Group #28302137 — subgroups

Subgroups — #28302137

Order 13568.

398 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 132 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	C₄×#13133285	C₂	1	1	yes	maximal, characteristic
2	#20290512	C₂	1	2	yes	maximal
3	#20289909	C₂	1	1	yes	maximal, characteristic
4	#20289549	C₂	1	2	yes	maximal
5	#335×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133285	C₄	1	2	yes	—
7	C₂×#13133271	—	2	2	no	—
8	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊#6157	C₂²	1	2	yes	—
11	#16811497	C₄	1	2	yes	—
12	(C₂×#6157)⋊C₈	—	2	2	no	—
13	#16811464	C₄	1	2	yes	complemented
14	#16811464	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810376	C₄	1	1	yes	characteristic, complemented
18	#16810375	C₄	1	2	yes	complemented
19	C₅₃×#335	C₄	1	1	yes	characteristic, complemented
20	C₂×D₅₃⋊C₈	—	2	1	no	—
21	C₂×D₅₃⋊C₈	D₄	1	2	yes	—
22	C₂×C₄⋊#6157	C₂×C₄	1	4	yes	—
23	C₂×C₄×#6157	—	2	2	no	—
24	#13133285	—	2	4	no	—
25	#13133271	—	4	4	no	—
26	C₄×#88324	—	2	1	no	—
27	C₄×#88324	C₂×C₄	1	2	yes	—
28	C₄×#88324	C₂×C₄	1	2	yes	—
29	C₄×#88324	Q₈	1	2	yes	—
30	(C₂×#6157)⋊C₄	—	2	2	no	—
31	C₂×C₈×D₅₃	—	2	2	no	—
32	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊#88322	C₂×C₄	1	2	yes	—
36	#13133146	—	2	2	no	—
37	C₄×#88322	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×#6157	—	2	4	no	—
43	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
46	C₂×#88324	—	2	2	no	—
47	C₂×#88324	C₄⋊C₄	1	4	yes	—
48	C₄⋊#6157	—	2	4	no	—
49	C₄⋊#6157	—	2	4	no	—
50	C₄×#6157	—	4	2	no	—
51	C₄×#6157	—	4	2	no	—
52	D₅₃⋊C₈	—	2	2	no	—
53	D₅₃⋊C₈	—	2	2	no	—
54	D₅₃⋊C₈	—	4	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
58	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₅₃	#41	1	1	yes	characteristic
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	C₄²	1	1	yes	characteristic
64	C₂×#88322	C₄²	1	1	yes	characteristic
65	C₈×D₅₃	—	2	2	no	—
66	C₈×D₅₃	—	4	2	no	—
67	C₈×D₅₃	—	4	2	no	—
68	C₈×D₅₃	M₁₆	1	4	yes	—
69	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×#6157	—	4	4	no	—
71	C₂×#6157	—	4	4	no	—
72	C₂×#6157	—	4	4	no	—
73	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₅₃	#117	1	1	yes	characteristic
78	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
79	C₄×D₅₃	—	2	2	no	—
80	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
81	C₄×D₅₃	#131	1	2	yes	—
82	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₄×D₅₃	#137	1	2	yes	—
84	#88324	—	2	4	no	—
85	#88324	—	4	2	no	—
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	—	4	2	no	—
88	C₈×C₅₃	#97	1	2	yes	complemented
89	#88322	—	2	1	no	—
90	#88322	—	4	2	no	—
91	#88322	#97	1	2	yes	complemented
92	#8286	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂×D₅₃	#436	1	2	yes	—
95	C₂×D₅₃	#440	1	2	yes	—
96	C₂×D₅₃	#444	1	2	yes	—
97	#6157	—	8	4	no	—
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
100	C₄×C₅₃	#405	1	1	yes	characteristic
101	C₄×C₅₃	#441	1	1	yes	characteristic
102	C₄×C₅₃	#442	1	1	yes	characteristic
103	Dic₅₃	—	2	1	no	—
104	Dic₅₃	#428	1	2	yes	—
105	Dic₅₃	#446	1	2	yes	—
106	C₄×#106	—	53	1	no	—
107	#1770	—	53	2	no	—
108	C₂×#335	—	53	1	no	—
109	C₂×#322	—	53	1	no	—
110	#1279	—	53	2	no	—
111	C₂×C₅₃	#1770	1	1	yes	characteristic
112	C₂×C₅₃	#1773	1	1	yes	characteristic
113	C₂×C₅₃	#1776	1	1	yes	characteristic
114	D₅₃	#1592	1	4	yes	—
115	C₂×#106	—	53	2	no	—
116	C₂×C₄⋊C₈	—	53	2	no	—
117	C₂×#97	—	106	2	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	C₄×C₄⋊C₄	—	53	2	no	—
122	#340	—	53	2	no	—
123	#336	—	106	2	no	—
124	#335	—	53	2	no	—
125	#335	—	53	1	no	—
126	#335	#6157	1	1	yes	characteristic, complemented
127	#322	—	53	2	no	—
128	#322	—	53	2	no	—
129	C₅₃	#8286	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	—	53	2	no	—
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	—	106	2	no	—
134	C₂²×C₈	—	106	1	no	—
135	C₂×C₄⋊C₄	—	53	4	no	—
136	C₂×C₄²	—	53	1	no	—
137	C₂×C₄²	—	106	2	no	—
138	#106	—	106	4	no	—
139	C₄⋊C₈	—	53	2	no	—
140	C₄⋊C₈	—	106	2	no	—
141	C₄⋊C₈	C₂×#6157	1	2	yes	—
142	#97	—	212	4	no	—
143	C₄×C₈	—	53	2	no	—
144	C₄×C₈	—	53	2	no	—
145	C₄×C₈	—	53	1	no	—
146	C₄×C₈	—	53	1	no	—
147	C₄×C₈	—	53	2	no	—
148	C₄×C₈	—	53	1	no	—
149	C₄×C₈	—	106	1	no	—
150	C₄×C₈	C₂×#6157	1	1	yes	characteristic
151	#95	—	106	2	no	—
152	C₂²×C₄	—	53	1	no	—
153	C₂²×C₄	—	53	1	no	—
154	C₂²×C₄	—	53	1	no	—
155	C₂²×C₄	—	106	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	53	1	no	—
158	C₂×C₈	—	53	4	no	—
159	C₂×C₈	—	53	4	no	—
160	C₂×C₈	—	53	1	no	—
161	C₂×C₈	—	106	2	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	—	106	2	no	—
165	C₂×C₈	—	106	2	no	—
166	C₂×C₈	—	212	2	no	—
167	C₂×C₈	—	212	2	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	C₄×#6157	1	1	yes	characteristic
170	C₂×C₈	C₄×#6157	1	1	yes	characteristic
171	C₄⋊C₄	—	106	4	no	—
172	C₄⋊C₄	—	106	4	no	—
173	C₄²	—	53	1	no	—
174	C₄²	—	53	1	no	—
175	C₄²	—	53	1	no	—
176	C₄²	—	212	2	no	—
177	C₄²	—	212	2	no	—
178	C₄²	C₂²×#6157	1	1	yes	characteristic
179	C₂³	—	53	1	no	—
180	C₂×C₄	—	53	1	no	—
181	C₂×C₄	—	53	1	no	—
182	C₂×C₄	—	53	2	no	—
183	C₂×C₄	—	53	2	no	—
184	C₂×C₄	—	53	2	no	—
185	C₂×C₄	—	53	2	no	—
186	C₂×C₄	—	53	1	no	—
187	C₂×C₄	—	106	2	no	—
188	C₂×C₄	—	212	4	no	—
189	C₂×C₄	—	212	4	no	—
190	C₂×C₄	—	212	4	no	—
191	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
192	C₂×C₄	D₄×#6157	1	1	yes	characteristic
193	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	4	2	no	—
196	C₈	—	53	2	no	—
197	C₈	—	106	4	no	—
198	C₈	—	106	1	no	—
199	C₈	—	212	2	no	—
200	C₈	—	212	2	no	—
201	C₈	#13133271	1	2	yes	complemented
202	C₂²	—	53	2	no	—
203	C₂²	—	53	2	no	—
204	C₂²	—	53	2	no	—
205	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	53	2	no	—
208	C₄	—	53	2	no	—
209	C₄	—	106	1	no	—
210	C₄	—	424	4	no	—
211	C₄	C₂×#13133271	1	1	yes	characteristic
212	C₄	#16811589	1	1	yes	characteristic
213	C₄	#16811663	1	1	yes	characteristic
214	C₄	#16811664	1	1	yes	characteristic
215	C₂	—	53	4	no	—
216	C₂	#20290512	1	1	yes	characteristic
217	C₂	#20290524	1	1	yes	characteristic
218	C₂	#20290530	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133285

C₂

yes

maximal, characteristic

#20290512

C₂

yes

maximal

#20289909

C₂

yes

maximal, characteristic

#20289549

C₂

yes

maximal

#335×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133285

C₄

yes

—

C₂×#13133271

—

C₄×D₅₃⋊C₈

C₂²

yes

characteristic

C₄×D₅₃⋊C₈

C₂²

yes

characteristic

C₄×C₄⋊#6157

C₂²

yes

—

#16811497

C₄

yes

—

(C₂×#6157)⋊C₈

—

#16811464

C₄

yes

complemented

#16811464

C₄

yes

complemented

C₄⋊C₈×D₅₃

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic, complemented

#16810375

C₄

yes

complemented

C₅₃×#335

C₄

yes

characteristic, complemented

C₂×D₅₃⋊C₈

—

C₂×D₅₃⋊C₈

D₄

yes

—

C₂×C₄⋊#6157

C₂×C₄

yes

—

C₂×C₄×#6157

—

—

—

—

C₂×C₄

yes

—

C₄×#88324

C₂×C₄

yes

—

C₄×#88324

Q₈

yes

—

(C₂×#6157)⋊C₄

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₄²×D₅₃

C₂³

yes

characteristic

C₄⋊#88322

C₂×C₄

yes

—

#13133146

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×D₄

yes

C₂×C₄×D₅₃

yes

—

C₄⋊C₄

yes

—

—

—

—

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

C₄×Dic₅₃

yes

—

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

M₁₆

yes

—

C₂²×D₅₃

C₄×D₄

yes

—

—

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

characteristic

C₂×Dic₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

#117

yes

characteristic

C₂×Dic₅₃

C₄×Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂×M₁₆

yes

—

C₄×D₅₃

#131

yes

—

C₄×D₅₃

AGL(1,ℤ/8)

yes

—

C₄×D₅₃

#137

yes

—

#88324

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

—

—

yes

—

C₂²×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₅₃

#436

yes

—

C₂×D₅₃

#440

yes

—

C₂×D₅₃

#444

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#97

yes

characteristic

100

C₄×C₅₃

#405

yes

characteristic

101

C₄×C₅₃

#441

yes

characteristic

102

C₄×C₅₃

#442

yes

characteristic

103

Dic₅₃

—

104

Dic₅₃

#428

yes

—

105

Dic₅₃

#446

yes

—

106

C₄×#106

—

107

#1770

—

108

C₂×#335

—

109

C₂×#322

—

110

#1279

—

111

C₂×C₅₃

#1770

yes

characteristic

112

C₂×C₅₃

#1773

yes

characteristic

113

C₂×C₅₃

#1776

yes

characteristic

114

D₅₃

#1592

yes

—

115

C₂×#106

—

116

C₂×C₄⋊C₈

—

117

C₂×#97

—

106

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₄×C₄⋊C₄

—

122

#340

—

123

#336

—

106

—

124

#335

—

125

#335

—

126

#335

#6157

yes

characteristic, complemented

127

#322

—

128

#322

—

129

C₅₃

#8286

yes

characteristic, complemented, Sylow

130

C₂²×C₈

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

C₂²×C₈

—

106

—

134

C₂²×C₈

—

106

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

—

137

C₂×C₄²

—

106

—

138

#106

—

106

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

106

—

141

C₄⋊C₈

C₂×#6157

yes

—

142

#97

—

212

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

—

106

—

150

C₄×C₈

C₂×#6157

yes

characteristic

151

#95

—

106

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

106

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

106

—

162

C₂×C₈

—

106

—

163

C₂×C₈

—

106

—

164

C₂×C₈

—

106

—

165

C₂×C₈

—

106

—

166

C₂×C₈

—

212

—

167

C₂×C₈

—

212

—

168

C₂×C₈

—

212

—

169

C₂×C₈

C₄×#6157

yes

characteristic

170

C₂×C₈

C₄×#6157

yes

characteristic

171

C₄⋊C₄

—

106

—

172

C₄⋊C₄

—

106

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

212

—

177

C₄²

—

212

—

178

C₄²

C₂²×#6157

yes

characteristic

179

C₂³

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

106

—

188

C₂×C₄

—

212

—

189

C₂×C₄

—

212

—

190

C₂×C₄

—

212

—

191

C₂×C₄

C₂×C₄×#6157

yes

characteristic

192

C₂×C₄

D₄×#6157

yes

characteristic

193

C₂×C₄

Q₈×#6157

yes

characteristic

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

106

—

198

C₈

—

106

—

199

C₈

—

212

—

200

C₈

—

212

—

201

C₈

#13133271

yes

complemented

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

C₄⋊C₄×#6157

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

106

—

210

C₄

—

424

—

211

C₄

C₂×#13133271

yes

characteristic

212

C₄

#16811589

yes

characteristic

213

C₄

#16811663

yes

characteristic

214

C₄

#16811664

yes

characteristic

215

C₂

—

216

C₂

#20290512

yes

characteristic

217

C₂

#20290524

yes

characteristic

218

C₂

#20290530

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×D₅₃	C₂³
hypercenter	#335	#6157
hypocenter	C₅₃	#8286
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#335

C₄

socle

C₂²×C₅₃

C₄×C₄⋊C₄

prime radical

C₂²×D₅₃

C₄×D₄

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
13. #16811464	C₄	semidirect
14. #16811464	C₄	semidirect
17. #16810376	C₄	semidirect
18. #16810375	C₄	semidirect
19. C₅₃×#335	C₄	semidirect
88. C₈×C₅₃	#97	semidirect
91. #88322	#97	semidirect
126. #335	#6157	semidirect
129. C₅₃	#8286	semidirect
201. C₈	#13133271	semidirect

H ≅ G/N

type

13. #16811464

C₄

semidirect

14. #16811464

C₄

semidirect

17. #16810376

C₄

semidirect

18. #16810375

C₄

semidirect

19. C₅₃×#335

C₄

semidirect

88. C₈×C₅₃

#97

semidirect

91. #88322

#97

semidirect

126. #335

#6157

semidirect

129. C₅₃

#8286

semidirect

201. C₈

#13133271

semidirect

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

subgroup	index
C₄×#13133285	2
#20290512	2
#20289909	2
#20289549	2
#335×D₅₃	2
#8286	53

subgroup

index

#335×D₅₃

Subgroups — #28302137

Conjugacy classes of subgroups

Complements

Maximal subgroups