Group #28306118 — subgroups

Subgroups — #28306118

Order 13568.

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

normal subgroups only

This group has 558 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	#20290896	C₂	1	4	yes	maximal
2	#20290651	C₂	1	1	yes	maximal, characteristic
3	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#494×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133286	—	2	2	no	—
6	C₄×D₅₃⋊C₈	C₂²	1	2	yes	—
7	#16811569	C₂²	1	2	yes	—
8	#16811496	—	2	8	no	—
9	C₈×#88324	—	2	2	no	—
10	C₂×Q₁₆×D₅₃	—	2	2	no	—
11	C₄⋊Q₈×D₅₃	C₂²	1	2	yes	—
12	#16811003	C₄	1	1	yes	characteristic, complemented
13	#16811002	C₄	1	2	yes	complemented
14	#16811001	—	2	2	no	—
15	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#494	C₄	1	1	yes	characteristic, complemented
17	C₂×D₅₃⋊C₈	D₄	1	4	yes	—
18	C₂×C₄⋊#6157	—	2	2	no	—
19	C₂×C₄⋊#6157	—	2	4	no	—
20	#13133286	—	4	4	no	—
21	C₄×#88324	—	2	2	no	—
22	C₄×#88324	—	2	2	no	—
23	C₂×Q₈×D₅₃	—	2	2	no	—
24	C₂×Dic₂₁₂	—	2	2	no	—
25	C₂×#11719925	—	2	4	no	—
26	Q₁₆×D₅₃	—	4	4	no	—
27	Q₁₆×D₅₃	—	4	4	no	—
28	C₂×C₈×D₅₃	D₄	1	2	yes	—
29	#13133217	C₂×C₄	1	2	yes	—
30	#13133216	—	2	4	no	—
31	#13133215	—	2	2	no	—
32	C₄⋊C₄×D₅₃	—	2	4	no	—
33	C₄²×D₅₃	C₂³	1	1	yes	characteristic
34	C₄×#88322	C₂×C₄	1	1	yes	characteristic
35	C₈×Dic₅₃	D₄	1	2	yes	—
36	C₂×Q₁₆×C₅₃	—	2	2	no	—
37	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×#6157	—	4	2	no	—
40	C₂×Q₈×C₅₃	—	2	2	no	—
41	Q₈×D₅₃	—	4	4	no	—
42	Q₈×D₅₃	—	4	4	no	—
43	C₂×C₄×D₅₃	—	4	2	no	—
44	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
46	C₂×Dic₁₀₆	—	2	2	no	—
47	C₂×Dic₁₀₆	—	4	2	no	—
48	C₂×#88324	—	2	4	no	—
49	C₄⋊#6157	—	4	4	no	—
50	C₄⋊#6157	—	4	8	no	—
51	D₅₃⋊C₈	—	2	4	no	—
52	D₅₃⋊C₈	Q₁₆	1	8	yes	complemented
53	Q₁₆×C₅₃	—	4	4	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	#11719925	—	4	8	no	—
58	C₄⋊Dic₅₃	—	2	4	no	—
59	#11719919	—	4	4	no	—
60	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄×Dic₅₃	#41	1	2	yes	—
62	C₂×#88322	C₂²⋊C₄	1	2	yes	—
63	Dic₂₁₂	—	4	4	no	—
64	C₈×D₅₃	D₈	1	4	yes	—
65	C₈×D₅₃	SD₁₆	1	4	yes	—
66	C₂²×D₅₃	#127	1	1	yes	characteristic
67	C₂×#6157	—	8	2	no	—
68	C₂×#6157	—	8	4	no	—
69	Q₈×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
73	C₂×Dic₅₃	—	4	2	no	—
74	C₂×Dic₅₃	#121	1	2	yes	—
75	C₂×Dic₅₃	#124	1	1	yes	characteristic
76	C₄×D₅₃	—	8	2	no	—
77	C₄×D₅₃	—	8	2	no	—
78	C₄×D₅₃	C₂×D₈	1	2	yes	—
79	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
80	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
81	C₄×D₅₃	#135	1	4	yes	—
82	Dic₁₀₆	—	4	4	no	—
83	Dic₁₀₆	—	8	4	no	—
84	#88324	—	2	8	no	—
85	C₈×C₅₃	#102	1	4	yes	—
86	#88322	#102	1	4	yes	—
87	#9825	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	#390	1	1	yes	characteristic
89	C₂×D₅₃	#492	1	1	yes	characteristic
90	C₂×D₅₃	#493	1	1	yes	characteristic
91	C₂×D₅₃	#494	1	2	yes	—
92	C₂×D₅₃	#495	1	2	yes	—
93	#6157	—	16	2	no	—
94	C₄×C₅₃	—	8	2	no	—
95	C₄×C₅₃	C₂×#102	1	2	yes	—
96	C₄×C₅₃	C₄×Q₁₆	1	4	yes	—
97	Dic₅₃	—	8	2	no	—
98	Dic₅₃	#467	1	4	yes	—
99	Dic₅₃	#486	1	2	yes	—
100	C₂×#494	—	53	1	no	—
101	#1990	—	53	4	no	—
102	#1845	—	53	1	no	—
103	C₂×C₈²	—	53	1	no	—
104	C₂×C₅₃	#1957	1	1	yes	characteristic
105	C₂×C₅₃	#1969	1	2	yes	—
106	D₅₃	#1707	1	2	yes	—
107	D₅₃	#1709	1	1	yes	characteristic
108	D₅₃	#1713	1	1	yes	characteristic
109	C₂²×Q₁₆	—	106	2	no	—
110	C₂×C₄⋊Q₈	—	53	2	no	—
111	#494	—	53	2	no	—
112	#494	—	53	1	no	—
113	#494	—	106	2	no	—
114	#494	#6157	1	1	yes	characteristic, complemented
115	C₂×#107	—	106	2	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	2	no	—
118	#384	—	53	2	no	—
119	#340	—	106	8	no	—
120	C₈²	—	106	2	no	—
121	C₅₃	#9825	1	1	yes	characteristic, complemented, Sylow
122	C₂²×Q₈	—	106	2	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	—	106	4	no	—
125	C₂×Q₁₆	—	106	2	no	—
126	C₂×Q₁₆	—	212	4	no	—
127	C₂×Q₁₆	—	212	4	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	—	53	4	no	—
130	C₄⋊Q₈	—	53	2	no	—
131	C₄⋊Q₈	—	106	4	no	—
132	C₄⋊Q₈	—	106	2	no	—
133	C₄⋊Q₈	C₂×#6157	1	2	yes	—
134	C₂×C₄⋊C₄	—	106	4	no	—
135	C₂×C₄⋊C₄	—	106	4	no	—
136	C₂×C₄⋊C₄	—	106	2	no	—
137	C₂×C₄²	—	53	1	no	—
138	#107	—	212	4	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	53	1	no	—
141	C₄×C₈	—	106	2	no	—
142	C₄×C₈	—	106	2	no	—
143	C₄×C₈	C₂×#6157	1	1	yes	characteristic
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	—	106	2	no	—
146	C₂×Q₈	—	212	2	no	—
147	C₂×Q₈	—	212	4	no	—
148	C₂×Q₈	—	212	4	no	—
149	C₂²×C₄	—	53	1	no	—
150	C₂²×C₄	—	53	2	no	—
151	C₂²×C₄	—	212	2	no	—
152	C₂²×C₄	—	212	2	no	—
153	Q₁₆	—	4	4	no	—
154	Q₁₆	—	212	8	no	—
155	Q₁₆	—	212	4	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	53	4	no	—
158	C₂×C₈	—	53	4	no	—
159	C₂×C₈	—	53	8	no	—
160	C₂×C₈	—	106	4	no	—
161	C₂×C₈	—	106	4	no	—
162	C₂×C₈	C₂²⋊#6157	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	106	4	no	—
165	C₄⋊C₄	—	212	4	no	—
166	C₄⋊C₄	—	212	4	no	—
167	C₄⋊C₄	—	212	8	no	—
168	C₄²	—	53	1	no	—
169	C₄²	—	53	2	no	—
170	C₄²	C₂²×#6157	1	1	yes	characteristic
171	C₂³	—	53	1	no	—
172	Q₈	—	4	4	no	—
173	Q₈	—	424	4	no	—
174	Q₈	—	212	4	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	53	2	no	—
177	C₂×C₄	—	53	1	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	53	2	no	—
180	C₂×C₄	—	53	4	no	—
181	C₂×C₄	—	53	4	no	—
182	C₂×C₄	—	424	2	no	—
183	C₂×C₄	—	424	2	no	—
184	C₂×C₄	—	424	2	no	—
185	C₂×C₄	—	424	4	no	—
186	C₂×C₄	—	212	2	no	—
187	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
188	C₂×C₄	D₄×#6157	1	2	yes	—
189	C₈	—	53	4	no	—
190	C₈	—	106	8	no	—
191	C₈	#13133279	1	4	yes	—
192	C₂²	—	53	1	no	—
193	C₂²	—	53	1	no	—
194	C₂²	—	53	2	no	—
195	C₂²	—	53	2	no	—
196	C₂²	#16811575	1	1	yes	characteristic
197	C₄	—	8	2	no	—
198	C₄	—	53	2	no	—
199	C₄	—	53	4	no	—
200	C₄	—	848	2	no	—
201	C₄	—	424	2	no	—
202	C₄	C₂×#13133279	1	2	yes	—
203	C₄	Q₁₆×#6157	1	4	yes	—
204	C₂	—	53	1	no	—
205	C₂	—	53	2	no	—
206	C₂	—	53	1	no	—
207	C₂	#20290843	1	1	yes	characteristic
208	C₂	#20290861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290896

C₂

yes

maximal

#20290651

C₂

yes

maximal, characteristic

C₈×D₅₃⋊C₈

C₂

yes

maximal, characteristic

#494×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133286

—

C₄×D₅₃⋊C₈

C₂²

yes

—

#16811569

C₂²

yes

—

#16811496

—

C₈×#88324

—

C₂×Q₁₆×D₅₃

—

C₄⋊Q₈×D₅₃

C₂²

yes

—

#16811003

C₄

yes

characteristic, complemented

#16811002

C₄

yes

complemented

#16811001

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₅₃×#494

C₄

yes

characteristic, complemented

C₂×D₅₃⋊C₈

D₄

yes

—

C₂×C₄⋊#6157

—

C₂×C₄⋊#6157

—

#13133286

—

C₄×#88324

—

C₄×#88324

—

C₂×Q₈×D₅₃

—

C₂×Dic₂₁₂

—

C₂×#11719925

—

Q₁₆×D₅₃

—

Q₁₆×D₅₃

—

C₂×C₈×D₅₃

D₄

yes

—

#13133217

C₂×C₄

yes

—

#13133216

—

#13133215

—

C₄⋊C₄×D₅₃

—

C₄²×D₅₃

C₂³

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

—

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×Q₈×C₅₃

—

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×#88324

—

C₄⋊#6157

—

C₄⋊#6157

—

D₅₃⋊C₈

—

D₅₃⋊C₈

Q₁₆

yes

complemented

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

#41

yes

—

C₂×#88322

C₂²⋊C₄

yes

—

Dic₂₁₂

—

C₈×D₅₃

D₈

yes

—

C₈×D₅₃

SD₁₆

yes

—

C₂²×D₅₃

yes

—

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#121

yes

—

C₂×Dic₅₃

#124

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₈

yes

—

C₄×D₅₃

C₂×SD₁₆

yes

—

C₄×D₅₃

C₂×Q₁₆

yes

—

C₄×D₅₃

#135

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

C₈×C₅₃

#102

yes

—

#88322

#102

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

yes

C₂×D₅₃

yes

C₂×D₅₃

yes

—

C₂×D₅₃

#495

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#102

yes

—

C₄×C₅₃

C₄×Q₁₆

yes

—

Dic₅₃

—

Dic₅₃

#467

yes

—

Dic₅₃

#486

yes

—

100

C₂×#494

—

101

#1990

—

102

#1845

—

103

C₂×C₈²

—

104

C₂×C₅₃

#1957

yes

characteristic

105

C₂×C₅₃

#1969

yes

—

106

D₅₃

#1707

yes

—

107

D₅₃

#1709

yes

characteristic

108

D₅₃

#1713

yes

characteristic

109

C₂²×Q₁₆

—

106

—

110

C₂×C₄⋊Q₈

—

111

#494

—

112

#494

—

113

#494

—

106

—

114

#494

#6157

yes

characteristic, complemented

115

C₂×#107

—

106

—

116

C₂×C₄×C₈

—

117

C₂×C₄×C₈

—

118

#384

—

119

#340

—

106

—

120

C₈²

—

106

—

121

C₅₃

#9825

yes

characteristic, complemented, Sylow

122

C₂²×Q₈

—

106

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

106

—

125

C₂×Q₁₆

—

106

—

126

C₂×Q₁₆

—

212

—

127

C₂×Q₁₆

—

212

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

106

—

132

C₄⋊Q₈

—

106

—

133

C₄⋊Q₈

C₂×#6157

yes

—

134

C₂×C₄⋊C₄

—

106

—

135

C₂×C₄⋊C₄

—

106

—

136

C₂×C₄⋊C₄

—

106

—

137

C₂×C₄²

—

138

#107

—

212

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

106

—

142

C₄×C₈

—

106

—

143

C₄×C₈

C₂×#6157

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

—

106

—

146

C₂×Q₈

—

212

—

147

C₂×Q₈

—

212

—

148

C₂×Q₈

—

212

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

212

—

152

C₂²×C₄

—

212

—

153

Q₁₆

—

154

Q₁₆

—

212

—

155

Q₁₆

—

212

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

106

—

161

C₂×C₈

—

106

—

162

C₂×C₈

C₂²⋊#6157

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

106

—

165

C₄⋊C₄

—

212

—

166

C₄⋊C₄

—

212

—

167

C₄⋊C₄

—

212

—

168

C₄²

—

169

C₄²

—

170

C₄²

C₂²×#6157

yes

characteristic

171

C₂³

—

172

Q₈

—

173

Q₈

—

424

—

174

Q₈

—

212

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

424

—

183

C₂×C₄

—

424

—

184

C₂×C₄

—

424

—

185

C₂×C₄

—

424

—

186

C₂×C₄

—

212

—

187

C₂×C₄

C₂×C₂²⋊#6157

yes

characteristic

188

C₂×C₄

D₄×#6157

yes

—

189

C₈

—

190

C₈

—

106

—

191

C₈

#13133279

yes

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

#16811575

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

848

—

201

C₄

—

424

—

202

C₄

C₂×#13133279

yes

—

203

C₄

Q₁₆×#6157

yes

—

204

C₂

—

205

C₂

—

206

C₂

—

207

C₂

#20290843

yes

characteristic

208

C₂

#20290861

yes

—

	subgroup	G/H
center	C₂²	#16811575
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#494	C₄
socle	C₂²×C₅₃	#390
prime radical	C₂²×D₅₃	#127
cosocle	C₄²×D₅₃	C₂³
hypercenter	#494	#6157
hypocenter	C₅₃	#9825
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#16811575

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#494

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
12. #16811003	C₄	semidirect
13. #16811002	C₄	semidirect
16. C₅₃×#494	C₄	semidirect
52. D₅₃⋊C₈	Q₁₆	semidirect
114. #494	#6157	semidirect
121. C₅₃	#9825	semidirect

H ≅ G/N

type

12. #16811003

C₄

semidirect

13. #16811002

C₄

semidirect

16. C₅₃×#494

C₄

semidirect

52. D₅₃⋊C₈

Q₁₆

semidirect

114. #494

#6157

semidirect

121. C₅₃

#9825

semidirect

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

subgroup	index
#20290896	2
#20290651	2
C₈×D₅₃⋊C₈	2
#494×D₅₃	2
#9825	53

subgroup

index

#20290896

#20290651

C₈×D₅₃⋊C₈

#494×D₅₃

#9825

Subgroups — #28306118

Conjugacy classes of subgroups

Complements

Maximal subgroups