Group #28302071 — subgroups

Subgroups — #28302071

Order 13568.

390 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 148 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×#13133271	C₂	1	1	yes	maximal, characteristic
2	#20290265	C₂	1	1	yes	maximal, characteristic
3	#20289921	C₂	1	2	yes	maximal
4	#20289542	C₂	1	2	yes	maximal
5	#20273460	C₂	1	1	yes	maximal, characteristic
6	#16811642	C₂²	1	2	yes	—
7	C₂×#13133271	C₄	1	2	yes	—
8	C₂×#13133271	C₄	1	2	yes	—
9	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄²×#6157	C₂²	1	1	yes	characteristic
11	#16811486	C₄	1	2	yes	complemented
12	#16811485	C₄	1	2	yes	complemented
13	#88324⋊C₈	—	2	2	no	—
14	C₈×#88324	—	2	2	no	—
15	#16810675	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810375	C₄	1	2	yes	complemented
18	#16810374	C₄	1	1	yes	characteristic, complemented
19	C₅₃×#334	C₄	1	1	yes	characteristic, complemented
20	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×#6157	—	2	1	no	—
22	C₂×C₄×#6157	—	2	1	no	—
23	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
24	C₄⋊#88324	C₂×C₄	1	4	yes	—
25	#13133274	—	2	2	no	—
26	#13133271	—	2	4	no	—
27	#13133271	C₈	1	8	yes	complemented
28	C₄×#88324	—	2	2	no	—
29	C₄×#88324	D₄	1	2	yes	—
30	C₄×#88324	Q₈	1	2	yes	—
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	C₂²×#6157	—	2	2	no	—
43	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
45	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
46	C₂×#88324	—	2	4	no	—
47	C₂×#88324	C₄⋊C₄	1	4	yes	—
48	C₄×#6157	—	2	2	no	—
49	C₄×#6157	—	4	2	no	—
50	C₄×#6157	—	4	2	no	—
51	C₄×#6157	C₂×C₈	1	4	yes	—
52	D₅₃⋊C₈	C₂×C₈	1	4	yes	—
53	D₅₃⋊C₈	M₁₆	1	4	yes	—
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	C₂×Q₈	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	#117	1	1	yes	characteristic
67	C₂×#6157	—	2	4	no	—
68	C₂×#6157	—	4	2	no	—
69	C₂×#6157	—	4	2	no	—
70	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
75	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
76	C₄×D₅₃	—	2	2	no	—
77	C₄×D₅₃	—	2	2	no	—
78	C₄×D₅₃	C₂²×C₈	1	2	yes	—
79	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
80	#88324	—	4	4	no	—
81	#88324	C₄⋊C₈	1	8	yes	—
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	#8268	—	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	#431	1	2	yes	—
93	#6157	—	4	4	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₅₃	#444	1	2	yes	—
98	Dic₅₃	—	2	1	no	—
99	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
100	Dic₅₃	#444	1	2	yes	—
101	C₄×#97	—	53	1	no	—
102	#1564	—	53	1	no	—
103	#1564	—	53	1	no	—
104	#1450	—	53	2	no	—
105	#1276	—	53	2	no	—
106	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
107	C₂×C₅₃	#1773	1	1	yes	characteristic
108	C₂×C₅₃	#1773	1	1	yes	characteristic
109	D₅₃	—	2	2	no	—
110	#431	—	53	2	no	—
111	#431	—	53	2	no	—
112	C₂×#97	—	53	2	no	—
113	C₂×#97	—	53	2	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₄³	—	53	1	no	—
117	#335	—	53	2	no	—
118	#335	—	53	2	no	—
119	#334	—	53	2	no	—
120	#334	—	53	1	no	—
121	#334	#6157	1	1	yes	characteristic, complemented
122	#322	—	106	2	no	—
123	C₈²	—	106	2	no	—
124	C₅₃	#8268	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₈	—	53	2	no	—
126	C₂²×C₈	—	53	1	no	—
127	C₂²×C₈	—	53	1	no	—
128	C₂×C₄²	—	53	1	no	—
129	C₂×C₄²	—	53	2	no	—
130	C₂×C₄²	—	106	1	no	—
131	C₂×C₄²	—	106	1	no	—
132	C₄⋊C₈	—	53	4	no	—
133	C₄⋊C₈	—	53	2	no	—
134	C₄⋊C₈	C₂×#6157	1	2	yes	—
135	C₂²⋊C₈	—	106	2	no	—
136	C₂²⋊C₈	—	106	2	no	—
137	#97	—	53	8	no	—
138	#97	—	106	4	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	53	2	no	—
141	C₄×C₈	—	53	1	no	—
142	C₄×C₈	—	53	1	no	—
143	C₄×C₈	—	53	1	no	—
144	C₄×C₈	—	106	2	no	—
145	C₄×C₈	—	106	2	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	C₂²×C₄	—	106	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	53	4	no	—
153	C₂×C₈	—	53	4	no	—
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	53	1	no	—
156	C₂×C₈	—	53	4	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	C₄×#6157	1	1	yes	characteristic
162	C₂×C₈	C₄×#6157	1	1	yes	characteristic
163	C₄²	—	53	1	no	—
164	C₄²	—	53	1	no	—
165	C₄²	—	53	1	no	—
166	C₄²	—	53	4	no	—
167	C₄²	—	106	2	no	—
168	C₄²	—	212	2	no	—
169	C₄²	—	212	2	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	2	no	—
179	C₂×C₄	—	106	4	no	—
180	C₂×C₄	—	212	2	no	—
181	C₂×C₄	—	212	2	no	—
182	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
183	C₂×C₄	D₄×#6157	1	1	yes	characteristic
184	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	4	2	no	—
187	C₈	—	53	8	no	—
188	C₈	—	53	2	no	—
189	C₈	—	106	1	no	—
190	C₈	—	212	4	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	—	212	4	no	—
202	C₄	C₂×C₈×#6157	1	1	yes	characteristic
203	C₄	M₁₆×#6157	1	1	yes	characteristic
204	C₄	#16811668	1	2	yes	—
205	C₂	—	106	2	no	—
206	C₂	C₄⋊C₈×#6157	1	1	yes	characteristic
207	C₂	#20290523	1	1	yes	characteristic
208	C₂	#20290524	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133271

C₂

yes

maximal, characteristic

#20290265

C₂

yes

maximal, characteristic

#20289921

C₂

yes

maximal

#20289542

C₂

yes

maximal

#20273460

C₂

yes

maximal, characteristic

#16811642

C₂²

yes

—

C₂×#13133271

C₄

yes

—

C₂×#13133271

C₄

yes

—

C₄×D₅₃⋊C₈

C₂²

yes

characteristic

C₄²×#6157

C₂²

yes

characteristic

#16811486

C₄

yes

complemented

#16811485

C₄

yes

—

—

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810375

C₄

yes

complemented

#16810374

C₄

yes

characteristic, complemented

C₅₃×#334

C₄

yes

characteristic, complemented

C₂×D₅₃⋊C₈

C₂×C₄

yes

—

C₂×C₄×#6157

—

C₂×C₄×#6157

—

C₂×C₄×#6157

C₂×C₄

yes

—

C₄⋊#88324

C₂×C₄

yes

—

#13133274

—

#13133271

—

#13133271

C₈

yes

complemented

C₄×#88324

—

C₄×#88324

D₄

yes

—

C₄×#88324

Q₈

yes

—

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

C₂²×#6157

—

C₂×C₄×D₅₃

C₂²×C₄

yes

C₂×C₄×D₅₃

yes

C₂×C₄×D₅₃

yes

—

C₄⋊C₄

yes

—

—

—

—

C₂×C₈

yes

—

D₅₃⋊C₈

C₂×C₈

yes

—

D₅₃⋊C₈

M₁₆

yes

—

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

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

characteristic

C₂×#88322

—

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

C₂×C₈

yes

—

C₂²×D₅₃

yes

—

—

—

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

C₄×D₄

yes

characteristic

C₂×Dic₅₃

C₄×Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂²×C₈

yes

—

C₄×D₅₃

C₂×M₁₆

yes

—

#88324

—

#88324

C₄⋊C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

—

#88322

—

#88322

—

#88322

C₄×C₈

yes

—

#8268

—

maximal, Sylow

C₂²×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#431

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

C₄×M₁₆

yes

characteristic

C₄×C₅₃

#444

yes

—

Dic₅₃

—

Dic₅₃

C₂×C₄⋊C₈

yes

—

100

Dic₅₃

#444

yes

—

101

C₄×#97

—

102

#1564

—

103

#1564

—

104

#1450

—

105

#1276

—

106

C₂×C₅₃

C₄×C₄⋊C₈

yes

characteristic

107

C₂×C₅₃

#1773

yes

characteristic

108

C₂×C₅₃

#1773

yes

characteristic

109

D₅₃

—

110

#431

—

111

#431

—

112

C₂×#97

—

113

C₂×#97

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

—

116

C₄³

—

117

#335

—

118

#335

—

119

#334

—

120

#334

—

121

#334

#6157

yes

characteristic, complemented

122

#322

—

106

—

123

C₈²

—

106

—

124

C₅₃

#8268

yes

characteristic, complemented, Sylow

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂×C₄²

—

129

C₂×C₄²

—

130

C₂×C₄²

—

106

—

131

C₂×C₄²

—

106

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

C₂×#6157

yes

—

135

C₂²⋊C₈

—

106

—

136

C₂²⋊C₈

—

106

—

137

#97

—

138

#97

—

106

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

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

C₂²×C₄

—

106

—

151

C₂×C₈

—

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

C₄×#6157

yes

characteristic

162

C₂×C₈

C₄×#6157

yes

characteristic

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

106

—

168

C₄²

—

212

—

169

C₄²

—

212

—

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

—

212

—

181

C₂×C₄

—

212

—

182

C₂×C₄

C₂×C₄×#6157

yes

characteristic

183

C₂×C₄

D₄×#6157

yes

characteristic

184

C₂×C₄

Q₈×#6157

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

—

106

—

190

C₈

—

212

—

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

—

212

—

202

C₄

C₂×C₈×#6157

yes

characteristic

203

C₄

M₁₆×#6157

yes

characteristic

204

C₄

#16811668

yes

—

205

C₂

—

106

—

206

C₂

C₄⋊C₈×#6157

yes

characteristic

207

C₂

#20290523

yes

characteristic

208

C₂

#20290524

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#334	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂²×D₅₃	#117
cosocle	C₄²×D₅₃	C₂³
hypercenter	#334	#6157
hypocenter	C₅₃	#8268
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#334

C₄

socle

C₂²×C₅₃

C₄×C₄⋊C₄

prime radical

C₂²×D₅₃

#117

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
11. #16811486	C₄	semidirect
12. #16811485	C₄	semidirect
17. #16810375	C₄	semidirect
18. #16810374	C₄	semidirect
19. C₅₃×#334	C₄	semidirect
27. #13133271	C₈	semidirect
121. #334	#6157	semidirect
124. C₅₃	#8268	semidirect

H ≅ G/N

type

11. #16811486

C₄

semidirect

12. #16811485

C₄

semidirect

17. #16810375

C₄

semidirect

18. #16810374

C₄

semidirect

19. C₅₃×#334

C₄

semidirect

27. #13133271

C₈

semidirect

121. #334

#6157

semidirect

124. C₅₃

#8268

semidirect

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

subgroup	index
C₄×#13133271	2
#20290265	2
#20289921	2
#20289542	2
#20273460	2
#8268	53

subgroup

index

Subgroups — #28302071

Conjugacy classes of subgroups

Complements

Maximal subgroups