Group #28298706 — subgroups

Subgroups — #28298706

Order 13568.

246 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 84 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	D₅₃⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#20290652	C₂	1	1	yes	maximal, characteristic
3	#334×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊C₁₆	—	2	1	no	—
5	C₂×#13133286	C₄	1	2	yes	—
6	C₂×#13133285	—	2	1	no	—
7	#16811567	—	2	1	no	—
8	#16811544	C₄	1	2	yes	complemented
9	C₄⋊#11719910	—	2	1	no	—
10	C₄⋊C₈×D₅₃	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810374	C₄	1	1	yes	characteristic, complemented
13	#16810373	—	2	1	no	—
14	C₅₃×#334	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#6157	—	2	2	no	—
16	D₅₃⋊C₁₆	—	4	1	no	—
17	D₅₃⋊C₁₆	—	4	1	no	—
18	C₂×#11719910	—	2	2	no	—
19	#13133286	—	4	2	no	—
20	#13133285	—	4	1	no	—
21	#13133285	—	4	1	no	—
22	(C₂×#6157)⋊C₄	—	4	1	no	—
23	(C₂×#6157)⋊C₄	—	4	1	no	—
24	C₂×C₈×D₅₃	—	4	1	no	—
25	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
27	C₄²×D₅₃	D₄	1	1	yes	characteristic
28	C₄⋊#88322	—	2	1	no	—
29	#13133146	—	4	1	no	—
30	C₄×#88322	D₄	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#6157	—	4	2	no	—
36	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
38	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
39	C₄⋊#6157	—	4	2	no	—
40	C₄⋊#6157	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	1	no	—
42	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
47	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
48	C₂×#88322	—	4	1	no	—
49	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
51	C₈×D₅₃	—	2	1	no	—
52	C₈×D₅₃	—	2	1	no	—
53	C₈×D₅₃	—	8	2	no	—
54	C₈×D₅₃	D₈	1	1	yes	characteristic
55	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
56	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
57	C₈×D₅₃	Q₁₆	1	1	yes	characteristic
58	#11719910	—	4	2	no	—
59	C₂²×D₅₃	#102	1	1	yes	characteristic
60	C₂×#6157	—	8	2	no	—
61	C₂×#6157	—	8	2	no	—
62	C₂×#6157	—	8	2	no	—
63	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
66	C₂×Dic₅₃	#103	1	1	yes	characteristic
67	C₂×Dic₅₃	#106	1	1	yes	characteristic
68	C₂×Dic₅₃	#107	1	1	yes	characteristic
69	C₄×D₅₃	—	2	2	no	—
70	C₄×D₅₃	—	2	2	no	—
71	C₄×D₅₃	#100	1	1	yes	characteristic
72	C₄×D₅₃	#101	1	1	yes	characteristic
73	C₄×D₅₃	#102	1	1	yes	characteristic
74	C₄×D₅₃	#103	1	1	yes	characteristic
75	C₈×C₅₃	—	2	1	no	—
76	C₈×C₅₃	—	8	1	no	—
77	C₈×C₅₃	#106	1	1	yes	characteristic
78	C₈×C₅₃	#107	1	1	yes	characteristic
79	#88322	—	2	1	no	—
80	#88322	—	8	1	no	—
81	#88322	#102	1	1	yes	characteristic
82	#88322	#103	1	1	yes	characteristic
83	#7275	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	#340	1	1	yes	characteristic
85	C₂×D₅₃	#331	1	1	yes	characteristic
86	C₂×D₅₃	#332	1	1	yes	characteristic
87	C₂×D₅₃	#359	1	2	yes	—
88	C₂×D₅₃	#360	1	2	yes	—
89	#6157	—	16	2	no	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	#340	1	1	yes	characteristic
93	C₄×C₅₃	#343	1	1	yes	characteristic
94	Dic₅₃	—	2	1	no	—
95	Dic₅₃	—	2	1	no	—
96	Dic₅₃	#332	1	1	yes	characteristic
97	Dic₅₃	#333	1	1	yes	characteristic
98	C₂×C₄⋊C₁₆	—	53	1	no	—
99	#1846	—	53	1	no	—
100	C₂×#334	—	53	1	no	—
101	C₂×C₅₃	#1296	1	1	yes	characteristic
102	C₂×C₅₃	#1382	1	1	yes	characteristic
103	C₂×C₅₃	#1386	1	1	yes	characteristic
104	D₅₃	#1360	1	2	yes	—
105	D₅₃	#1363	1	2	yes	—
106	C₂²×C₁₆	—	106	1	no	—
107	C₂×#107	—	53	2	no	—
108	C₂×#106	—	106	1	no	—
109	C₂×C₄⋊C₈	—	106	1	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	#382	—	106	1	no	—
112	C₄⋊C₁₆	—	53	2	no	—
113	C₄⋊C₁₆	—	106	1	no	—
114	#334	—	53	1	no	—
115	#334	—	106	1	no	—
116	#334	#6157	1	1	yes	characteristic, complemented
117	C₅₃	#7275	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₈	—	53	1	no	—
119	C₂²×C₈	—	53	1	no	—
120	C₂²×C₈	—	212	1	no	—
121	C₂×C₄⋊C₄	—	106	2	no	—
122	C₂×C₄²	—	53	1	no	—
123	C₂×C₁₆	—	106	2	no	—
124	C₂×C₁₆	—	212	1	no	—
125	C₂×C₁₆	—	212	1	no	—
126	#107	—	212	2	no	—
127	#106	—	212	1	no	—
128	#106	—	212	1	no	—
129	C₄⋊C₈	—	2	1	no	—
130	C₄⋊C₈	—	106	1	no	—
131	C₄⋊C₈	—	212	1	no	—
132	C₄×C₈	—	53	1	no	—
133	C₄×C₈	—	53	1	no	—
134	C₄×C₈	—	53	1	no	—
135	C₄×C₈	C₂×#6157	1	1	yes	characteristic
136	#95	—	212	1	no	—
137	#95	—	212	1	no	—
138	C₂²×C₄	—	53	1	no	—
139	C₂²×C₄	—	53	1	no	—
140	C₂²×C₄	—	53	1	no	—
141	C₂²×C₄	—	212	2	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	53	1	no	—
144	C₂×C₈	—	53	1	no	—
145	C₂×C₈	—	53	1	no	—
146	C₂×C₈	—	53	1	no	—
147	C₂×C₈	—	53	1	no	—
148	C₂×C₈	—	53	1	no	—
149	C₂×C₈	—	106	1	no	—
150	C₂×C₈	—	106	1	no	—
151	C₂×C₈	—	424	2	no	—
152	C₂×C₈	—	212	1	no	—
153	C₂×C₈	C₄×#6157	1	1	yes	characteristic
154	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
155	C₄⋊C₄	—	212	2	no	—
156	C₄⋊C₄	—	212	2	no	—
157	C₄²	—	53	1	no	—
158	C₄²	—	53	1	no	—
159	C₄²	—	53	1	no	—
160	C₄²	C₂²⋊#6157	1	1	yes	characteristic
161	C₁₆	—	212	2	no	—
162	C₂³	—	53	1	no	—
163	C₂×C₄	—	53	1	no	—
164	C₂×C₄	—	53	1	no	—
165	C₂×C₄	—	53	1	no	—
166	C₂×C₄	—	53	1	no	—
167	C₂×C₄	—	53	1	no	—
168	C₂×C₄	—	53	1	no	—
169	C₂×C₄	—	53	1	no	—
170	C₂×C₄	—	106	2	no	—
171	C₂×C₄	—	106	2	no	—
172	C₂×C₄	—	424	2	no	—
173	C₂×C₄	—	424	2	no	—
174	C₂×C₄	—	424	2	no	—
175	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#13133279	1	1	yes	characteristic
177	C₂×C₄	#13133280	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	8	1	no	—
180	C₈	—	53	1	no	—
181	C₈	—	53	1	no	—
182	C₈	—	106	1	no	—
183	C₈	—	424	1	no	—
184	C₈	#13133285	1	1	yes	characteristic
185	C₈	#13133286	1	1	yes	characteristic
186	C₂²	—	53	1	no	—
187	C₂²	—	53	1	no	—
188	C₂²	—	53	2	no	—
189	C₂²	—	53	2	no	—
190	C₂²	#16811496	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	53	1	no	—
194	C₄	—	53	1	no	—
195	C₄	—	848	2	no	—
196	C₄	—	106	1	no	—
197	C₄	—	106	1	no	—
198	C₄	#16811497	1	1	yes	characteristic
199	C₄	#16811502	1	1	yes	characteristic
200	C₂	—	53	2	no	—
201	C₂	—	53	2	no	—
202	C₂	#20289594	1	1	yes	characteristic
203	C₂	#20289778	1	1	yes	characteristic
204	C₂	#20289788	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊(C₄⋊C₁₆)

C₂

yes

maximal, characteristic

#20290652

C₂

yes

maximal, characteristic

#334×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

—

C₂×#13133286

C₄

yes

—

C₂×#13133285

—

#16811567

—

#16811544

C₄

yes

complemented

C₄⋊#11719910

—

C₄⋊C₈×D₅₃

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810374

C₄

yes

characteristic, complemented

#16810373

—

C₅₃×#334

C₄

yes

characteristic, complemented

C₂×C₄⋊#6157

—

D₅₃⋊C₁₆

—

D₅₃⋊C₁₆

—

—

—

—

—

—

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₅₃

D₈

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

characteristic

C₄⋊#6157

—

C₄⋊#6157

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₄×Dic₅₃

Q₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

D₈

yes

characteristic

C₈×D₅₃

SD₁₆

yes

characteristic

C₈×D₅₃

SD₁₆

yes

characteristic

C₈×D₅₃

Q₁₆

yes

characteristic

#11719910

—

C₂²×D₅₃

yes

—

—

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

yes

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

yes

C₂×D₅₃

yes

C₂×D₅₃

yes

—

C₂×D₅₃

#360

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

C₂×C₄⋊C₁₆

—

#1846

—

100

C₂×#334

—

101

C₂×C₅₃

#1296

yes

characteristic

102

C₂×C₅₃

#1382

yes

characteristic

103

C₂×C₅₃

#1386

yes

characteristic

104

D₅₃

#1360

yes

—

105

D₅₃

#1363

yes

—

106

C₂²×C₁₆

—

106

—

107

C₂×#107

—

108

C₂×#106

—

106

—

109

C₂×C₄⋊C₈

—

106

—

110

C₂×C₄×C₈

—

111

#382

—

106

—

112

C₄⋊C₁₆

—

113

C₄⋊C₁₆

—

106

—

114

#334

—

115

#334

—

106

—

116

#334

#6157

yes

characteristic, complemented

117

C₅₃

#7275

yes

characteristic, complemented, Sylow

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

212

—

121

C₂×C₄⋊C₄

—

106

—

122

C₂×C₄²

—

123

C₂×C₁₆

—

106

—

124

C₂×C₁₆

—

212

—

125

C₂×C₁₆

—

212

—

126

#107

—

212

—

127

#106

—

212

—

128

#106

—

212

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

106

—

131

C₄⋊C₈

—

212

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

C₂×#6157

yes

characteristic

136

#95

—

212

—

137

#95

—

212

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

212

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

106

—

150

C₂×C₈

—

106

—

151

C₂×C₈

—

424

—

152

C₂×C₈

—

212

—

153

C₂×C₈

C₄×#6157

yes

characteristic

154

C₂×C₈

C₄⋊#6157

yes

characteristic

155

C₄⋊C₄

—

212

—

156

C₄⋊C₄

—

212

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂²⋊#6157

yes

characteristic

161

C₁₆

—

212

—

162

C₂³

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

106

—

171

C₂×C₄

—

106

—

172

C₂×C₄

—

424

—

173

C₂×C₄

—

424

—

174

C₂×C₄

—

424

—

175

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

176

C₂×C₄

#13133279

yes

characteristic

177

C₂×C₄

#13133280

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

106

—

183

C₈

—

424

—

184

C₈

#13133285

yes

characteristic

185

C₈

#13133286

yes

characteristic

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#16811496

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

848

—

196

C₄

—

106

—

197

C₄

—

106

—

198

C₄

#16811497

yes

characteristic

199

C₄

#16811502

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

#20289594

yes

characteristic

203

C₂

#20289778

yes

characteristic

204

C₂

#20289788

yes

characteristic

	subgroup	G/H
center	C₂²	#16811496
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#334	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#102
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#334	#6157
hypocenter	C₅₃	#7275
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#16811496

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#334

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
8. #16811544	C₄	semidirect
12. #16810374	C₄	semidirect
14. C₅₃×#334	C₄	semidirect
116. #334	#6157	semidirect
117. C₅₃	#7275	semidirect

H ≅ G/N

type

8. #16811544

C₄

semidirect

12. #16810374

C₄

semidirect

14. C₅₃×#334

C₄

semidirect

116. #334

#6157

semidirect

117. C₅₃

#7275

semidirect

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

subgroup	index
D₅₃⋊(C₄⋊C₁₆)	2
#20290652	2
#334×D₅₃	2
#7275	53

subgroup

index

D₅₃⋊(C₄⋊C₁₆)

#20290652

#334×D₅₃

#7275

Subgroups — #28298706

Conjugacy classes of subgroups

Complements

Maximal subgroups