Group #28310792 — subgroups

Subgroups — #28310792

Order 13568.

334 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 128 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	#20290927	C₂	1	2	yes	maximal
2	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20290517	C₂	1	1	yes	maximal, characteristic
4	#20289771	C₂	1	2	yes	maximal
5	#364×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×D₅₃⋊C₁₆	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	#16811510	—	2	2	no	—
12	#16811509	C₄	1	2	yes	complemented
13	C₄×#11719910	C₄	1	2	yes	complemented
14	C₁₆×#6157	—	2	2	no	—
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	D₅₃⋊C₁₆	—	2	1	no	—
24	D₅₃⋊C₁₆	—	2	1	no	—
25	D₅₃⋊C₁₆	D₄	1	1	yes	characteristic
26	D₅₃⋊C₁₆	D₄	1	1	yes	characteristic
27	D₅₃⋊C₁₆	Q₈	1	2	yes	—
28	C₂×#11719910	C₂×C₄	1	2	yes	—
29	C₂×#11719910	C₂×C₄	1	2	yes	—
30	#13133288	C₈	1	2	yes	—
31	#13133287	C₈	1	2	yes	—
32	#13133286	C₈	1	2	yes	—
33	#13133285	C₈	1	2	yes	—
34	#13133271	—	2	2	no	—
35	C₈×#6157	—	2	2	no	—
36	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
38	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
39	#13133157	C₂×C₄	1	2	yes	—
40	#13133156	—	2	2	no	—
41	C₁₆×D₅₃	—	2	2	no	—
42	C₄×#88322	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅₃	—	2	1	no	—
48	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²⋊#6157	—	2	2	no	—
50	#11719940	—	2	2	no	—
51	C₄⋊#6157	C₂×C₈	1	4	yes	—
52	C₄×#6157	—	2	2	no	—
53	#11719936	C₂×C₈	1	4	yes	—
54	D₅₃⋊C₈	—	2	2	no	—
55	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₄²	1	1	yes	characteristic
58	C₄×Dic₅₃	—	2	1	no	—
59	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
60	C₂×#88322	C₄²	1	1	yes	characteristic
61	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
62	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
63	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
64	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
65	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
66	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
67	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₈×D₅₃	#41	1	1	yes	characteristic
69	C₈×D₅₃	#41	1	1	yes	characteristic
70	#11719910	—	2	2	no	—
71	#11719910	C₄⋊C₄	1	2	yes	—
72	#11719910	C₄⋊C₄	1	2	yes	—
73	C₁₆×C₅₃	—	2	2	no	—
74	#11719908	—	2	2	no	—
75	C₂²×D₅₃	#131	1	1	yes	characteristic
76	C₂×#6157	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×Dic₅₃	—	2	1	no	—
80	C₂×Dic₅₃	#131	1	1	yes	characteristic
81	C₄×D₅₃	—	2	4	no	—
82	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
83	C₄×D₅₃	C₄×Q₈	1	1	yes	characteristic
84	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
86	#88324	—	2	4	no	—
87	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
88	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
89	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
90	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
91	#88322	C₄×C₈	1	1	yes	characteristic
92	#88322	C₄×C₈	1	1	yes	characteristic
93	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
94	#88322	#117	1	1	yes	characteristic
95	#12664	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#405	1	1	yes	characteristic
97	C₂×D₅₃	—	2	2	no	—
98	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
99	C₂×D₅₃	C₈×Q₈	1	1	yes	characteristic
100	#6157	—	4	4	no	—
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
104	Dic₅₃	—	2	2	no	—
105	Dic₅₃	C₈×D₄	1	1	yes	characteristic
106	Dic₅₃	C₈×Q₈	1	1	yes	characteristic
107	C₂×#364	—	53	1	no	—
108	#2108	—	53	2	no	—
109	C₂×C₄×C₁₆	—	53	1	no	—
110	#1772	—	53	1	no	—
111	#1379	—	53	2	no	—
112	C₂×C₅₃	—	2	1	no	—
113	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
114	D₅₃	—	2	1	no	—
115	D₅₃	#2170	1	2	yes	—
116	C₂×M₃₂	—	53	2	no	—
117	C₂²×C₁₆	—	53	1	no	—
118	C₂²×C₁₆	—	53	1	no	—
119	C₂×#108	—	53	1	no	—
120	#427	—	53	1	no	—
121	#405	—	53	2	no	—
122	C₂×C₄×C₈	—	53	1	no	—
123	#364	—	53	2	no	—
124	#364	—	53	1	no	—
125	#364	#6157	1	1	yes	characteristic, complemented
126	#346	—	106	2	no	—
127	C₄×C₁₆	—	53	2	no	—
128	C₄×C₁₆	—	53	2	no	—
129	C₄×C₁₆	—	106	2	no	—
130	#339	—	53	2	no	—
131	C₅₃	#12664	1	1	yes	characteristic, complemented, Sylow
132	C₂×M₁₆	—	53	2	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	53	1	no	—
135	#117	—	53	2	no	—
136	C₂×C₄²	—	53	1	no	—
137	M₃₂	—	53	2	no	—
138	M₃₂	—	106	2	no	—
139	M₃₂	C₂×#6157	1	2	yes	—
140	C₂×C₁₆	—	53	2	no	—
141	C₂×C₁₆	—	53	1	no	—
142	C₂×C₁₆	—	53	1	no	—
143	C₂×C₁₆	—	53	2	no	—
144	C₂×C₁₆	—	53	2	no	—
145	C₂×C₁₆	—	106	2	no	—
146	C₂×C₁₆	—	106	1	no	—
147	C₂×C₁₆	—	106	1	no	—
148	#108	—	53	2	no	—
149	#108	—	53	2	no	—
150	#107	—	53	2	no	—
151	#106	—	53	2	no	—
152	#97	—	106	2	no	—
153	C₄×C₈	—	53	1	no	—
154	C₄×C₈	—	53	1	no	—
155	C₄×C₈	—	53	1	no	—
156	C₄×C₈	—	106	2	no	—
157	C₄×C₈	C₂×#6157	1	1	yes	characteristic
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	106	1	no	—
160	M₁₆	—	53	4	no	—
161	M₁₆	—	106	2	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	—	53	1	no	—
170	C₂×C₈	—	53	1	no	—
171	C₂×C₈	—	53	1	no	—
172	C₂×C₈	—	106	2	no	—
173	C₂×C₈	C₄×#6157	1	1	yes	characteristic
174	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
175	C₄⋊C₄	—	53	4	no	—
176	C₂²⋊C₄	—	106	2	no	—
177	C₄²	—	53	1	no	—
178	C₄²	—	106	1	no	—
179	C₄²	—	106	2	no	—
180	C₄²	C₄×#6157	1	1	yes	characteristic
181	C₁₆	—	2	2	no	—
182	C₁₆	—	53	2	no	—
183	C₁₆	—	53	2	no	—
184	C₁₆	—	106	2	no	—
185	C₁₆	—	106	2	no	—
186	C₂³	—	53	1	no	—
187	C₂×C₄	—	2	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	—	53	1	no	—
193	C₂×C₄	—	106	4	no	—
194	C₂×C₄	—	106	1	no	—
195	C₂×C₄	—	106	4	no	—
196	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
197	C₈	—	53	1	no	—
198	C₈	—	53	1	no	—
199	C₈	—	53	1	no	—
200	C₈	—	53	1	no	—
201	C₈	—	106	4	no	—
202	C₈	C₈×#6157	1	1	yes	characteristic
203	C₈	C₈×#6157	1	1	yes	characteristic
204	C₈	D₄×#6157	1	1	yes	characteristic
205	C₈	Q₈×#6157	1	1	yes	characteristic
206	C₂²	—	53	1	no	—
207	C₂²	—	53	1	no	—
208	C₂²	—	106	2	no	—
209	C₂²	#16811589	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	53	1	no	—
212	C₄	—	53	1	no	—
213	C₄	—	106	2	no	—
214	C₄	—	212	4	no	—
215	C₄	C₄⋊C₄×#6157	1	1	yes	characteristic
216	C₄	C₂×C₈×#6157	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	53	2	no	—
219	C₂	—	106	1	no	—
220	C₂	#20290508	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290927

C₂

yes

maximal

C₄×D₅₃⋊C₁₆

C₂

yes

maximal, characteristic

#20290517

C₂

yes

maximal, characteristic

#20289771

C₂

yes

maximal

#364×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

C₂²

yes

characteristic

C₂×D₅₃⋊C₁₆

C₂²

yes

characteristic

D₅₃⋊#108

C₄

yes

characteristic

#16811633

C₄

yes

characteristic

#16811586

C₂²

yes

—

#16811510

—

#16811509

C₄

yes

complemented

C₄×#11719910

C₄

yes

complemented

C₁₆×#6157

—

#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

—

D₅₃⋊C₁₆

—

D₅₃⋊C₁₆

—

D₅₃⋊C₁₆

D₄

yes

characteristic

D₅₃⋊C₁₆

D₄

yes

characteristic

D₅₃⋊C₁₆

Q₈

yes

—

C₂×#11719910

C₂×C₄

yes

—

C₂×#11719910

C₂×C₄

yes

—

#13133288

C₈

yes

—

#13133287

C₈

yes

—

#13133286

C₈

yes

—

#13133285

C₈

yes

—

#13133271

—

C₈×#6157

—

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

—

#11719940

—

C₄⋊#6157

C₂×C₈

yes

—

C₄×#6157

—

#11719936

C₂×C₈

yes

—

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

C₂×C₈

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

characteristic

C₈×D₅₃

C₂×D₄

yes

characteristic

C₈×D₅₃

C₂×Q₈

yes

C₈×D₅₃

yes

C₈×D₅₃

yes

—

C₄⋊C₄

yes

—

#11719910

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

C₂²×C₈

yes

characteristic

C₄×D₅₃

C₂²×C₈

yes

characteristic

#88324

—

C₈×C₅₃

C₄×C₈

yes

characteristic

C₈×C₅₃

C₄×C₈

yes

characteristic

C₈×C₅₃

C₄×D₄

yes

characteristic

C₈×C₅₃

C₄×Q₈

yes

characteristic

#88322

C₄×C₈

yes

characteristic

#88322

C₄×C₈

yes

characteristic

#88322

C₂×C₄⋊C₄

yes

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

C₈×D₄

yes

characteristic

C₂×D₅₃

C₈×Q₈

yes

characteristic

100

#6157

—

101

C₄×C₅₃

—

102

C₄×C₅₃

C₄×C₄⋊C₄

yes

characteristic

103

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

104

Dic₅₃

—

105

Dic₅₃

C₈×D₄

yes

characteristic

106

Dic₅₃

C₈×Q₈

yes

characteristic

107

C₂×#364

—

108

#2108

—

109

C₂×C₄×C₁₆

—

110

#1772

—

111

#1379

—

112

C₂×C₅₃

—

113

C₂×C₅₃

C₈×C₄⋊C₄

yes

characteristic

114

D₅₃

—

115

D₅₃

#2170

yes

—

116

C₂×M₃₂

—

117

C₂²×C₁₆

—

118

C₂²×C₁₆

—

119

C₂×#108

—

120

#427

—

121

#405

—

122

C₂×C₄×C₈

—

123

#364

—

124

#364

—

125

#364

#6157

yes

characteristic, complemented

126

#346

—

106

—

127

C₄×C₁₆

—

128

C₄×C₁₆

—

129

C₄×C₁₆

—

106

—

130

#339

—

131

C₅₃

#12664

yes

characteristic, complemented, Sylow

132

C₂×M₁₆

—

133

C₂²×C₈

—

134

C₂²×C₈

—

135

#117

—

136

C₂×C₄²

—

137

M₃₂

—

138

M₃₂

—

106

—

139

M₃₂

C₂×#6157

yes

—

140

C₂×C₁₆

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

145

C₂×C₁₆

—

106

—

146

C₂×C₁₆

—

106

—

147

C₂×C₁₆

—

106

—

148

#108

—

149

#108

—

150

#107

—

151

#106

—

152

#97

—

106

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

—

106

—

157

C₄×C₈

C₂×#6157

yes

characteristic

158

C₂²×C₄

—

159

C₂²×C₄

—

106

—

160

M₁₆

—

161

M₁₆

—

106

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

106

—

173

C₂×C₈

C₄×#6157

yes

characteristic

174

C₂×C₈

C₂²×#6157

yes

characteristic

175

C₄⋊C₄

—

176

C₂²⋊C₄

—

106

—

177

C₄²

—

178

C₄²

—

106

—

179

C₄²

—

106

—

180

C₄²

C₄×#6157

yes

characteristic

181

C₁₆

—

182

C₁₆

—

183

C₁₆

—

184

C₁₆

—

106

—

185

C₁₆

—

106

—

186

C₂³

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

106

—

194

C₂×C₄

—

106

—

195

C₂×C₄

—

106

—

196

C₂×C₄

C₂×C₄×#6157

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

106

—

202

C₈

C₈×#6157

yes

characteristic

203

C₈

C₈×#6157

yes

characteristic

204

C₈

D₄×#6157

yes

characteristic

205

C₈

Q₈×#6157

yes

characteristic

206

C₂²

—

207

C₂²

—

208

C₂²

—

106

—

209

C₂²

#16811589

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

106

—

214

C₄

—

212

—

215

C₄

C₄⋊C₄×#6157

yes

characteristic

216

C₄

C₂×C₈×#6157

yes

characteristic

217

C₂

—

218

C₂

—

219

C₂

—

106

—

220

C₂

#20290508

yes

characteristic

	subgroup	G/H
center	C₈	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#364	C₄
socle	C₂×C₅₃	C₈×C₄⋊C₄
prime radical	C₂²×D₅₃	#131
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	#364	#6157
hypocenter	C₅₃	#12664
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#364

C₄

socle

C₂×C₅₃

C₈×C₄⋊C₄

prime radical

C₂²×D₅₃

#131

cosocle

C₂×C₈×D₅₃

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
12. #16811509	C₄	semidirect
13. C₄×#11719910	C₄	semidirect
18. #16810444	C₄	semidirect
19. #16810443	C₄	semidirect
20. C₅₃×#364	C₄	semidirect
125. #364	#6157	semidirect
131. C₅₃	#12664	semidirect

H ≅ G/N

type

12. #16811509

C₄

semidirect

13. C₄×#11719910

C₄

semidirect

18. #16810444

C₄

semidirect

19. #16810443

C₄

semidirect

20. C₅₃×#364

C₄

semidirect

125. #364

#6157

semidirect

131. C₅₃

#12664

semidirect

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

subgroup	index
#20290927	2
C₄×D₅₃⋊C₁₆	2
#20290517	2
#20289771	2
#364×D₅₃	2
#12664	53

subgroup

index

C₄×D₅₃⋊C₁₆

#364×D₅₃

Subgroups — #28310792

Conjugacy classes of subgroups

Complements

Maximal subgroups