Group #28310809 — subgroups

Subgroups — #28310809

Order 13568.

296 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 110 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	#20290935	C₂	1	2	yes	maximal
2	C₄×#13133292	C₂	1	1	yes	maximal, characteristic
3	#20290517	C₂	1	1	yes	maximal, characteristic
4	#20289767	C₂	1	2	yes	maximal
5	#364×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133292	C₂²	1	1	yes	characteristic
7	C₂×#13133292	C₂²	1	1	yes	characteristic
8	D₅₃⋊#108	C₄	1	1	yes	characteristic
9	#16811633	C₄	1	1	yes	characteristic
10	#16811586	C₂²	1	2	yes	—
11	#16811514	—	2	2	no	—
12	#16811513	—	2	2	no	—
13	#16811512	C₄	1	2	yes	complemented
14	C₄×#11719910	C₄	1	2	yes	complemented
15	#16811494	C₄	1	2	yes	—
16	M₃₂×D₅₃	C₂²	1	2	yes	—
17	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	#16810444	C₄	1	1	yes	characteristic, complemented
19	#16810443	C₄	1	2	yes	complemented
20	C₅₃×#364	C₄	1	1	yes	characteristic, complemented
21	D₅₃⋊M₁₆	C₂×C₄	1	2	yes	—
22	#13133296	C₂×C₄	1	2	yes	—
23	#13133292	—	2	1	no	—
24	#13133292	—	2	1	no	—
25	#13133292	D₄	1	2	yes	—
26	#13133292	Q₈	1	2	yes	—
27	C₂×#11719910	C₂×C₄	1	2	yes	—
28	C₂×#11719910	C₂×C₄	1	2	yes	—
29	#13133288	—	2	1	no	—
30	#13133287	—	2	1	no	—
31	#13133286	—	2	1	no	—
32	#13133285	—	2	1	no	—
33	#13133271	—	2	2	no	—
34	C₈×#6157	—	2	2	no	—
35	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
37	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
38	#13133157	C₂×C₄	1	2	yes	—
39	#13133156	—	2	2	no	—
40	C₁₆×D₅₃	—	2	2	no	—
41	C₄×#88322	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	—	2	1	no	—
47	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²⋊#6157	—	2	2	no	—
49	#11719940	—	2	2	no	—
50	C₄⋊#6157	—	2	2	no	—
51	C₄×#6157	—	2	2	no	—
52	#11719936	—	2	2	no	—
53	D₅₃⋊C₈	—	2	2	no	—
54	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₄²	1	1	yes	characteristic
57	C₄×Dic₅₃	—	2	1	no	—
58	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
59	C₂×#88322	C₄²	1	1	yes	characteristic
60	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
61	C₈×D₅₃	M₁₆	1	1	yes	characteristic
62	C₈×D₅₃	M₁₆	1	1	yes	characteristic
63	C₈×D₅₃	M₁₆	1	1	yes	characteristic
64	C₈×D₅₃	M₁₆	1	1	yes	characteristic
65	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
66	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₈×D₅₃	#41	1	1	yes	characteristic
68	C₈×D₅₃	#41	1	1	yes	characteristic
69	#11719910	—	2	2	no	—
70	#11719910	C₄⋊C₄	1	4	yes	—
71	C₁₆×C₅₃	—	2	2	no	—
72	#11719908	—	2	2	no	—
73	C₂²×D₅₃	#131	1	1	yes	characteristic
74	C₂×#6157	—	4	2	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	#131	1	1	yes	characteristic
79	C₄×D₅₃	—	4	2	no	—
80	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
81	C₄×D₅₃	C₄×Q₈	1	1	yes	characteristic
82	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
84	#88324	—	4	2	no	—
85	C₈×C₅₃	#97	1	1	yes	characteristic
86	C₈×C₅₃	#97	1	1	yes	characteristic
87	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
89	#88322	#97	1	1	yes	characteristic
90	#88322	#97	1	1	yes	characteristic
91	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
92	#88322	#117	1	1	yes	characteristic
93	#12667	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#405	1	1	yes	characteristic
95	C₂×D₅₃	—	4	1	no	—
96	C₂×D₅₃	#436	1	1	yes	characteristic
97	C₂×D₅₃	#446	1	1	yes	characteristic
98	#6157	—	8	2	no	—
99	C₄×C₅₃	—	2	2	no	—
100	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
102	Dic₅₃	—	4	1	no	—
103	Dic₅₃	#436	1	1	yes	characteristic
104	Dic₅₃	#446	1	1	yes	characteristic
105	C₂×#364	—	53	1	no	—
106	#2110	—	53	2	no	—
107	C₄×M₃₂	—	53	1	no	—
108	#1772	—	53	1	no	—
109	#1378	—	53	2	no	—
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	#1770	1	1	yes	characteristic
112	D₅₃	—	2	1	no	—
113	D₅₃	—	2	1	no	—
114	C₂×M₃₂	—	53	1	no	—
115	C₂×M₃₂	—	53	1	no	—
116	C₂×M₃₂	—	53	2	no	—
117	C₂×#108	—	53	1	no	—
118	#427	—	53	1	no	—
119	#405	—	53	2	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	#364	—	53	2	no	—
122	#364	—	53	1	no	—
123	#364	#6157	1	1	yes	characteristic, complemented
124	#347	—	106	2	no	—
125	#347	—	106	2	no	—
126	#346	—	53	2	no	—
127	C₄×C₁₆	—	53	2	no	—
128	#339	—	53	2	no	—
129	C₅₃	#12667	1	1	yes	characteristic, complemented, Sylow
130	C₂×M₁₆	—	53	2	no	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	1	no	—
133	#117	—	53	2	no	—
134	C₂×C₄²	—	53	1	no	—
135	M₃₂	—	53	2	no	—
136	M₃₂	—	53	2	no	—
137	M₃₂	—	53	2	no	—
138	M₃₂	—	106	2	no	—
139	M₃₂	—	106	1	no	—
140	M₃₂	—	106	1	no	—
141	M₃₂	C₂×#6157	1	2	yes	—
142	C₂×C₁₆	—	53	2	no	—
143	C₂×C₁₆	—	53	2	no	—
144	C₂×C₁₆	—	106	2	no	—
145	#108	—	106	1	no	—
146	#108	—	106	1	no	—
147	#107	—	106	1	no	—
148	#106	—	106	1	no	—
149	#97	—	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	C₂×#6157	1	1	yes	characteristic
155	C₂²×C₄	—	53	1	no	—
156	C₂²×C₄	—	106	1	no	—
157	M₁₆	—	106	2	no	—
158	M₁₆	—	106	2	no	—
159	C₂×C₈	—	53	1	no	—
160	C₂×C₈	—	53	1	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	53	1	no	—
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	—	53	1	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	1	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	C₄×#6157	1	1	yes	characteristic
171	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
172	C₄⋊C₄	—	106	2	no	—
173	C₂²⋊C₄	—	106	2	no	—
174	C₄²	—	53	1	no	—
175	C₄²	—	106	1	no	—
176	C₄²	—	106	2	no	—
177	C₄²	C₄×#6157	1	1	yes	characteristic
178	C₁₆	—	2	2	no	—
179	C₁₆	—	53	4	no	—
180	C₁₆	—	106	2	no	—
181	C₁₆	—	106	2	no	—
182	C₂³	—	53	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	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	—	106	1	no	—
190	C₂×C₄	—	212	2	no	—
191	C₂×C₄	—	212	2	no	—
192	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
193	C₈	—	53	1	no	—
194	C₈	—	53	1	no	—
195	C₈	—	53	1	no	—
196	C₈	—	53	1	no	—
197	C₈	—	212	2	no	—
198	C₈	#13133271	1	1	yes	characteristic
199	C₈	#13133271	1	1	yes	characteristic
200	C₈	D₄×#6157	1	1	yes	characteristic
201	C₈	Q₈×#6157	1	1	yes	characteristic
202	C₂²	—	53	1	no	—
203	C₂²	—	53	1	no	—
204	C₂²	—	212	1	no	—
205	C₂²	#16811589	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	53	1	no	—
208	C₄	—	53	1	no	—
209	C₄	—	424	2	no	—
210	C₄	—	212	1	no	—
211	C₄	C₄⋊C₄×#6157	1	1	yes	characteristic
212	C₄	C₂×#13133271	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	106	1	no	—
215	C₂	—	106	1	no	—
216	C₂	#20290512	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290935

C₂

yes

maximal

C₄×#13133292

C₂

yes

maximal, characteristic

#20290517

C₂

yes

maximal, characteristic

#20289767

C₂

yes

maximal

#364×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133292

C₂²

yes

characteristic

C₂×#13133292

C₂²

yes

characteristic

D₅₃⋊#108

C₄

yes

characteristic

#16811633

C₄

yes

characteristic

#16811586

C₂²

yes

—

#16811514

—

#16811513

—

#16811512

C₄

yes

complemented

C₄×#11719910

C₄

yes

complemented

#16811494

C₄

yes

—

M₃₂×D₅₃

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810444

C₄

yes

characteristic, complemented

#16810443

C₄

yes

complemented

C₅₃×#364

C₄

yes

characteristic, complemented

D₅₃⋊M₁₆

C₂×C₄

yes

—

#13133296

C₂×C₄

yes

—

#13133292

—

#13133292

—

#13133292

D₄

yes

—

#13133292

Q₈

yes

—

C₂×#11719910

C₂×C₄

yes

—

C₂×#11719910

C₂×C₄

yes

—

—

—

—

—

—

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

C₂³

yes

characteristic

C₄²×D₅₃

C₂×C₄

yes

characteristic

#13133157

C₂×C₄

yes

—

#13133156

—

C₁₆×D₅₃

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂²×C₄

yes

characteristic

C₂²⋊#6157

—

—

—

—

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₂²×C₄

yes

characteristic

C₈×D₅₃

M₁₆

yes

characteristic

C₈×D₅₃

M₁₆

yes

characteristic

C₈×D₅₃

M₁₆

yes

characteristic

C₈×D₅₃

M₁₆

yes

characteristic

C₈×D₅₃

C₂×D₄

yes

characteristic

C₈×D₅₃

C₂×Q₈

yes

C₈×D₅₃

yes

C₈×D₅₃

yes

—

C₄⋊C₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

#131

yes

characteristic

C₂×#6157

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#131

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₄×D₄

yes

characteristic

C₄×D₅₃

C₄×Q₈

yes

characteristic

C₄×D₅₃

C₂×M₁₆

yes

characteristic

C₄×D₅₃

C₂×M₁₆

yes

characteristic

#88324

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₈×C₅₃

C₄×D₄

yes

characteristic

C₈×C₅₃

C₄×Q₈

yes

yes

yes

C₂×C₄⋊C₄

yes

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

—

C₄×C₅₃

—

100

C₄×C₅₃

C₄×C₄⋊C₄

yes

characteristic

101

C₄×C₅₃

C₂×#97

yes

characteristic

102

Dic₅₃

—

103

Dic₅₃

#436

yes

characteristic

104

Dic₅₃

#446

yes

characteristic

105

C₂×#364

—

106

#2110

—

107

C₄×M₃₂

—

108

#1772

—

109

#1378

—

110

C₂×C₅₃

—

111

C₂×C₅₃

#1770

yes

characteristic

112

D₅₃

—

113

D₅₃

—

114

C₂×M₃₂

—

115

C₂×M₃₂

—

116

C₂×M₃₂

—

117

C₂×#108

—

118

#427

—

119

#405

—

120

C₂×C₄×C₈

—

121

#364

—

122

#364

—

123

#364

#6157

yes

characteristic, complemented

124

#347

—

106

—

125

#347

—

106

—

126

#346

—

127

C₄×C₁₆

—

128

#339

—

129

C₅₃

#12667

yes

characteristic, complemented, Sylow

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

#117

—

134

C₂×C₄²

—

135

M₃₂

—

136

M₃₂

—

137

M₃₂

—

138

M₃₂

—

106

—

139

M₃₂

—

106

—

140

M₃₂

—

106

—

141

M₃₂

C₂×#6157

yes

—

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

106

—

145

#108

—

106

—

146

#108

—

106

—

147

#107

—

106

—

148

#106

—

106

—

149

#97

—

106

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

106

—

154

C₄×C₈

C₂×#6157

yes

characteristic

155

C₂²×C₄

—

156

C₂²×C₄

—

106

—

157

M₁₆

—

106

—

158

M₁₆

—

106

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

106

—

170

C₂×C₈

C₄×#6157

yes

characteristic

171

C₂×C₈

C₂²×#6157

yes

characteristic

172

C₄⋊C₄

—

106

—

173

C₂²⋊C₄

—

106

—

174

C₄²

—

175

C₄²

—

106

—

176

C₄²

—

106

—

177

C₄²

C₄×#6157

yes

characteristic

178

C₁₆

—

179

C₁₆

—

180

C₁₆

—

106

—

181

C₁₆

—

106

—

182

C₂³

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

106

—

190

C₂×C₄

—

212

—

191

C₂×C₄

—

212

—

192

C₂×C₄

C₂×C₄×#6157

yes

characteristic

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

212

—

198

C₈

#13133271

yes

characteristic

199

C₈

#13133271

yes

characteristic

200

C₈

D₄×#6157

yes

characteristic

201

C₈

Q₈×#6157

yes

characteristic

202

C₂²

—

203

C₂²

—

204

C₂²

—

212

—

205

C₂²

#16811589

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

424

—

210

C₄

—

212

—

211

C₄

C₄⋊C₄×#6157

yes

characteristic

212

C₄

C₂×#13133271

yes

characteristic

213

C₂

—

214

C₂

—

106

—

215

C₂

—

106

—

216

C₂

#20290512

yes

characteristic

	subgroup	G/H
center	C₄	C₄⋊C₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#364	C₄
socle	C₂×C₅₃	#1770
prime radical	C₂²×D₅₃	#131
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	#364	#6157
hypocenter	C₅₃	#12667
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₄

C₄⋊C₄×#6157

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#364

C₄

C₂×C₅₃

C₂²×D₅₃

C₂×C₈×D₅₃

C₂³

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
13. #16811512	C₄	semidirect
14. C₄×#11719910	C₄	semidirect
18. #16810444	C₄	semidirect
19. #16810443	C₄	semidirect
20. C₅₃×#364	C₄	semidirect
123. #364	#6157	semidirect
129. C₅₃	#12667	semidirect

H ≅ G/N

type

13. #16811512

C₄

semidirect

14. C₄×#11719910

C₄

semidirect

18. #16810444

C₄

semidirect

19. #16810443

C₄

semidirect

20. C₅₃×#364

C₄

semidirect

123. #364

#6157

semidirect

129. C₅₃

#12667

semidirect

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

subgroup	index
#20290935	2
C₄×#13133292	2
#20290517	2
#20289767	2
#364×D₅₃	2
#12667	53

subgroup

index

#364×D₅₃

Subgroups — #28310809

Conjugacy classes of subgroups

Complements

Maximal subgroups