Group #28316259 — subgroups

Subgroups — #28316259

Order 13568.

702 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 270 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	#20291361	C₂	1	2	yes	maximal
2	#20290283	C₂	1	6	yes	maximal, complemented
3	#20290271	C₂	1	3	yes	maximal, complemented
4	D₅₃⋊#334	C₂	1	3	yes	maximal, complemented
5	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133305	—	2	2	no	—
7	#16811630	—	2	6	no	—
8	#16811627	C₂²	1	6	yes	—
9	D₅₃⋊(C₄⋊C₈)	C₂²	1	6	yes	—
10	#16811485	C₂²	1	1	yes	characteristic
11	#16811485	C₂²	1	3	yes	—
12	#16811485	C₂²	1	6	yes	—
13	#16811484	C₂²	1	6	yes	—
14	C₄×#11719949	C₂²	1	1	yes	characteristic
15	Q₈×#88322	C₄	1	1	yes	characteristic
16	C₈×D₂₁₂	C₄	1	3	yes	—
17	#16810688	C₄	1	3	yes	—
18	#16810675	C₂²	1	3	yes	—
19	C₄×C₈×D₅₃	C₂²	1	3	yes	—
20	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
21	#13133305	—	4	8	no	—
22	C₂×#11719936	—	2	6	no	—
23	C₂×D₅₃⋊C₈	—	2	6	no	—
24	#13133284	—	2	6	no	—
25	C₄⋊#88324	—	2	6	no	—
26	C₄⋊#88324	C₂³	1	2	yes	—
27	C₄⋊#88324	C₂³	1	6	yes	—
28	C₂×#11719949	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
31	Q₈×Dic₅₃	Q₈	1	1	yes	characteristic
32	C₄×D₂₁₂	C₂×C₄	1	3	yes	—
33	#13133179	Q₈	1	3	yes	—
34	#13133175	D₄	1	3	yes	—
35	C₄²×D₅₃	C₂³	1	3	yes	—
36	C₄⋊#88322	C₂×C₄	1	3	yes	—
37	#13133129	C₂×C₄	1	6	yes	—
38	C₄×#88322	C₂×C₄	1	3	yes	—
39	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
40	C₈×Dic₅₃	C₂³	1	3	yes	—
41	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
44	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	#11719949	SD₁₆	1	8	yes	—
46	C₂×D₂₁₂	C₄⋊C₄	1	3	yes	—
47	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
48	C₂×C₄×D₅₃	C₂×D₄	1	3	yes	—
49	C₂×C₄×D₅₃	C₂×Q₈	1	3	yes	—
50	C₂×#88324	—	2	6	no	—
51	C₂×#88324	—	2	2	no	—
52	#11719936	—	4	12	no	—
53	D₅₃⋊C₈	—	4	12	no	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
56	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
57	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
58	#11719921	C₄⋊C₄	1	6	yes	—
59	C₄⋊Dic₅₃	C₄⋊C₄	1	3	yes	—
60	C₄×Dic₅₃	C₂×D₄	1	3	yes	—
61	C₄×Dic₅₃	C₂×Q₈	1	3	yes	—
62	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
63	C₂×#88322	C₂²×C₄	1	3	yes	—
64	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
65	C₈×D₅₃	—	2	6	no	—
66	C₈×D₅₃	—	4	3	no	—
67	C₂²×D₅₃	C₂×C₄⋊C₄	1	3	yes	—
68	Q₈×C₅₃	#106	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×Dic₅₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
75	D₂₁₂	#106	1	12	yes	—
76	C₄×D₅₃	—	2	3	no	—
77	C₄×D₅₃	—	2	3	no	—
78	C₄×D₅₃	—	4	3	no	—
79	C₄×D₅₃	C₂×SD₁₆	1	12	yes	—
80	#88324	—	4	6	no	—
81	#88324	—	4	2	no	—
82	C₈×C₅₃	—	2	3	no	—
83	C₈×C₅₃	—	2	1	no	—
84	#88322	—	2	3	no	—
85	#88322	#131	1	2	yes	complemented
86	#17826	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₅₃	—	2	3	no	—
89	C₂×D₅₃	—	2	3	no	—
90	C₂×D₅₃	C₂×#106	1	6	yes	—
91	C₄×C₅₃	—	2	3	no	—
92	C₄×C₅₃	C₂×#106	1	6	yes	—
93	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
94	C₄×C₅₃	#568	1	1	yes	characteristic
95	Dic₅₃	—	2	3	no	—
96	Dic₅₃	—	2	1	no	—
97	Dic₅₃	C₂²×SD₁₆	1	2	yes	—
98	C₈×#41	—	53	1	no	—
99	#2906	—	53	2	no	—
100	#1570	—	53	6	no	—
101	#1566	—	53	3	no	—
102	C₂×#334	—	53	3	no	—
103	C₂×C₅₃	#2906	1	1	yes	characteristic
104	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
105	C₂×C₅₃	#2916	1	1	yes	characteristic
106	D₅₃	—	2	6	no	—
107	C₂×#131	—	106	2	no	—
108	C₄×#41	—	53	1	no	—
109	C₈×Q₈	—	53	1	no	—
110	C₈×Q₈	#6157	1	1	yes	characteristic
111	C₈×D₄	—	53	3	no	—
112	C₈×D₄	—	53	3	no	—
113	#431	—	53	3	no	—
114	#424	—	106	6	no	—
115	#423	—	53	6	no	—
116	C₂×C₄⋊C₈	—	53	6	no	—
117	C₂×C₄×C₈	—	53	3	no	—
118	#334	—	53	1	no	—
119	#334	—	53	3	no	—
120	#334	—	53	6	no	—
121	#334	—	53	6	no	—
122	C₅₃	#17826	1	1	yes	characteristic, complemented, Sylow
123	C₂×#41	—	53	1	no	—
124	#131	—	212	8	no	—
125	C₂×M₁₆	—	106	6	no	—
126	C₂²×C₈	—	53	3	no	—
127	C₂²×C₈	—	53	3	no	—
128	C₂²×C₈	—	106	6	no	—
129	C₄×Q₈	—	53	1	no	—
130	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
131	C₄×D₄	—	53	3	no	—
132	C₄×D₄	—	53	3	no	—
133	#117	—	53	3	no	—
134	C₂×C₄²	—	53	3	no	—
135	C₄⋊C₈	—	53	2	no	—
136	C₄⋊C₈	—	53	6	no	—
137	C₄⋊C₈	—	53	3	no	—
138	C₄⋊C₈	—	106	6	no	—
139	C₄⋊C₈	—	106	6	no	—
140	C₄⋊C₈	C₂×#6157	1	3	yes	—
141	C₂²⋊C₈	—	53	6	no	—
142	C₄×C₈	—	53	1	no	—
143	C₄×C₈	—	53	3	no	—
144	C₄×C₈	—	53	3	no	—
145	C₄×C₈	C₂×#6157	1	3	yes	—
146	#41	—	53	8	no	—
147	C₂×Q₈	C₄⋊#6157	1	1	yes	characteristic
148	C₂×D₄	—	53	3	no	—
149	C₂²×C₄	—	53	3	no	—
150	C₂²×C₄	—	53	3	no	—
151	C₂²×C₄	—	53	3	no	—
152	M₁₆	—	212	12	no	—
153	C₂×C₈	—	53	1	no	—
154	C₂×C₈	—	53	3	no	—
155	C₂×C₈	—	106	6	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	106	6	no	—
158	C₂×C₈	—	212	3	no	—
159	C₂×C₈	—	212	12	no	—
160	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
161	C₂×C₈	C₂²×#6157	1	3	yes	—
162	C₄⋊C₄	—	53	3	no	—
163	C₄⋊C₄	C₄⋊#6157	1	3	yes	—
164	C₂²⋊C₄	—	53	6	no	—
165	C₄²	—	53	3	no	—
166	C₄²	—	53	3	no	—
167	C₄²	—	53	1	no	—
168	C₄²	C₂²×#6157	1	3	yes	—
169	C₂³	—	53	3	no	—
170	Q₈	#13133285	1	4	yes	—
171	D₄	—	53	12	no	—
172	C₂×C₄	—	53	1	no	—
173	C₂×C₄	—	53	12	no	—
174	C₂×C₄	—	53	3	no	—
175	C₂×C₄	—	53	1	no	—
176	C₂×C₄	—	106	3	no	—
177	C₂×C₄	—	106	3	no	—
178	C₂×C₄	—	212	3	no	—
179	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
180	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
181	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
182	C₈	—	2	3	no	—
183	C₈	—	2	1	no	—
184	C₈	—	53	2	no	—
185	C₈	—	106	3	no	—
186	C₈	—	212	2	no	—
187	C₈	—	212	6	no	—
188	C₂²	—	53	6	no	—
189	C₂²	—	106	3	no	—
190	C₂²	—	106	3	no	—
191	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	53	2	no	—
194	C₄	—	106	3	no	—
195	C₄	—	106	1	no	—
196	C₄	C₂×#13133285	1	6	yes	—
197	C₄	D₅₃⋊#131	1	1	yes	characteristic
198	C₄	#16811703	1	1	yes	characteristic
199	C₂	—	106	6	no	—
200	C₂	#20291363	1	1	yes	characteristic
201	C₂	C₂²×#13133285	1	1	yes	characteristic
202	C₂	#20291377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291361

C₂

yes

maximal

#20290283

C₂

yes

maximal, complemented

#20290271

C₂

yes

maximal, complemented

D₅₃⋊#334

C₂

yes

maximal, complemented

C₈×#11719949

C₂

yes

maximal, characteristic

C₂×#13133305

—

#16811630

—

#16811627

C₂²

yes

—

D₅₃⋊(C₄⋊C₈)

C₂²

yes

—

#16811485

C₂²

yes

characteristic

#16811485

C₂²

yes

—

#16811485

C₂²

yes

—

#16811484

C₂²

yes

—

C₄×#11719949

C₂²

yes

characteristic

Q₈×#88322

C₄

yes

characteristic

C₈×D₂₁₂

C₄

yes

—

#16810688

C₄

yes

—

#16810675

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

—

C₈×Q₈×C₅₃

C₄

yes

characteristic

#13133305

—

C₂×#11719936

—

C₂×D₅₃⋊C₈

—

#13133284

—

C₄⋊#88324

—

C₄⋊#88324

C₂³

yes

—

C₄⋊#88324

C₂³

yes

—

C₂×#11719949

D₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

Q₈×Dic₅₃

Q₈

yes

characteristic

C₄×D₂₁₂

C₂×C₄

yes

—

#13133179

Q₈

yes

—

#13133175

D₄

yes

—

C₄²×D₅₃

C₂³

yes

—

C₄⋊#88322

C₂×C₄

yes

—

#13133129

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₈×Dic₅₃

C₂³

yes

characteristic

C₈×Dic₅₃

C₂³

yes

—

C₄×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₄⋊C₄

yes

characteristic

#11719949

SD₁₆

yes

—

C₂×D₂₁₂

C₄⋊C₄

yes

—

C₂×C₄×D₅₃

C₂²×C₄

yes

—

C₂×C₄×D₅₃

C₂×D₄

yes

—

C₂×C₄×D₅₃

C₂×Q₈

yes

—

C₂×#88324

—

C₂×#88324

—

#11719936

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄⋊C₄×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

#11719921

C₄⋊C₄

yes

—

C₄⋊Dic₅₃

C₄⋊C₄

yes

—

C₄×Dic₅₃

C₂×D₄

yes

—

C₄×Dic₅₃

C₂×Q₈

yes

—

C₄×Dic₅₃

C₂⁴

yes

characteristic

C₂×#88322

C₂²×C₄

yes

—

C₂×#88322

C₂²×C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₂²×D₅₃

C₂×C₄⋊C₄

yes

—

Q₈×C₅₃

#106

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂³×C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂²×D₄

yes

characteristic

C₂×Dic₅₃

C₂²×Q₈

yes

characteristic

D₂₁₂

#106

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×SD₁₆

yes

—

#88324

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

—

yes

—

C₂²×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×#106

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#106

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²×SD₁₆

yes

—

C₈×#41

—

#2906

—

100

#1570

—

101

#1566

—

102

C₂×#334

—

103

C₂×C₅₃

#2906

yes

characteristic

104

C₂×C₅₃

C₂²×#106

yes

characteristic

105

C₂×C₅₃

#2916

yes

characteristic

106

D₅₃

—

107

C₂×#131

—

106

—

108

C₄×#41

—

109

C₈×Q₈

—

110

C₈×Q₈

#6157

yes

characteristic

111

C₈×D₄

—

112

C₈×D₄

—

113

#431

—

114

#424

—

106

—

115

#423

—

116

C₂×C₄⋊C₈

—

117

C₂×C₄×C₈

—

118

#334

—

119

#334

—

120

#334

—

121

#334

—

122

C₅₃

#17826

yes

characteristic, complemented, Sylow

123

C₂×#41

—

124

#131

—

212

—

125

C₂×M₁₆

—

106

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

106

—

129

C₄×Q₈

—

130

C₄×Q₈

C₂×#6157

yes

characteristic

131

C₄×D₄

—

132

C₄×D₄

—

133

#117

—

134

C₂×C₄²

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

106

—

139

C₄⋊C₈

—

106

—

140

C₄⋊C₈

C₂×#6157

yes

—

141

C₂²⋊C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

C₂×#6157

yes

—

146

#41

—

147

C₂×Q₈

C₄⋊#6157

yes

characteristic

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

M₁₆

—

212

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

106

—

156

C₂×C₈

—

106

—

157

C₂×C₈

—

106

—

158

C₂×C₈

—

212

—

159

C₂×C₈

—

212

—

160

C₂×C₈

C₂²×#6157

yes

characteristic

161

C₂×C₈

C₂²×#6157

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

C₄⋊#6157

yes

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₂²×#6157

yes

—

169

C₂³

—

170

Q₈

#13133285

yes

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

106

—

177

C₂×C₄

—

106

—

178

C₂×C₄

—

212

—

179

C₂×C₄

C₂×C₄⋊#6157

yes

—

180

C₂×C₄

C₂×C₄⋊#6157

yes

—

181

C₂×C₄

C₂³×#6157

yes

characteristic

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

106

—

186

C₈

—

212

—

187

C₈

—

212

—

188

C₂²

—

189

C₂²

—

106

—

190

C₂²

—

106

—

191

C₂²

C₂²×C₄⋊#6157

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

106

—

195

C₄

—

106

—

196

C₄

C₂×#13133285

yes

—

197

C₄

D₅₃⋊#131

yes

characteristic

198

C₄

#16811703

yes

characteristic

199

C₂

—

106

—

200

C₂

#20291363

yes

characteristic

201

C₂

C₂²×#13133285

yes

characteristic

202

C₂

#20291377

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₈×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719949	D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	#17826
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×#6157

Fitting

C₈×Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂²×C₄⋊C₄

prime radical

C₂×#11719949

D₄

cosocle

C₄×Dic₅₃

C₂⁴

C₈×Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
2. #20290283	C₂	semidirect
3. #20290271	C₂	semidirect
4. D₅₃⋊#334	C₂	semidirect
85. #88322	#131	semidirect
122. C₅₃	#17826	semidirect

H ≅ G/N

type

2. #20290283

C₂

semidirect

3. #20290271

C₂

semidirect

4. D₅₃⋊#334

C₂

semidirect

85. #88322

#131

semidirect

122. C₅₃

#17826

semidirect

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

subgroup	index
#20291361	2
#20290283	2
#20290271	2
D₅₃⋊#334	2
C₈×#11719949	2
#17826	53

subgroup

index

D₅₃⋊#334

Subgroups — #28316259

Conjugacy classes of subgroups

Complements

Maximal subgroups