Group #28305666 — subgroups

Subgroups — #28305666

Order 13568.

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

normal subgroups only

This group has 310 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	#20290509	C₂	1	1	yes	maximal, characteristic
3	#20290285	C₂	1	2	yes	maximal
4	#20289612	C₂	1	2	yes	maximal
5	#20273462	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×#13133271	—	2	1	no	—
11	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
12	(C₂×#6157)⋊C₈	—	2	1	no	—
13	#16811487	—	2	2	no	—
14	#16811486	—	2	2	no	—
15	#16810675	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810376	C₄	1	1	yes	characteristic, complemented
18	#16810373	C₄	1	2	yes	complemented
19	C₅₃×#335	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	#13133271	—	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	—	4	1	no	—
50	C₄⋊#6157	—	4	1	no	—
51	C₄×#6157	—	4	1	no	—
52	C₄×#6157	—	4	1	no	—
53	#11719936	—	4	2	no	—
54	#11719936	—	4	2	no	—
55	D₅₃⋊C₈	—	4	1	no	—
56	D₅₃⋊C₈	—	4	1	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₄×Dic₅₃	#41	1	1	yes	characteristic
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
66	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
67	C₈×D₅₃	—	2	1	no	—
68	C₈×D₅₃	—	2	1	no	—
69	C₈×D₅₃	D₈	1	2	yes	—
70	C₈×D₅₃	Q₁₆	1	2	yes	—
71	C₂²×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×#6157	—	4	2	no	—
73	C₂×#6157	—	4	2	no	—
74	C₂×#6157	—	4	2	no	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×Dic₅₃	#121	1	1	yes	characteristic
79	C₂×Dic₅₃	#125	1	1	yes	characteristic
80	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₄×D₅₃	—	2	1	no	—
82	C₄×D₅₃	—	2	1	no	—
83	C₄×D₅₃	—	4	1	no	—
84	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
85	C₄×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₄×D₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₄×D₅₃	#137	1	1	yes	characteristic
88	#88324	—	4	2	no	—
89	#88324	—	8	2	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	—	4	2	no	—
92	C₈×C₅₃	#107	1	2	yes	—
93	#88322	—	2	1	no	—
94	#88322	—	4	2	no	—
95	#88322	#107	1	2	yes	—
96	#9630	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	#389	1	1	yes	characteristic
98	C₂×D₅₃	—	2	1	no	—
99	C₂×D₅₃	—	2	1	no	—
100	C₂×D₅₃	#474	1	1	yes	characteristic
101	C₂×D₅₃	#477	1	1	yes	characteristic
102	#6157	—	8	2	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
105	C₄×C₅₃	#428	1	1	yes	characteristic
106	C₄×C₅₃	#444	1	1	yes	characteristic
107	C₄×C₅₃	#444	1	1	yes	characteristic
108	Dic₅₃	—	2	1	no	—
109	Dic₅₃	—	2	1	no	—
110	Dic₅₃	#459	1	1	yes	characteristic
111	Dic₅₃	#462	1	1	yes	characteristic
112	#1848	—	53	1	no	—
113	#1770	—	53	1	no	—
114	#1571	—	53	2	no	—
115	#1564	—	53	1	no	—
116	#1303	—	53	2	no	—
117	C₂×C₅₃	#1946	1	1	yes	characteristic
118	C₂×C₅₃	#1947	1	1	yes	characteristic
119	C₂×C₅₃	#1953	1	1	yes	characteristic
120	D₅₃	—	2	1	no	—
121	D₅₃	—	2	1	no	—
122	#431	—	53	2	no	—
123	C₂×#108	—	106	1	no	—
124	C₂×#108	—	106	1	no	—
125	#423	—	53	2	no	—
126	C₂×C₄⋊C₈	—	53	1	no	—
127	C₂×#97	—	106	1	no	—
128	C₂×C₄×C₈	—	53	1	no	—
129	C₄×C₄⋊C₄	—	53	1	no	—
130	#336	—	106	1	no	—
131	#335	—	53	1	no	—
132	#335	—	106	2	no	—
133	#335	—	106	2	no	—
134	#335	#6157	1	1	yes	characteristic, complemented
135	#334	—	53	2	no	—
136	C₅₃	#9630	1	1	yes	characteristic, complemented, Sylow
137	C₂×M₁₆	—	106	2	no	—
138	C₂²×C₈	—	53	1	no	—
139	C₂²×C₈	—	53	1	no	—
140	C₂²×C₈	—	106	1	no	—
141	C₂×C₄⋊C₄	—	106	1	no	—
142	C₂×C₄²	—	53	1	no	—
143	C₂×C₄²	—	106	1	no	—
144	#108	—	212	2	no	—
145	#108	—	212	2	no	—
146	C₄⋊C₈	—	53	2	no	—
147	C₄⋊C₈	—	53	2	no	—
148	C₄⋊C₈	—	106	1	no	—
149	C₄⋊C₈	—	106	2	no	—
150	C₄⋊C₈	—	106	2	no	—
151	C₄⋊C₈	C₂×#6157	1	2	yes	—
152	C₂²⋊C₈	—	106	2	no	—
153	#97	—	212	2	no	—
154	C₄×C₈	—	53	1	no	—
155	C₄×C₈	—	53	1	no	—
156	C₄×C₈	—	53	1	no	—
157	C₄×C₈	—	106	2	no	—
158	C₄×C₈	C₂×#6157	1	1	yes	characteristic
159	#95	—	53	2	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	—	53	1	no	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	—	106	2	no	—
164	M₁₆	—	212	2	no	—
165	M₁₆	—	212	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	—	53	1	no	—
171	C₂×C₈	—	106	1	no	—
172	C₂×C₈	—	106	1	no	—
173	C₂×C₈	—	106	2	no	—
174	C₂×C₈	—	106	2	no	—
175	C₂×C₈	—	212	2	no	—
176	C₂×C₈	—	212	1	no	—
177	C₂×C₈	—	212	1	no	—
178	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
179	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
180	C₄⋊C₄	—	212	1	no	—
181	C₄⋊C₄	—	212	1	no	—
182	C₄²	—	53	1	no	—
183	C₄²	—	53	1	no	—
184	C₄²	—	53	1	no	—
185	C₄²	—	212	1	no	—
186	C₄²	—	212	1	no	—
187	C₄²	C₂²×#6157	1	1	yes	characteristic
188	C₂³	—	53	1	no	—
189	C₂×C₄	—	53	1	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	1	no	—
196	C₂×C₄	—	106	1	no	—
197	C₂×C₄	—	106	1	no	—
198	C₂×C₄	—	212	1	no	—
199	C₂×C₄	—	212	2	no	—
200	C₂×C₄	—	212	2	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
203	C₂×C₄	D₄×#6157	1	1	yes	characteristic
204	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	4	2	no	—
207	C₈	—	53	2	no	—
208	C₈	—	106	1	no	—
209	C₈	—	424	2	no	—
210	C₈	—	212	2	no	—
211	C₈	—	212	2	no	—
212	C₈	#13133286	1	2	yes	—
213	C₂²	—	53	1	no	—
214	C₂²	—	53	1	no	—
215	C₂²	—	106	1	no	—
216	C₂²	—	106	1	no	—
217	C₂²	#16811574	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	53	1	no	—
220	C₄	—	53	1	no	—
221	C₄	—	106	1	no	—
222	C₄	—	106	1	no	—
223	C₄	—	424	2	no	—
224	C₄	C₂×#13133286	1	1	yes	characteristic
225	C₄	#16811634	1	1	yes	characteristic
226	C₄	#16811669	1	1	yes	characteristic
227	C₄	#16811670	1	1	yes	characteristic
228	C₂	—	106	1	no	—
229	C₂	—	106	1	no	—
230	C₂	#20290824	1	1	yes	characteristic
231	C₂	#20290826	1	1	yes	characteristic
232	C₂	#20290834	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290654

C₂

yes

maximal, characteristic

#20290509

C₂

yes

maximal, characteristic

#20290285

C₂

yes

maximal

#20289612

C₂

yes

maximal

#20273462

C₂

yes

maximal, characteristic

C₂×#13133287

—

C₂×#13133287

—

#16811627

C₂²

yes

—

D₅₃⋊(C₄⋊C₈)

C₂²

yes

characteristic

C₂×#13133271

—

C₄×C₄⋊#6157

C₂²

yes

—

—

—

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic, complemented

#16810373

C₄

yes

complemented

C₅₃×#335

C₄

yes

characteristic, complemented

C₂×#11719936

—

C₂×D₅₃⋊C₈

—

C₂×C₄⋊#6157

—

C₂×C₄×#6157

—

—

—

—

—

—

D₄

yes

—

#13133271

—

(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

D₈

yes

—

C₈×D₅₃

Q₁₆

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

—

C₄×D₅₃

C₂×D₈

yes

characteristic

C₄×D₅₃

C₂×Q₁₆

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

—

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#107

yes

—

—

—

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

100

C₂×D₅₃

#474

yes

characteristic

101

C₂×D₅₃

#477

yes

characteristic

102

#6157

—

103

C₄×C₅₃

—

104

C₄×C₅₃

C₂×#107

yes

characteristic

105

C₄×C₅₃

#428

yes

characteristic

106

C₄×C₅₃

#444

yes

characteristic

107

C₄×C₅₃

#444

yes

characteristic

108

Dic₅₃

—

109

Dic₅₃

—

110

Dic₅₃

#459

yes

characteristic

111

Dic₅₃

#462

yes

characteristic

112

#1848

—

113

#1770

—

114

#1571

—

115

#1564

—

116

#1303

—

117

C₂×C₅₃

#1946

yes

characteristic

118

C₂×C₅₃

#1947

yes

characteristic

119

C₂×C₅₃

#1953

yes

characteristic

120

D₅₃

—

121

D₅₃

—

122

#431

—

123

C₂×#108

—

106

—

124

C₂×#108

—

106

—

125

#423

—

126

C₂×C₄⋊C₈

—

127

C₂×#97

—

106

—

128

C₂×C₄×C₈

—

129

C₄×C₄⋊C₄

—

130

#336

—

106

—

131

#335

—

132

#335

—

106

—

133

#335

—

106

—

134

#335

#6157

yes

characteristic, complemented

135

#334

—

136

C₅₃

#9630

yes

characteristic, complemented, Sylow

137

C₂×M₁₆

—

106

—

138

C₂²×C₈

—

139

C₂²×C₈

—

140

C₂²×C₈

—

106

—

141

C₂×C₄⋊C₄

—

106

—

142

C₂×C₄²

—

143

C₂×C₄²

—

106

—

144

#108

—

212

—

145

#108

—

212

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

106

—

149

C₄⋊C₈

—

106

—

150

C₄⋊C₈

—

106

—

151

C₄⋊C₈

C₂×#6157

yes

—

152

C₂²⋊C₈

—

106

—

153

#97

—

212

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

106

—

158

C₄×C₈

C₂×#6157

yes

characteristic

159

#95

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

106

—

164

M₁₆

—

212

—

165

M₁₆

—

212

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

106

—

172

C₂×C₈

—

106

—

173

C₂×C₈

—

106

—

174

C₂×C₈

—

106

—

175

C₂×C₈

—

212

—

176

C₂×C₈

—

212

—

177

C₂×C₈

—

212

—

178

C₂×C₈

C₄⋊#6157

yes

characteristic

179

C₂×C₈

C₄⋊#6157

yes

characteristic

180

C₄⋊C₄

—

212

—

181

C₄⋊C₄

—

212

—

182

C₄²

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

212

—

186

C₄²

—

212

—

187

C₄²

C₂²×#6157

yes

characteristic

188

C₂³

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

106

—

197

C₂×C₄

—

106

—

198

C₂×C₄

—

212

—

199

C₂×C₄

—

212

—

200

C₂×C₄

—

212

—

201

C₂×C₄

—

212

—

202

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

203

C₂×C₄

D₄×#6157

yes

characteristic

204

C₂×C₄

Q₈×#6157

yes

characteristic

205

C₈

—

206

C₈

—

207

C₈

—

208

C₈

—

106

—

209

C₈

—

424

—

210

C₈

—

212

—

211

C₈

—

212

—

212

C₈

#13133286

yes

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

106

—

216

C₂²

—

106

—

217

C₂²

#16811574

yes

characteristic

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

106

—

222

C₄

—

106

—

223

C₄

—

424

—

224

C₄

C₂×#13133286

yes

characteristic

225

C₄

#16811634

yes

characteristic

226

C₄

#16811669

yes

characteristic

227

C₄

#16811670

yes

characteristic

228

C₂

—

106

—

229

C₂

—

106

—

230

C₂

#20290824

yes

characteristic

231

C₂

#20290826

yes

characteristic

232

C₂

#20290834

yes

characteristic

	subgroup	G/H
center	C₂²	#16811574
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	#389
prime radical	C₂²×D₅₃	C₂²⋊Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	#335	#6157
hypocenter	C₅₃	#9630
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#16811574

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#335

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 ◁ #335

N	H ≅ G/N	type
17. #16810376	C₄	semidirect
18. #16810373	C₄	semidirect
19. C₅₃×#335	C₄	semidirect
134. #335	#6157	semidirect
136. C₅₃	#9630	semidirect

H ≅ G/N

type

17. #16810376

C₄

semidirect

18. #16810373

C₄

semidirect

19. C₅₃×#335

C₄

semidirect

134. #335

#6157

semidirect

136. C₅₃

#9630

semidirect

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

subgroup	index
#20290654	2
#20290509	2
#20290285	2
#20289612	2
#20273462	2
#9630	53

subgroup

index

Subgroups — #28305666

Conjugacy classes of subgroups

Complements

Maximal subgroups