Group #28298402 — subgroups

Subgroups — #28298402

Order 13568.

258 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 78 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⋊#364	C₂	1	1	yes	maximal, characteristic
2	#20290517	C₂	1	1	yes	maximal, characteristic
3	#322×D₅₃	C₂	1	1	yes	maximal, characteristic
4	D₅₃⋊M₃₂	—	2	1	no	—
5	D₅₃⋊#108	C₄	1	1	yes	characteristic
6	#16811633	C₄	1	1	yes	characteristic
7	#16811586	—	2	1	no	—
8	#16811546	—	2	1	no	—
9	#16811545	C₄	1	2	yes	complemented
10	#16811494	—	2	1	no	—
11	C₄×C₈×D₅₃	—	2	1	no	—
12	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#16810345	—	2	1	no	—
14	#16810344	C₄	1	1	yes	characteristic, complemented
15	C₅₃×#322	C₄	1	1	yes	characteristic, complemented
16	D₅₃⋊M₁₆	—	2	1	no	—
17	#13133296	—	2	1	no	—
18	#13133292	—	4	1	no	—
19	#13133291	—	2	2	no	—
20	D₅₃⋊C₁₆	—	4	1	no	—
21	#13133288	C₈	1	2	yes	complemented
22	#13133287	—	2	1	no	—
23	#13133286	C₈	1	2	yes	complemented
24	#13133285	—	2	1	no	—
25	#13133271	—	4	1	no	—
26	C₈×#6157	—	4	1	no	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
30	C₄²×D₅₃	D₄	1	1	yes	characteristic
31	C₄×#88322	—	2	1	no	—
32	C₄×#88322	D₄	1	1	yes	characteristic
33	C₈×Dic₅₃	—	2	2	no	—
34	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
36	C₄×C₈×C₅₃	—	2	1	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₅₃	—	2	1	no	—
39	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²⋊#6157	—	4	1	no	—
41	#11719940	—	4	1	no	—
42	C₄⋊#6157	—	2	2	no	—
43	C₄×#6157	—	4	1	no	—
44	#11719936	—	2	2	no	—
45	D₅₃⋊C₈	—	4	1	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₄×Dic₅₃	—	2	1	no	—
51	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
55	C₈×D₅₃	—	4	4	no	—
56	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
57	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
58	C₈×D₅₃	M₁₆	1	1	yes	characteristic
59	C₈×D₅₃	M₁₆	1	1	yes	characteristic
60	C₈×D₅₃	D₈	1	1	yes	characteristic
61	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
62	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
63	C₈×D₅₃	Q₁₆	1	1	yes	characteristic
64	#11719910	—	4	2	no	—
65	C₂²×D₅₃	#104	1	1	yes	characteristic
66	C₂×#6157	—	4	2	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
69	C₂×Dic₅₃	—	2	1	no	—
70	C₂×Dic₅₃	#104	1	1	yes	characteristic
71	C₄×D₅₃	—	2	4	no	—
72	C₄×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
73	C₄×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×D₅₃	#102	1	1	yes	characteristic
75	C₄×D₅₃	#103	1	1	yes	characteristic
76	#88324	—	4	2	no	—
77	C₈×C₅₃	—	4	2	no	—
78	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
79	C₈×C₅₃	#97	1	1	yes	characteristic
80	C₈×C₅₃	#106	1	1	yes	characteristic
81	C₈×C₅₃	#107	1	1	yes	characteristic
82	#88322	—	4	2	no	—
83	#88322	#102	1	1	yes	characteristic
84	#88322	#103	1	1	yes	characteristic
85	#88322	C₄⋊C₈	1	1	yes	characteristic
86	#88322	C₄⋊C₈	1	1	yes	characteristic
87	#7182	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	#339	1	1	yes	characteristic
89	C₂×D₅₃	—	2	2	no	—
90	C₂×D₅₃	D₄⋊C₈	1	1	yes	characteristic
91	C₂×D₅₃	Q₈⋊C₈	1	1	yes	characteristic
92	#6157	—	8	2	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	#336	1	1	yes	characteristic
95	C₄×C₅₃	#340	1	1	yes	characteristic
96	Dic₅₃	—	2	2	no	—
97	Dic₅₃	D₄⋊C₈	1	1	yes	characteristic
98	Dic₅₃	Q₈⋊C₈	1	1	yes	characteristic
99	C₂×#364	—	53	1	no	—
100	#1772	—	53	1	no	—
101	C₂×#322	—	53	1	no	—
102	C₂×C₅₃	—	2	1	no	—
103	C₂×C₅₃	#1279	1	1	yes	characteristic
104	D₅₃	—	2	1	no	—
105	D₅₃	#1336	1	2	yes	—
106	C₂×M₃₂	—	106	1	no	—
107	C₂×#108	—	53	1	no	—
108	#427	—	53	1	no	—
109	#405	—	106	1	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	106	1	no	—
112	#364	—	53	2	no	—
113	#364	—	106	1	no	—
114	#339	—	106	1	no	—
115	#322	—	53	1	no	—
116	#322	—	106	1	no	—
117	#322	#6157	1	1	yes	characteristic, complemented
118	C₅₃	#7182	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	106	1	no	—
120	C₂²×C₈	—	53	1	no	—
121	C₂²×C₈	—	53	1	no	—
122	C₂²×C₈	—	106	2	no	—
123	#117	—	106	1	no	—
124	C₂×C₄²	—	53	1	no	—
125	M₃₂	—	106	2	no	—
126	M₃₂	—	212	1	no	—
127	C₂×C₁₆	—	212	1	no	—
128	#108	—	53	2	no	—
129	#108	—	106	1	no	—
130	#107	—	53	2	no	—
131	#106	—	106	1	no	—
132	#97	—	212	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	53	1	no	—
135	C₄×C₈	—	53	1	no	—
136	C₄×C₈	—	53	1	no	—
137	C₄×C₈	—	106	2	no	—
138	C₄×C₈	—	106	1	no	—
139	C₄×C₈	—	212	1	no	—
140	C₄×C₈	C₂×#6157	1	1	yes	characteristic
141	C₂²×C₄	—	53	1	no	—
142	C₂²×C₄	—	106	1	no	—
143	M₁₆	—	106	2	no	—
144	M₁₆	—	212	1	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	53	1	no	—
147	C₂×C₈	—	53	1	no	—
148	C₂×C₈	—	53	1	no	—
149	C₂×C₈	—	53	1	no	—
150	C₂×C₈	—	53	1	no	—
151	C₂×C₈	—	53	1	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	1	no	—
154	C₂×C₈	—	53	1	no	—
155	C₂×C₈	—	53	1	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	212	4	no	—
158	C₂×C₈	—	212	1	no	—
159	C₂×C₈	C₄×#6157	1	1	yes	characteristic
160	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
161	C₄⋊C₄	—	106	2	no	—
162	C₂²⋊C₄	—	212	1	no	—
163	C₄²	—	53	1	no	—
164	C₄²	—	106	1	no	—
165	C₄²	—	212	1	no	—
166	C₄²	C₂²⋊#6157	1	1	yes	characteristic
167	C₁₆	—	212	2	no	—
168	C₂³	—	53	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	53	1	no	—
171	C₂×C₄	—	53	1	no	—
172	C₂×C₄	—	53	1	no	—
173	C₂×C₄	—	53	1	no	—
174	C₂×C₄	—	53	1	no	—
175	C₂×C₄	—	106	4	no	—
176	C₂×C₄	—	106	1	no	—
177	C₂×C₄	—	212	2	no	—
178	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	—	53	1	no	—
181	C₈	—	53	1	no	—
182	C₈	—	53	1	no	—
183	C₈	—	53	1	no	—
184	C₈	—	212	2	no	—
185	C₈	—	212	2	no	—
186	C₈	C₈×#6157	1	1	yes	characteristic
187	C₈	#13133271	1	1	yes	characteristic
188	C₈	#13133285	1	1	yes	characteristic
189	C₈	#13133286	1	1	yes	characteristic
190	C₂²	—	53	1	no	—
191	C₂²	—	53	1	no	—
192	C₂²	—	106	2	no	—
193	C₂²	#16811494	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	53	1	no	—
196	C₄	—	53	1	no	—
197	C₄	—	106	2	no	—
198	C₄	—	424	2	no	—
199	C₄	(C₂×#6157)⋊C₈	1	1	yes	characteristic
200	C₄	#16811497	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	53	2	no	—
203	C₂	—	106	1	no	—
204	C₂	#20289550	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊#364

C₂

yes

maximal, characteristic

#20290517

C₂

yes

maximal, characteristic

#322×D₅₃

C₂

yes

maximal, characteristic

D₅₃⋊M₃₂

—

D₅₃⋊#108

C₄

yes

characteristic

#16811633

C₄

yes

—

—

C₄

yes

complemented

#16811494

—

C₄×C₈×D₅₃

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810345

—

#16810344

C₄

yes

characteristic, complemented

C₅₃×#322

C₄

yes

characteristic, complemented

D₅₃⋊M₁₆

—

#13133296

—

#13133292

—

#13133291

—

D₅₃⋊C₁₆

—

#13133288

C₈

yes

complemented

#13133287

—

#13133286

C₈

yes

—

—

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

D₄

yes

characteristic

C₄×#88322

—

C₄×#88322

D₄

yes

characteristic

C₈×Dic₅₃

—

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₂²⋊#6157

—

—

—

—

—

D₅₃⋊C₈

—

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×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

C₂×C₈

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

characteristic

C₈×D₅₃

M₁₆

yes

characteristic

C₈×D₅₃

M₁₆

yes

characteristic

C₈×D₅₃

D₈

yes

characteristic

C₈×D₅₃

SD₁₆

yes

characteristic

C₈×D₅₃

SD₁₆

yes

characteristic

C₈×D₅₃

Q₁₆

yes

characteristic

#11719910

—

C₂²×D₅₃

#104

yes

characteristic

C₂×#6157

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#104

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂²⋊C₈

yes

characteristic

C₄×D₅₃

C₂²⋊C₈

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

—

yes

yes

C₄⋊C₈

yes

characteristic

#88322

C₄⋊C₈

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

D₄⋊C₈

yes

characteristic

C₂×D₅₃

Q₈⋊C₈

yes

characteristic

#6157

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

D₄⋊C₈

yes

characteristic

Dic₅₃

Q₈⋊C₈

yes

characteristic

C₂×#364

—

100

#1772

—

101

C₂×#322

—

102

C₂×C₅₃

—

103

C₂×C₅₃

#1279

yes

characteristic

104

D₅₃

—

105

D₅₃

#1336

yes

—

106

C₂×M₃₂

—

106

—

107

C₂×#108

—

108

#427

—

109

#405

—

106

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

106

—

112

#364

—

113

#364

—

106

—

114

#339

—

106

—

115

#322

—

116

#322

—

106

—

117

#322

#6157

yes

characteristic, complemented

118

C₅₃

#7182

yes

characteristic, complemented, Sylow

119

C₂×M₁₆

—

106

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

106

—

123

#117

—

106

—

124

C₂×C₄²

—

125

M₃₂

—

106

—

126

M₃₂

—

212

—

127

C₂×C₁₆

—

212

—

128

#108

—

129

#108

—

106

—

130

#107

—

131

#106

—

106

—

132

#97

—

212

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

106

—

138

C₄×C₈

—

106

—

139

C₄×C₈

—

212

—

140

C₄×C₈

C₂×#6157

yes

characteristic

141

C₂²×C₄

—

142

C₂²×C₄

—

106

—

143

M₁₆

—

106

—

144

M₁₆

—

212

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

106

—

157

C₂×C₈

—

212

—

158

C₂×C₈

—

212

—

159

C₂×C₈

C₄×#6157

yes

characteristic

160

C₂×C₈

C₄⋊#6157

yes

characteristic

161

C₄⋊C₄

—

106

—

162

C₂²⋊C₄

—

212

—

163

C₄²

—

164

C₄²

—

106

—

165

C₄²

—

212

—

166

C₄²

C₂²⋊#6157

yes

characteristic

167

C₁₆

—

212

—

168

C₂³

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

106

—

177

C₂×C₄

—

212

—

178

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

212

—

185

C₈

—

212

—

186

C₈

C₈×#6157

yes

characteristic

187

C₈

#13133271

yes

characteristic

188

C₈

#13133285

yes

characteristic

189

C₈

#13133286

yes

characteristic

190

C₂²

—

191

C₂²

—

192

C₂²

—

106

—

193

C₂²

#16811494

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

106

—

198

C₄

—

424

—

199

C₄

(C₂×#6157)⋊C₈

yes

characteristic

200

C₄

#16811497

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

—

106

—

204

C₂

#20289550

yes

characteristic

	subgroup	G/H
center	C₄	#16811497
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#322	C₄
socle	C₂×C₅₃	#1279
prime radical	C₂²×D₅₃	#104
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#322	#6157
hypocenter	C₅₃	#7182
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₄

#16811497

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#322

C₄

C₂×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
9. #16811545	C₄	semidirect
14. #16810344	C₄	semidirect
15. C₅₃×#322	C₄	semidirect
21. #13133288	C₈	semidirect
23. #13133286	C₈	semidirect
117. #322	#6157	semidirect
118. C₅₃	#7182	semidirect

H ≅ G/N

type

9. #16811545

C₄

semidirect

14. #16810344

C₄

semidirect

15. C₅₃×#322

C₄

semidirect

21. #13133288

C₈

semidirect

23. #13133286

C₈

semidirect

117. #322

#6157

semidirect

118. C₅₃

#7182

semidirect

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

subgroup	index
D₅₃⋊#364	2
#20290517	2
#322×D₅₃	2
#7182	53

subgroup

index

D₅₃⋊#364

#20290517

#322×D₅₃

#7182

Subgroups — #28298402

Conjugacy classes of subgroups

Complements

Maximal subgroups