	—
16	C₈×#88322	—	2	2	no	—
17	C₅₃×#532	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#329	C₄	1	4	yes	—
20	#13133224	—	4	4	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	C₂×C₄	1	2	yes	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	C₂×C₄	1	4	yes	—
28	C₂²⋊#88322	—	2	4	no	—
29	C₂²⋊#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	no	—
31	C₄×#88322	D₄	1	2	yes	—
32	C₄×#88322	D₄	1	2	yes	—
33	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
37	#124×C₅₃	C₂×C₄	1	2	yes	—
38	#121×C₅₃	—	2	2	no	—
39	C₄×D₄×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#97×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	#41×C₅₃	—	4	2	no	—
44	#41×C₅₃	—	4	2	no	—
45	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
46	C₂×D₄×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	—	2	2	no	—
48	C₂×D₄×C₅₃	C₄²	1	2	yes	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	2	yes	—
52	M₁₆×C₅₃	—	2	8	no	—
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂²⋊C₄×C₅₃	—	2	4	no	—
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×D₄	1	2	yes	—
59	#11719917	—	4	4	no	—
60	#11719917	—	4	2	no	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	4	no	—
64	C₂×#88322	—	4	1	no	—
65	C₂×#88322	—	4	4	no	—
66	C₂×#88322	C₂²⋊C₄	1	2	yes	—
67	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	#117	1	1	yes	characteristic
71	Q₈×C₅₃	—	4	2	no	—
72	D₄×C₅₃	—	4	2	no	—
73	D₄×C₅₃	—	4	4	no	—
74	D₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₈×C₅₃	—	2	8	no	—
86	#88322	—	2	1	no	—
87	#88322	—	2	2	no	—
88	#88322	—	4	2	no	—
89	#88322	—	4	4	no	—
90	#88322	#100	1	2	yes	—
91	#8241	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	4	2	no	—
95	C₂²×C₅₃	—	4	4	no	—
96	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	4	2	no	—
100	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
101	C₄×C₅₃	#413	1	1	yes	characteristic
102	C₄×C₅₃	#443	1	4	yes	—
103	#2931	—	53	1	no	—
104	#1543	—	53	1	no	—
105	#1525	D₅₃	1	1	yes	characteristic
106	C₈×M₁₆	—	53	2	no	—
107	#1299	—	53	2	no	—
108	C₂×C₅₃	—	4	1	no	—
109	C₂×C₅₃	—	8	2	no	—
110	C₂×C₅₃	C₄×#100	1	1	yes	characteristic
111	C₂×C₅₃	#1764	1	2	yes	—
112	C₂×#131	—	53	2	no	—
113	#532	C₂×D₅₃	1	1	yes	characteristic
114	#436	—	106	2	no	—
115	C₈×D₄	—	106	2	no	—
116	#431	—	53	2	no	—
117	#423	—	53	1	no	—
118	#408	—	53	2	no	—
119	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
120	C₂×C₄×C₈	—	53	1	no	—
121	#333	—	53	2	no	—
122	#331	—	53	2	no	—
123	#329	Dic₅₃	1	4	yes	—
124	#322	—	106	2	no	—
125	C₈²	—	106	2	no	—
126	C₅₃	#8241	1	1	yes	characteristic, complemented, Sylow
127	C₂×#41	C₄×D₅₃	1	2	yes	—
128	#131	—	212	4	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	53	2	no	—
131	C₂×M₁₆	—	106	2	no	—
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	106	2	no	—
135	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
136	#127	C₄×D₅₃	1	1	yes	characteristic
137	#124	C₂×Dic₅₃	1	2	yes	—
138	#121	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
141	C₄⋊C₈	—	53	4	no	—
142	C₄⋊C₈	—	106	2	no	—
143	C₂²⋊C₈	—	106	4	no	—
144	C₂²⋊C₈	—	106	2	no	—
145	#97	C₂×Dic₅₃	1	4	yes	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	53	2	no	—
148	C₄×C₈	—	53	2	no	—
149	C₄×C₈	—	106	2	no	—
150	#41	—	4	2	no	—
151	#41	—	4	2	no	—
152	C₂×Q₈	C₄×Dic₅₃	1	2	yes	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₄×Dic₅₃	1	2	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	#11719947	1	2	yes	—
159	M₁₆	—	2	8	no	—
160	M₁₆	—	212	4	no	—
161	M₁₆	—	212	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	106	2	no	—
167	C₂×C₈	—	106	2	no	—
168	C₂×C₈	—	106	4	no	—
169	C₂×C₈	—	212	4	no	—
170	C₂×C₈	—	212	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
174	C₄²	D₄×D₅₃	1	2	yes	—
175	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
176	C₂³	—	2	2	no	—
177	C₂³	#13133173	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
190	C₂×C₄	#13133177	1	2	yes	—
191	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
192	C₈	—	2	8	no	—
193	C₈	—	53	2	no	—
194	C₈	—	106	1	no	—
195	C₈	—	106	2	no	—
196	C₈	—	212	2	no	—
197	C₈	—	212	4	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	4	no	—
202	C₂²	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	4	2	no	—
206	C₄	#100×D₅₃	1	1	yes	characteristic
207	C₄	#16810604	1	1	yes	characteristic
208	C₄	#16810737	1	4	yes	—
209	C₂	—	4	1	no	—
210	C₂	—	8	2	no	—
211	C₂	#100×Dic₅₃	1	1	yes	characteristic
212	C₂	#20274553	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283427

C₂

yes

maximal, characteristic

#20273377

C₂

yes

maximal, characteristic

M₁₆×#88322

C₂

yes

maximal, complemented

#20272605

C₂

yes

maximal, complemented

C₅₃×#1525

C₂

yes

maximal, characteristic

C₂×#13133224

C₄

yes

—

#16810701

—

D₄×#88322

—

#16810676

C₂²

yes

—

#16810647

C₂²

yes

characteristic

#16810590

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

#16810372

C₄

yes

—

#16810367

C₄

yes

—

#16810344

—

C₈×#88322

—

C₅₃×#532

C₂²

yes

characteristic

C₄×M₁₆×C₅₃

C₂²

yes

—

C₅₃×#329

C₄

yes

—

#13133224

—

C₂×#11719917

—

C₂×#11719917

C₂×C₄

yes

—

C₂²×#88322

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₄⋊#88322

—

C₄⋊#88322

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₄×#88322

D₄

yes

—

C₂×#41×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

characteristic

#127×C₅₃

C₂×C₄

yes

characteristic

#124×C₅₃

C₂×C₄

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#97×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₄²

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

—

—

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

characteristic

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×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₈×C₅₃

—

—

—

—

—

yes

—

#8241

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

100

C₄×C₅₃

C₂×#100

yes

characteristic

101

C₄×C₅₃

#413

yes

characteristic

102

C₄×C₅₃

#443

yes

—

103

#2931

—

104

#1543

—

105

#1525

D₅₃

yes

characteristic

106

C₈×M₁₆

—

107

#1299

—

108

C₂×C₅₃

—

109

C₂×C₅₃

—

110

C₂×C₅₃

C₄×#100

yes

characteristic

111

C₂×C₅₃

#1764

yes

—

112

C₂×#131

—

113

#532

C₂×D₅₃

yes

characteristic

114

#436

—

106

—

115

C₈×D₄

—

106

—

116

#431

—

117

#423

—

118

#408

—

119

C₄×M₁₆

C₂×D₅₃

yes

—

120

C₂×C₄×C₈

—

121

#333

—

122

#331

—

123

#329

Dic₅₃

yes

—

124

#322

—

106

—

125

C₈²

—

106

—

126

C₅₃

#8241

yes

characteristic, complemented, Sylow

127

C₂×#41

C₄×D₅₃

yes

—

128

#131

—

212

—

129

C₂×M₁₆

—

130

C₂×M₁₆

—

131

C₂×M₁₆

—

106

—

132

C₂²×C₈

—

133

C₂²×C₈

—

134

C₂²×C₈

—

106

—

135

C₄⋊Q₈

C₄×D₅₃

yes

characteristic

136

#127

C₄×D₅₃

yes

characteristic

137

#124

C₂×Dic₅₃

yes

—

138

#121

—

139

C₄×D₄

—

140

C₂×C₄²

C₂²×D₅₃

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

106

—

143

C₂²⋊C₈

—

106

—

144

C₂²⋊C₈

—

106

—

145

#97

C₂×Dic₅₃

yes

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

—

106

—

150

#41

—

151

#41

—

152

C₂×Q₈

C₄×Dic₅₃

yes

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

C₄×Dic₅₃

yes

—

156

C₂²×C₄

—

157

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

158

C₂²×C₄

#11719947

yes

—

159

M₁₆

—

160

M₁₆

—

212

—

161

M₁₆

—

212

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

106

—

167

C₂×C₈

—

106

—

168

C₂×C₈

—

106

—

169

C₂×C₈

—

212

—

170

C₂×C₈

—

212

—

171

C₄⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

C₂×C₄×D₅₃

yes

characteristic

174

C₄²

D₄×D₅₃

yes

—

175

C₄²

C₂²×Dic₅₃

yes

characteristic

176

C₂³

—

177

C₂³

#13133173

yes

characteristic

178

Q₈

—

179

D₄

—

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

C₂×C₄×Dic₅₃

yes

characteristic

189

C₂×C₄

C₂²⋊C₄×D₅₃

yes

characteristic

190

C₂×C₄

#13133177

yes

—

191

C₂×C₄

D₄×Dic₅₃

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

106

—

195

C₈

—

106

—

196

C₈

—

212

—

197

C₈

—

212

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

C₂²⋊C₄×Dic₅₃

yes

characteristic

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

#100×D₅₃

yes

characteristic

207

C₄

#16810604

yes

characteristic

208

C₄

#16810737

yes

—

209

C₂

—

210

C₂

—

211

C₂

#100×Dic₅₃

yes

characteristic

212

C₂

#20274553

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1525	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₅₃×#532	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1525	D₅₃
hypocenter	C₅₃	#8241
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1525

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1525

C₂

socle

C₂²×C₅₃

C₄×C₂²⋊C₄

prime radical

C₅₃×#532

C₂²

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1525

N	H ≅ G/N	type
3. M₁₆×#88322	C₂	semidirect
4. #20272605	C₂	semidirect
126. C₅₃	#8241	semidirect

H ≅ G/N

type

3. M₁₆×#88322

C₂

semidirect

4. #20272605

C₂

semidirect

126. C₅₃

#8241

semidirect

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

subgroup	index
#20283427	2
#20273377	2
M₁₆×#88322	2
#20272605	2
C₅₃×#1525	2
#8241	53

subgroup

index

M₁₆×#88322

C₅₃×#1525

Subgroups — #27275097

Conjugacy classes of subgroups

Complements

Maximal subgroups