Group #28302163 — subgroups

Subgroups — #28302163

Order 13568.

374 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 140 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×#13133288	C₂	1	1	yes	maximal, characteristic
2	#20289917	C₂	1	2	yes	maximal
3	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20289546	C₂	1	2	yes	maximal
5	#20273462	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133288	C₄	1	2	yes	—
7	C₄×#11719936	C₂²	1	2	yes	—
8	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
10	#16811499	C₄	1	2	yes	—
11	#88324⋊C₈	—	2	2	no	—
12	C₈×#88324	—	2	2	no	—
13	C₈×#88324	C₄	1	2	yes	—
14	C₈×#88324	C₄	1	2	yes	—
15	#16810675	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810376	C₄	1	1	yes	characteristic
18	#16810373	C₄	1	2	yes	—
19	C₅₃×#335	C₄	1	1	yes	characteristic
20	C₂×#11719936	C₂×C₄	1	4	yes	—
21	C₂×D₅₃⋊C₈	—	2	1	no	—
22	C₂×D₅₃⋊C₈	Q₈	1	2	yes	—
23	#13133288	C₈	1	8	yes	complemented
24	#13133272	—	2	2	no	—
25	C₄×#88324	—	2	2	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	D₄	1	1	yes	characteristic
30	C₄×#88324	D₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×D₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊#88322	C₂×C₄	1	2	yes	—
35	#13133129	—	2	2	no	—
36	C₄×#88322	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₅₃	—	2	2	no	—
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	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
45	C₂×#88324	—	2	2	no	—
46	C₂×#88324	—	2	4	no	—
47	C₂×#88324	C₄⋊C₄	1	2	yes	—
48	C₂×#88324	C₄⋊C₄	1	2	yes	—
49	#11719936	—	2	4	no	—
50	#11719936	C₂×C₈	1	8	yes	—
51	D₅₃⋊C₈	—	2	2	no	—
52	D₅₃⋊C₈	—	2	2	no	—
53	D₅₃⋊C₈	—	2	4	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₅₃	#41	1	1	yes	characteristic
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	C₄²	1	1	yes	characteristic
63	C₂×#88322	C₄²	1	1	yes	characteristic
64	C₈×D₅₃	—	2	2	no	—
65	C₈×D₅₃	C₂×C₈	1	4	yes	—
66	C₂²×D₅₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×Dic₅₃	#117	1	1	yes	characteristic
72	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
73	C₄×D₅₃	—	2	1	no	—
74	C₄×D₅₃	—	2	1	no	—
75	C₄×D₅₃	—	2	2	no	—
76	C₄×D₅₃	C₂²×C₈	1	2	yes	—
77	C₄×D₅₃	#131	1	2	yes	—
78	#88324	—	2	4	no	—
79	#88324	—	2	2	no	—
80	#88324	—	2	8	no	—
81	#88324	#108	1	4	yes	complemented
82	C₈×C₅₃	—	2	1	no	—
83	C₈×C₅₃	—	4	2	no	—
84	C₈×C₅₃	C₄×C₈	1	2	yes	—
85	#88322	—	2	1	no	—
86	#88322	—	4	2	no	—
87	#88322	C₄×C₈	1	2	yes	—
88	C₈×#108	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₅₃	—	2	1	no	—
91	C₂×D₅₃	—	2	1	no	—
92	C₂×D₅₃	C₈×Q₈	1	2	yes	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
95	C₄×C₅₃	#405	1	1	yes	characteristic
96	C₄×C₅₃	#443	1	1	yes	characteristic
97	C₄×C₅₃	#443	1	1	yes	characteristic
98	Dic₅₃	—	2	1	no	—
99	Dic₅₃	C₂×#108	1	2	yes	—
100	Dic₅₃	C₈×D₄	1	2	yes	—
101	C₄×#108	—	53	1	no	—
102	#1564	—	53	1	no	—
103	C₈×M₁₆	—	53	2	no	—
104	C₂×C₈²	—	53	1	no	—
105	#1278	—	53	2	no	—
106	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₅₃	#1772	1	1	yes	characteristic
108	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
109	D₅₃	—	2	2	no	—
110	#431	—	53	2	no	—
111	C₂×#108	—	53	2	no	—
112	C₄×M₁₆	—	53	2	no	—
113	C₂×C₄×C₈	—	53	1	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	#341	—	53	2	no	—
117	#335	—	53	1	no	—
118	#335	#6157	1	1	yes	characteristic
119	#334	—	53	2	no	—
120	#322	—	106	2	no	—
121	C₈²	—	53	2	no	—
122	C₈²	—	53	2	no	—
123	C₈²	—	106	2	no	—
124	C₅₃	C₈×#108	1	1	yes	characteristic, complemented, Sylow
125	C₂×M₁₆	—	53	4	no	—
126	C₂²×C₈	—	53	1	no	—
127	C₂²×C₈	—	53	2	no	—
128	C₂²×C₈	—	53	1	no	—
129	C₂²×C₈	—	106	1	no	—
130	C₂×C₄²	—	53	1	no	—
131	#108	—	53	8	no	—
132	C₄⋊C₈	—	53	2	no	—
133	C₄⋊C₈	C₂×#6157	1	2	yes	—
134	C₂²⋊C₈	—	106	2	no	—
135	#97	—	106	2	no	—
136	C₄×C₈	—	53	2	no	—
137	C₄×C₈	—	53	1	no	—
138	C₄×C₈	—	53	1	no	—
139	C₄×C₈	—	53	1	no	—
140	C₄×C₈	—	53	1	no	—
141	C₄×C₈	—	53	2	no	—
142	C₄×C₈	—	53	1	no	—
143	C₄×C₈	—	106	2	no	—
144	C₄×C₈	—	106	2	no	—
145	C₄×C₈	—	106	1	no	—
146	C₄×C₈	C₂×#6157	1	1	yes	characteristic
147	C₂²×C₄	—	53	1	no	—
148	C₂²×C₄	—	53	1	no	—
149	C₂²×C₄	—	53	1	no	—
150	M₁₆	—	53	8	no	—
151	M₁₆	—	106	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	1	no	—
155	C₂×C₈	—	53	4	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	53	1	no	—
158	C₂×C₈	—	106	2	no	—
159	C₂×C₈	—	106	4	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	4	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	—	106	2	no	—
165	C₂×C₈	C₄×#6157	1	1	yes	characteristic
166	C₂×C₈	C₄×#6157	1	1	yes	characteristic
167	C₄²	—	53	1	no	—
168	C₄²	—	53	1	no	—
169	C₄²	—	53	1	no	—
170	C₄²	C₂²×#6157	1	1	yes	characteristic
171	C₂³	—	53	1	no	—
172	C₂×C₄	—	53	1	no	—
173	C₂×C₄	—	53	1	no	—
174	C₂×C₄	—	53	2	no	—
175	C₂×C₄	—	53	2	no	—
176	C₂×C₄	—	53	1	no	—
177	C₂×C₄	—	106	2	no	—
178	C₂×C₄	—	106	1	no	—
179	C₂×C₄	—	106	1	no	—
180	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
181	C₂×C₄	D₄×#6157	1	1	yes	characteristic
182	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	4	2	no	—
185	C₈	—	53	4	no	—
186	C₈	—	53	2	no	—
187	C₈	—	106	2	no	—
188	C₈	—	106	8	no	—
189	C₈	—	106	4	no	—
190	C₈	—	106	1	no	—
191	C₈	—	212	2	no	—
192	C₈	C₈×#6157	1	2	yes	—
193	C₂²	—	53	2	no	—
194	C₂²	—	106	1	no	—
195	C₂²	—	106	1	no	—
196	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	53	2	no	—
199	C₄	—	53	2	no	—
200	C₄	—	106	1	no	—
201	C₄	C₂×C₈×#6157	1	1	yes	characteristic
202	C₄	#16811589	1	1	yes	characteristic
203	C₄	#16811665	1	1	yes	characteristic
204	C₄	#16811667	1	1	yes	characteristic
205	C₂	—	106	2	no	—
206	C₂	#20290508	1	1	yes	characteristic
207	C₂	#20290519	1	1	yes	characteristic
208	C₂	#108×#6157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133288

C₂

yes

maximal, characteristic

#20289917

C₂

yes

maximal

C₈×D₅₃⋊C₈

C₂

yes

maximal, characteristic

#20289546

C₂

yes

maximal

#20273462

C₂

yes

maximal, characteristic

C₂×#13133288

C₄

yes

—

C₄×#11719936

C₂²

yes

—

C₄×D₅₃⋊C₈

C₂²

yes

characteristic

C₄×D₅₃⋊C₈

C₂²

yes

characteristic

#16811499

C₄

yes

—

#88324⋊C₈

—

C₈×#88324

—

C₈×#88324

C₄

yes

—

C₈×#88324

C₄

yes

—

#16810675

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic

#16810373

C₄

yes

—

C₅₃×#335

C₄

yes

characteristic

C₂×#11719936

C₂×C₄

yes

—

C₂×D₅₃⋊C₈

—

C₂×D₅₃⋊C₈

Q₈

yes

—

#13133288

C₈

yes

—

—

—

C₂×C₄

yes

—

C₄×#88324

C₂×C₄

yes

—

C₄×#88324

D₄

yes

characteristic

C₄×#88324

D₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₄²×D₅₃

C₂³

yes

characteristic

C₄⋊#88322

C₂×C₄

yes

—

#13133129

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

—

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

C₂×C₄×D₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

C₂×C₄×D₅₃

yes

—

—

C₄⋊C₄

yes

—

C₂×#88324

C₄⋊C₄

yes

—

#11719936

—

#11719936

C₂×C₈

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×D₄

yes

C₄×Dic₅₃

yes

—

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

C₂×C₈

yes

—

C₂²×D₅₃

C₄×Q₈

yes

characteristic

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×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂²×C₈

yes

—

C₄×D₅₃

#131

yes

—

—

—

—

yes

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

—

#88322

—

#88322

—

#88322

C₄×C₈

yes

—

C₈×#108

—

maximal, Sylow

C₂²×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₈×Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄×C₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

C₂×#108

yes

—

100

Dic₅₃

C₈×D₄

yes

—

101

C₄×#108

—

102

#1564

—

103

C₈×M₁₆

—

104

C₂×C₈²

—

105

#1278

—

106

C₂×C₅₃

C₈×C₄⋊C₄

yes

characteristic

107

C₂×C₅₃

#1772

yes

characteristic

108

C₂×C₅₃

C₄×#108

yes

characteristic

109

D₅₃

—

110

#431

—

111

C₂×#108

—

112

C₄×M₁₆

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

—

116

#341

—

117

#335

—

118

yes

119

—

120

#322

—

106

—

121

C₈²

—

122

C₈²

—

123

C₈²

—

106

—

124

C₅₃

C₈×#108

yes

characteristic, complemented, Sylow

125

C₂×M₁₆

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

—

106

—

130

C₂×C₄²

—

131

#108

—

132

C₄⋊C₈

—

133

C₄⋊C₈

C₂×#6157

yes

—

134

C₂²⋊C₈

—

106

—

135

#97

—

106

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

106

—

144

C₄×C₈

—

106

—

145

C₄×C₈

—

106

—

146

C₄×C₈

C₂×#6157

yes

characteristic

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

M₁₆

—

151

M₁₆

—

106

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

106

—

159

C₂×C₈

—

106

—

160

C₂×C₈

—

106

—

161

C₂×C₈

—

106

—

162

C₂×C₈

—

106

—

163

C₂×C₈

—

106

—

164

C₂×C₈

—

106

—

165

C₂×C₈

C₄×#6157

yes

characteristic

166

C₂×C₈

C₄×#6157

yes

characteristic

167

C₄²

—

168

C₄²

—

169

C₄²

—

170

C₄²

C₂²×#6157

yes

characteristic

171

C₂³

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

106

—

178

C₂×C₄

—

106

—

179

C₂×C₄

—

106

—

180

C₂×C₄

C₂×C₄×#6157

yes

characteristic

181

C₂×C₄

D₄×#6157

yes

characteristic

182

C₂×C₄

Q₈×#6157

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

106

—

188

C₈

—

106

—

189

C₈

—

106

—

190

C₈

—

106

—

191

C₈

—

212

—

192

C₈

C₈×#6157

yes

—

193

C₂²

—

194

C₂²

—

106

—

195

C₂²

—

106

—

196

C₂²

C₄⋊C₄×#6157

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

106

—

201

C₄

C₂×C₈×#6157

yes

characteristic

202

C₄

#16811589

yes

characteristic

203

C₄

#16811665

yes

characteristic

204

C₄

#16811667

yes

characteristic

205

C₂

—

106

—

206

C₂

#20290508

yes

characteristic

207

C₂

#20290519

yes

characteristic

208

C₂

#108×#6157

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂²×D₅₃	C₄×Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	#335	#6157
hypocenter	C₅₃	C₈×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#335

C₄

socle

C₂²×C₅₃

C₄×C₄⋊C₄

prime radical

C₂²×D₅₃

C₄×Q₈

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
23. #13133288	C₈	semidirect
81. #88324	#108	semidirect
124. C₅₃	C₈×#108	semidirect

H ≅ G/N

type

23. #13133288

C₈

semidirect

81. #88324

#108

semidirect

124. C₅₃

C₈×#108

semidirect

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

subgroup	index
C₄×#13133288	2
#20289917	2
C₈×D₅₃⋊C₈	2
#20289546	2
#20273462	2
C₈×#108	53

subgroup

index

C₈×D₅₃⋊C₈

Subgroups — #28302163

Conjugacy classes of subgroups

Complements

Maximal subgroups