Group #28305662 — subgroups

Subgroups — #28305662

Order 13568.

322 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 98 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	#20290654	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic
3	#20290283	C₂	1	2	yes	maximal
4	#20289612	C₂	1	2	yes	maximal
5	#20273460	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133287	—	2	1	no	—
7	C₂×#13133287	—	2	1	no	—
8	#16811627	C₂²	1	2	yes	—
9	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₈×#6157	—	2	1	no	—
11	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
12	(C₂×#6157)⋊C₈	—	2	1	no	—
13	#16811485	—	2	2	no	—
14	#16811484	—	2	2	no	—
15	#16810675	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810375	C₄	1	2	yes	complemented
18	#16810374	C₄	1	1	yes	characteristic, complemented
19	C₅₃×#334	C₄	1	1	yes	characteristic, complemented
20	C₂×#11719936	—	2	2	no	—
21	C₂×D₅₃⋊C₈	—	2	1	no	—
22	C₂×C₄⋊#6157	—	2	1	no	—
23	C₂×C₄×#6157	—	2	1	no	—
24	#13133287	—	4	2	no	—
25	#13133287	—	4	2	no	—
26	#13133284	—	2	1	no	—
27	C₄⋊#88324	—	2	2	no	—
28	C₄⋊#88324	—	2	2	no	—
29	C₄⋊#88324	D₄	1	2	yes	—
30	C₈×#6157	—	4	2	no	—
31	(C₂×#6157)⋊C₄	Q₈	1	2	yes	—
32	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
33	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
34	C₄²×D₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊#88322	C₂×C₄	1	2	yes	—
36	#13133129	—	2	2	no	—
37	C₄×#88322	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	—	2	2	no	—
39	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
40	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#6157	—	2	2	no	—
44	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
47	C₂×#88324	—	2	2	no	—
48	C₂×#88324	—	4	2	no	—
49	C₄⋊#6157	—	2	4	no	—
50	C₄×#6157	—	4	1	no	—
51	C₄×#6157	—	4	1	no	—
52	#11719936	—	4	2	no	—
53	#11719936	—	4	2	no	—
54	D₅₃⋊C₈	—	4	1	no	—
55	D₅₃⋊C₈	—	4	1	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₄×Dic₅₃	#41	1	1	yes	characteristic
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
65	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
66	C₈×D₅₃	—	2	1	no	—
67	C₈×D₅₃	—	2	1	no	—
68	C₈×D₅₃	SD₁₆	1	2	yes	—
69	C₈×D₅₃	SD₁₆	1	2	yes	—
70	C₂²×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×#6157	—	2	4	no	—
72	C₂×#6157	—	4	2	no	—
73	C₂×#6157	—	4	2	no	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×Dic₅₃	#121	1	1	yes	characteristic
78	C₂×Dic₅₃	#125	1	1	yes	characteristic
79	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₄×D₅₃	—	2	2	no	—
81	C₄×D₅₃	—	4	1	no	—
82	C₄×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₄×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₄×D₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₄×D₅₃	#137	1	1	yes	characteristic
86	#88324	—	4	2	no	—
87	#88324	—	8	2	no	—
88	C₈×C₅₃	—	2	1	no	—
89	C₈×C₅₃	—	4	2	no	—
90	C₈×C₅₃	#106	1	2	yes	—
91	#88322	—	2	1	no	—
92	#88322	—	4	2	no	—
93	#88322	#106	1	2	yes	—
94	#9628	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	#389	1	1	yes	characteristic
96	C₂×D₅₃	—	2	1	no	—
97	C₂×D₅₃	—	2	1	no	—
98	C₂×D₅₃	#475	1	1	yes	characteristic
99	C₂×D₅₃	#476	1	1	yes	characteristic
100	#6157	—	4	4	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
103	C₄×C₅₃	#428	1	1	yes	characteristic
104	C₄×C₅₃	#443	1	2	yes	—
105	Dic₅₃	—	2	1	no	—
106	Dic₅₃	—	2	1	no	—
107	Dic₅₃	#460	1	1	yes	characteristic
108	Dic₅₃	#461	1	1	yes	characteristic
109	#1848	—	53	1	no	—
110	C₈×C₄⋊C₄	—	53	1	no	—
111	#1570	—	53	2	no	—
112	#1564	—	53	1	no	—
113	#1303	—	53	2	no	—
114	C₂×C₅₃	#1945	1	1	yes	characteristic
115	C₂×C₅₃	#1948	1	1	yes	characteristic
116	C₂×C₅₃	#1953	1	1	yes	characteristic
117	D₅₃	—	2	1	no	—
118	D₅₃	—	2	1	no	—
119	#431	—	53	2	no	—
120	C₂×#108	—	106	1	no	—
121	C₂×#108	—	106	1	no	—
122	#423	—	53	2	no	—
123	C₂×C₄⋊C₈	—	53	1	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	C₂×C₄×C₈	—	106	1	no	—
126	C₄×C₄⋊C₄	—	53	1	no	—
127	#336	—	106	1	no	—
128	#335	—	53	2	no	—
129	#334	—	53	1	no	—
130	#334	—	106	2	no	—
131	#334	—	106	2	no	—
132	#334	#6157	1	1	yes	characteristic, complemented
133	C₅₃	#9628	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	106	2	no	—
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	106	1	no	—
138	C₂×C₄⋊C₄	—	106	1	no	—
139	C₂×C₄²	—	53	1	no	—
140	C₂×C₄²	—	106	1	no	—
141	#108	—	212	2	no	—
142	#108	—	212	2	no	—
143	C₄⋊C₈	—	53	2	no	—
144	C₄⋊C₈	—	53	2	no	—
145	C₄⋊C₈	—	106	1	no	—
146	C₄⋊C₈	—	106	2	no	—
147	C₄⋊C₈	—	106	2	no	—
148	C₄⋊C₈	C₂×#6157	1	2	yes	—
149	C₂²⋊C₈	—	106	2	no	—
150	C₄×C₈	—	53	1	no	—
151	C₄×C₈	—	53	1	no	—
152	C₄×C₈	—	53	1	no	—
153	C₄×C₈	—	106	2	no	—
154	C₄×C₈	—	212	2	no	—
155	C₄×C₈	C₂×#6157	1	1	yes	characteristic
156	#95	—	53	2	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	106	2	no	—
161	M₁₆	—	212	2	no	—
162	M₁₆	—	212	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	53	1	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	1	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	106	2	no	—
172	C₂×C₈	—	212	2	no	—
173	C₂×C₈	—	212	1	no	—
174	C₂×C₈	—	212	1	no	—
175	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
176	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
177	C₄⋊C₄	—	106	4	no	—
178	C₄²	—	53	1	no	—
179	C₄²	—	53	1	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	212	1	no	—
182	C₄²	—	212	1	no	—
183	C₄²	C₂²×#6157	1	1	yes	characteristic
184	C₂³	—	53	1	no	—
185	C₂×C₄	—	53	1	no	—
186	C₂×C₄	—	53	1	no	—
187	C₂×C₄	—	53	1	no	—
188	C₂×C₄	—	53	1	no	—
189	C₂×C₄	—	53	1	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	106	2	no	—
193	C₂×C₄	—	106	4	no	—
194	C₂×C₄	—	212	1	no	—
195	C₂×C₄	—	212	2	no	—
196	C₂×C₄	—	212	2	no	—
197	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
198	C₂×C₄	D₄×#6157	1	1	yes	characteristic
199	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	—	53	2	no	—
203	C₈	—	106	1	no	—
204	C₈	—	424	2	no	—
205	C₈	—	212	2	no	—
206	C₈	—	212	2	no	—
207	C₈	#13133285	1	2	yes	—
208	C₂²	—	53	1	no	—
209	C₂²	—	53	1	no	—
210	C₂²	—	106	1	no	—
211	C₂²	—	106	1	no	—
212	C₂²	#16811574	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	53	1	no	—
215	C₄	—	53	1	no	—
216	C₄	—	106	1	no	—
217	C₄	—	106	1	no	—
218	C₄	—	212	4	no	—
219	C₄	C₂×#13133285	1	1	yes	characteristic
220	C₄	#16811634	1	1	yes	characteristic
221	C₄	#16811666	1	2	yes	—
222	C₂	—	106	1	no	—
223	C₂	—	106	1	no	—
224	C₂	#20290822	1	1	yes	characteristic
225	C₂	#20290827	1	1	yes	characteristic
226	C₂	#20290834	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290654

C₂

yes

maximal, characteristic

C₈×C₄⋊#6157

C₂

yes

maximal, characteristic

#20290283

C₂

yes

maximal

#20289612

C₂

yes

maximal

#20273460

C₂

yes

maximal, characteristic

C₂×#13133287

—

C₂×#13133287

—

#16811627

C₂²

yes

—

D₅₃⋊(C₄⋊C₈)

C₂²

yes

characteristic

C₂×C₈×#6157

—

C₄×C₄⋊#6157

C₂²

yes

—

—

—

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810375

C₄

yes

complemented

#16810374

C₄

yes

characteristic, complemented

C₅₃×#334

C₄

yes

characteristic, complemented

C₂×#11719936

—

C₂×D₅₃⋊C₈

—

C₂×C₄⋊#6157

—

C₂×C₄×#6157

—

—

—

—

—

—

D₄

yes

—

C₈×#6157

—

(C₂×#6157)⋊C₄

Q₈

yes

—

C₂×C₈×D₅₃

D₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

C₂³

yes

characteristic

C₄⋊#88322

C₂×C₄

yes

—

#13133129

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

—

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

C₂×C₄×D₅₃

yes

—

—

—

—

—

—

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

C₄×Dic₅₃

yes

—

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

SD₁₆

yes

—

C₈×D₅₃

SD₁₆

yes

—

C₂²×D₅₃

C₂²⋊Q₈

yes

—

—

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

C₄⋊Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×SD₁₆

yes

characteristic

C₄×D₅₃

C₂×SD₁₆

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

—

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#106

yes

—

—

—

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

100

—

101

C₄×C₅₃

—

102

C₄×C₅₃

C₂×#106

yes

characteristic

103

C₄×C₅₃

#428

yes

characteristic

104

C₄×C₅₃

#443

yes

—

105

Dic₅₃

—

106

Dic₅₃

—

107

Dic₅₃

#460

yes

characteristic

108

Dic₅₃

#461

yes

characteristic

109

#1848

—

110

C₈×C₄⋊C₄

—

111

#1570

—

112

#1564

—

113

#1303

—

114

C₂×C₅₃

#1945

yes

characteristic

115

C₂×C₅₃

#1948

yes

characteristic

116

C₂×C₅₃

#1953

yes

characteristic

117

D₅₃

—

118

D₅₃

—

119

#431

—

120

C₂×#108

—

106

—

121

C₂×#108

—

106

—

122

#423

—

123

C₂×C₄⋊C₈

—

124

C₂×C₄×C₈

—

125

C₂×C₄×C₈

—

106

—

126

C₄×C₄⋊C₄

—

127

#336

—

106

—

128

#335

—

129

#334

—

130

#334

—

106

—

131

#334

—

106

—

132

#334

#6157

yes

characteristic, complemented

133

C₅₃

#9628

yes

characteristic, complemented, Sylow

134

C₂×M₁₆

—

106

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

106

—

138

C₂×C₄⋊C₄

—

106

—

139

C₂×C₄²

—

140

C₂×C₄²

—

106

—

141

#108

—

212

—

142

#108

—

212

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

106

—

146

C₄⋊C₈

—

106

—

147

C₄⋊C₈

—

106

—

148

C₄⋊C₈

C₂×#6157

yes

—

149

C₂²⋊C₈

—

106

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

106

—

154

C₄×C₈

—

212

—

155

C₄×C₈

C₂×#6157

yes

characteristic

156

#95

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

106

—

161

M₁₆

—

212

—

162

M₁₆

—

212

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

106

—

169

C₂×C₈

—

106

—

170

C₂×C₈

—

106

—

171

C₂×C₈

—

106

—

172

C₂×C₈

—

212

—

173

C₂×C₈

—

212

—

174

C₂×C₈

—

212

—

175

C₂×C₈

C₄⋊#6157

yes

characteristic

176

C₂×C₈

C₄⋊#6157

yes

characteristic

177

C₄⋊C₄

—

106

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

212

—

182

C₄²

—

212

—

183

C₄²

C₂²×#6157

yes

characteristic

184

C₂³

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

106

—

193

C₂×C₄

—

106

—

194

C₂×C₄

—

212

—

195

C₂×C₄

—

212

—

196

C₂×C₄

—

212

—

197

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

198

C₂×C₄

D₄×#6157

yes

characteristic

199

C₂×C₄

Q₈×#6157

yes

characteristic

200

C₈

—

201

C₈

—

202

C₈

—

203

C₈

—

106

—

204

C₈

—

424

—

205

C₈

—

212

—

206

C₈

—

212

—

207

C₈

#13133285

yes

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

106

—

211

C₂²

—

106

—

212

C₂²

#16811574

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

106

—

217

C₄

—

106

—

218

C₄

—

212

—

219

C₄

C₂×#13133285

yes

characteristic

220

C₄

#16811634

yes

characteristic

221

C₄

#16811666

yes

—

222

C₂

—

106

—

223

C₂

—

106

—

224

C₂

#20290822

yes

characteristic

225

C₂

#20290827

yes

characteristic

226

C₂

#20290834

yes

characteristic

	subgroup	G/H
center	C₂²	#16811574
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#334	C₄
socle	C₂²×C₅₃	#389
prime radical	C₂²×D₅₃	C₂²⋊Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	#334	#6157
hypocenter	C₅₃	#9628
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#16811574

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#334

C₄

socle

C₂²×C₅₃

#389

prime radical

C₂²×D₅₃

C₂²⋊Q₈

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
17. #16810375	C₄	semidirect
18. #16810374	C₄	semidirect
19. C₅₃×#334	C₄	semidirect
132. #334	#6157	semidirect
133. C₅₃	#9628	semidirect

H ≅ G/N

type

17. #16810375

C₄

semidirect

18. #16810374

C₄

semidirect

19. C₅₃×#334

C₄

semidirect

132. #334

#6157

semidirect

133. C₅₃

#9628

semidirect

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

subgroup	index
#20290654	2
C₈×C₄⋊#6157	2
#20290283	2
#20289612	2
#20273460	2
#9628	53

subgroup

index

C₈×C₄⋊#6157

Subgroups — #28305662

Conjugacy classes of subgroups

Complements

Maximal subgroups