Group #28305628 — subgroups

Subgroups — #28305628

Order 13568.

306 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 118 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	#20290268	C₂	1	2	yes	maximal
2	C₂×#16811486	C₂	1	1	yes	maximal, characteristic
3	#20289918	C₂	1	1	yes	maximal, characteristic
4	#20289561	C₂	1	2	yes	maximal
5	#108×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₂²	1	1	yes	characteristic
7	#16811639	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#88324	C₂²	1	2	yes	—
9	#16811487	—	2	2	no	—
10	#16811486	—	2	1	no	—
11	#16811486	—	2	1	no	—
12	#16811485	—	2	2	no	—
13	#16811464	—	2	1	no	—
14	C₈×#88324	—	2	1	no	—
15	C₂×#13133153	C₄	1	1	yes	characteristic
16	M₁₆×Dic₅₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810387	C₄	1	2	yes	—
19	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×#88324	—	2	2	no	—
21	C₄⋊#88324	—	2	1	no	—
22	C₄⋊#88324	—	2	2	no	—
23	C₄⋊#88324	—	2	2	no	—
24	C₄⋊#88324	D₄	1	1	yes	characteristic
25	C₄⋊#88324	D₄	1	1	yes	characteristic
26	C₂²⋊#88324	—	2	1	no	—
27	C₂²⋊#88324	Q₈	1	2	yes	—
28	C₄×#88324	—	2	1	no	—
29	C₄×#88324	—	2	1	no	—
30	C₂×#11719917	C₂×C₄	1	2	yes	—
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
33	#13133153	C₈	1	4	yes	complemented
34	#13133127	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₈×Dic₅₃	D₄	1	1	yes	characteristic
37	C₈×Dic₅₃	D₄	1	1	yes	characteristic
38	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
39	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#108×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×Dic₅₃	#41	1	1	yes	characteristic
46	C₂×#88324	—	2	1	no	—
47	C₂×#88324	—	2	1	no	—
48	C₂×#88324	—	2	2	no	—
49	C₂×#88324	—	4	2	no	—
50	C₂×#88324	—	8	2	no	—
51	M₁₆×C₅₃	—	2	2	no	—
52	M₁₆×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
58	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₅₃	#41	1	1	yes	characteristic
62	#11719917	—	2	2	no	—
63	#11719917	C₂×C₈	1	4	yes	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
68	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	1	no	—
77	C₂×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	—	4	1	no	—
79	C₂×Dic₅₃	#121	1	1	yes	characteristic
80	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×Dic₅₃	#125	1	1	yes	characteristic
82	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
83	#88324	—	4	2	no	—
84	#88324	—	4	1	no	—
85	#88324	—	4	1	no	—
86	#88324	—	8	2	no	—
87	C₈×C₅₃	—	4	2	no	—
88	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
89	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
90	#88322	—	2	1	no	—
91	#88322	—	2	1	no	—
92	#88322	—	2	4	no	—
93	#9608	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	#389	1	1	yes	characteristic
97	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
98	C₂²×C₅₃	#446	1	1	yes	characteristic
99	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₅₃	#424	1	1	yes	characteristic
101	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
102	C₄×C₅₃	#436	1	1	yes	characteristic
103	Dic₅₃	—	2	1	no	—
104	Dic₅₃	—	2	1	no	—
105	Dic₅₃	—	2	1	no	—
106	Dic₅₃	#472	1	1	yes	characteristic
107	Dic₅₃	#473	1	1	yes	characteristic
108	C₄×#108	—	53	1	no	—
109	#1565	—	53	2	no	—
110	C₂×#335	—	53	1	no	—
111	#1450	—	53	1	no	—
112	#1283	—	53	2	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	#1939	1	1	yes	characteristic
116	C₂×C₅₃	#1952	1	1	yes	characteristic
117	C₂×C₅₃	#1953	1	1	yes	characteristic
118	#431	—	53	1	no	—
119	#431	—	53	1	no	—
120	C₂×#108	—	53	1	no	—
121	C₂×#108	#6157	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	53	2	no	—
123	C₄×M₁₆	—	53	2	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	#341	—	53	2	no	—
126	#335	—	106	1	no	—
127	#335	—	106	1	no	—
128	#335	—	106	2	no	—
129	#334	—	106	2	no	—
130	#322	—	106	1	no	—
131	C₈²	—	106	1	no	—
132	C₅₃	#9608	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	—	53	2	no	—
134	C₂×M₁₆	C₂×#6157	1	2	yes	—
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	106	2	no	—
137	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
138	C₂×C₄²	—	53	1	no	—
139	#108	—	2	2	no	—
140	#108	—	53	4	no	—
141	C₄⋊C₈	—	53	1	no	—
142	C₄⋊C₈	—	53	1	no	—
143	C₄⋊C₈	—	106	1	no	—
144	C₄⋊C₈	—	106	2	no	—
145	C₄⋊C₈	—	106	2	no	—
146	C₂²⋊C₈	—	53	2	no	—
147	C₂²⋊C₈	—	106	1	no	—
148	#97	—	106	2	no	—
149	C₄×C₈	—	53	1	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	—	106	1	no	—
155	C₄×C₈	—	106	1	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
159	M₁₆	—	2	2	no	—
160	M₁₆	—	2	2	no	—
161	M₁₆	—	53	4	no	—
162	M₁₆	—	106	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	53	1	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	106	2	no	—
167	C₂×C₈	—	106	1	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	106	1	no	—
171	C₂×C₈	—	106	1	no	—
172	C₂×C₈	—	424	2	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
175	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
176	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
177	C₂×C₈	#11719940	1	2	yes	—
178	C₄²	—	53	1	no	—
179	C₄²	—	53	1	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	53	1	no	—
182	C₂³	Q₈×#6157	1	1	yes	characteristic
183	C₂×C₄	—	53	1	no	—
184	C₂×C₄	—	53	1	no	—
185	C₂×C₄	—	53	1	no	—
186	C₂×C₄	—	53	1	no	—
187	C₂×C₄	—	106	1	no	—
188	C₂×C₄	—	106	1	no	—
189	C₂×C₄	—	212	1	no	—
190	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
191	C₂×C₄	D₄×#6157	1	1	yes	characteristic
192	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
193	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
194	C₂×C₄	#13133304	1	1	yes	characteristic
195	C₂×C₄	#13133305	1	1	yes	characteristic
196	C₈	—	4	2	no	—
197	C₈	—	106	1	no	—
198	C₈	—	106	1	no	—
199	C₈	—	106	4	no	—
200	C₈	—	424	2	no	—
201	C₈	—	212	1	no	—
202	C₈	—	212	1	no	—
203	C₈	—	212	2	no	—
204	C₈	#13133284	1	2	yes	—
205	C₈	#13133284	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16811574	1	1	yes	characteristic
209	C₂²	#16811672	1	1	yes	characteristic
210	C₂²	#16811674	1	1	yes	characteristic
211	C₄	—	53	1	no	—
212	C₄	—	53	1	no	—
213	C₄	—	106	1	no	—
214	C₄	—	106	1	no	—
215	C₄	—	106	1	no	—
216	C₄	C₂×#13133284	1	1	yes	characteristic
217	C₄	#16811630	1	1	yes	characteristic
218	C₄	#16811651	1	1	yes	characteristic
219	C₄	#16811658	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#20290813	1	1	yes	characteristic
223	C₂	#20290833	1	1	yes	characteristic
224	C₂	#20290835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290268

C₂

yes

maximal

C₂×#16811486

C₂

yes

maximal, characteristic

#20289918

C₂

yes

maximal, characteristic

#20289561

C₂

yes

maximal

#108×Dic₅₃

C₂

yes

maximal, characteristic

#16811639

C₂²

yes

characteristic

#16811639

C₂²

yes

characteristic

C₂×C₄⋊#88324

C₂²

yes

—

—

—

—

—

—

—

C₄

yes

characteristic

M₁₆×Dic₅₃

C₂²

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810387

C₄

yes

—

C₂×#108×C₅₃

C₄

yes

characteristic

C₂²×#88324

—

—

—

—

D₄

yes

characteristic

C₄⋊#88324

D₄

yes

characteristic

C₂²⋊#88324

—

C₂²⋊#88324

Q₈

yes

—

C₄×#88324

—

C₄×#88324

—

C₂×#11719917

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂³

yes

characteristic

#13133153

C₈

yes

complemented

#13133127

—

C₈×Dic₅₃

—

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

#108×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×Q₈

yes

C₂²×Dic₅₃

yes

—

—

—

—

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

C₄×Dic₅₃

yes

—

C₂×C₈

yes

—

—

—

—

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂³×C₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#121

yes

characteristic

C₂×Dic₅₃

C₂²⋊Q₈

yes

characteristic

C₂×Dic₅₃

#125

yes

characteristic

C₂×Dic₅₃

C₄⋊Q₈

yes

—

—

—

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₄⋊C₈

yes

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#389

yes

characteristic

C₂²×C₅₃

C₈×Q₈

yes

characteristic

C₂²×C₅₃

#446

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

100

C₄×C₅₃

#424

yes

characteristic

101

C₄×C₅₃

C₈×D₄

yes

characteristic

102

C₄×C₅₃

#436

yes

characteristic

103

Dic₅₃

—

104

Dic₅₃

—

105

Dic₅₃

—

106

Dic₅₃

#472

yes

characteristic

107

Dic₅₃

#473

yes

characteristic

108

C₄×#108

—

109

#1565

—

110

C₂×#335

—

111

#1450

—

112

#1283

—

113

C₂×C₅₃

—

114

C₂×C₅₃

—

115

C₂×C₅₃

#1939

yes

characteristic

116

C₂×C₅₃

#1952

yes

characteristic

117

C₂×C₅₃

#1953

yes

characteristic

118

#431

—

119

#431

—

120

C₂×#108

—

121

C₂×#108

#6157

yes

characteristic

122

C₂×C₄⋊C₈

—

123

C₄×M₁₆

—

124

C₂×C₄×C₈

—

125

#341

—

126

#335

—

106

—

127

#335

—

106

—

128

#335

—

106

—

129

#334

—

106

—

130

#322

—

106

—

131

C₈²

—

106

—

132

C₅₃

#9608

yes

characteristic, complemented, Sylow

133

C₂×M₁₆

—

134

C₂×M₁₆

C₂×#6157

yes

—

135

C₂²×C₈

—

136

C₂²×C₈

—

106

—

137

C₂²×C₈

C₂×#6157

yes

characteristic

138

C₂×C₄²

—

139

#108

—

140

#108

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

106

—

144

C₄⋊C₈

—

106

—

145

C₄⋊C₈

—

106

—

146

C₂²⋊C₈

—

147

C₂²⋊C₈

—

106

—

148

#97

—

106

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

106

—

154

C₄×C₈

—

106

—

155

C₄×C₈

—

106

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

C₂²×#6157

yes

characteristic

159

M₁₆

—

160

M₁₆

—

161

M₁₆

—

162

M₁₆

—

106

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

106

—

167

C₂×C₈

—

106

—

168

C₂×C₈

—

106

—

169

C₂×C₈

—

106

—

170

C₂×C₈

—

106

—

171

C₂×C₈

—

106

—

172

C₂×C₈

—

424

—

173

C₂×C₈

—

212

—

174

C₂×C₈

D₅₃⋊C₈

yes

—

175

C₂×C₈

C₄⋊#6157

yes

characteristic

176

C₂×C₈

C₄⋊#6157

yes

characteristic

177

C₂×C₈

#11719940

yes

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

182

C₂³

Q₈×#6157

yes

characteristic

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

106

—

188

C₂×C₄

—

106

—

189

C₂×C₄

—

212

—

190

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

191

C₂×C₄

D₄×#6157

yes

characteristic

192

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

193

C₂×C₄

C₂×#11719940

yes

characteristic

194

C₂×C₄

#13133304

yes

characteristic

195

C₂×C₄

#13133305

yes

characteristic

196

C₈

—

197

C₈

—

106

—

198

C₈

—

106

—

199

C₈

—

106

—

200

C₈

—

424

—

201

C₈

—

212

—

202

C₈

—

212

—

203

C₈

—

212

—

204

C₈

#13133284

yes

—

205

C₈

#13133284

yes

—

206

C₂²

—

207

C₂²

—

208

C₂²

#16811574

yes

characteristic

209

C₂²

#16811672

yes

characteristic

210

C₂²

#16811674

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

106

—

214

C₄

—

106

—

215

C₄

—

106

—

216

C₄

C₂×#13133284

yes

characteristic

217

C₄

#16811630

yes

characteristic

218

C₄

#16811651

yes

characteristic

219

C₄

#16811658

yes

characteristic

220

C₂

—

221

C₂

—

222

C₂

#20290813

yes

characteristic

223

C₂

#20290833

yes

characteristic

224

C₂

#20290835

yes

characteristic

	subgroup	G/H
center	C₂²	#16811574
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#108×C₅₃	C₄
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#108	#6157
hypocenter	C₅₃	#9608
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#16811574

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₂×#108×C₅₃

C₄

socle

C₂²×C₅₃

#389

prime radical

C₂³×C₅₃

C₄×Q₈

cosocle

C₂×C₄×Dic₅₃

C₂³

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
33. #13133153	C₈	semidirect
132. C₅₃	#9608	semidirect

H ≅ G/N

type

33. #13133153

C₈

semidirect

132. C₅₃

#9608

semidirect

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

subgroup	index
#20290268	2
C₂×#16811486	2
#20289918	2
#20289561	2
#108×Dic₅₃	2
#9608	53

subgroup

index

#108×Dic₅₃

Subgroups — #28305628

Conjugacy classes of subgroups

Complements

Maximal subgroups