Group #28302176 — subgroups

Subgroups — #28302176

Order 13568.

334 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 122 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×#13133287	C₂	1	1	yes	maximal, characteristic
2	#20290265	C₂	1	1	yes	maximal, characteristic
3	#20289925	C₂	1	2	yes	maximal
4	#20289547	C₂	1	2	yes	maximal
5	#322×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811642	C₂²	1	2	yes	—
7	C₂×#13133287	C₄	1	2	yes	—
8	C₄×#11719936	C₂²	1	2	yes	—
9	#16811499	C₄	1	2	yes	—
10	#16811486	C₄	1	2	yes	—
11	#16811485	C₄	1	2	yes	—
12	#88324⋊C₈	—	2	2	no	—
13	#16811464	—	2	2	no	—
14	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810345	C₄	1	1	yes	characteristic
18	#16810345	C₄	1	1	yes	characteristic
19	#16810344	C₄	1	1	yes	characteristic
20	C₅₃×#322	C₄	1	1	yes	characteristic
21	C₂×#11719936	C₂×C₄	1	4	yes	—
22	#13133287	—	2	4	no	—
23	C₄⋊#88324	C₂×C₄	1	4	yes	—
24	#13133274	—	2	2	no	—
25	#13133272	—	2	2	no	—
26	C₄×#88324	—	2	2	no	—
27	C₄×#88324	—	2	2	no	—
28	C₂×C₈×D₅₃	—	2	1	no	—
29	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	Q₈	1	2	yes	—
32	C₄²×D₅₃	C₂³	1	1	yes	characteristic
33	C₄×#88322	C₂×C₄	1	1	yes	characteristic
34	C₄×#88322	C₂×C₄	1	1	yes	characteristic
35	C₄×#88322	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₅₃	—	2	1	no	—
37	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₅₃	D₄	1	2	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
46	C₂×#88324	—	2	4	no	—
47	C₂×#88324	—	2	4	no	—
48	#11719936	—	2	4	no	—
49	#11719936	—	2	4	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₅₃	#41	1	1	yes	characteristic
58	C₂×#88322	—	2	1	no	—
59	C₂×#88322	C₄²	1	1	yes	characteristic
60	C₂×#88322	C₄²	1	1	yes	characteristic
61	C₂×#88322	C₄⋊C₄	1	2	yes	—
62	C₈×D₅₃	—	2	2	no	—
63	C₈×D₅₃	—	4	2	no	—
64	C₈×D₅₃	—	4	2	no	—
65	C₈×D₅₃	M₁₆	1	4	yes	—
66	C₂²×D₅₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
70	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
71	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
72	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₄×D₅₃	—	2	2	no	—
74	C₄×D₅₃	—	2	2	no	—
75	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
76	C₄×D₅₃	#131	1	2	yes	—
77	#88324	—	4	4	no	—
78	#88324	—	4	4	no	—
79	C₈×C₅₃	—	2	1	no	—
80	C₈×C₅₃	—	2	2	no	—
81	C₈×C₅₃	—	4	1	no	—
82	C₈×C₅₃	#97	1	2	yes	—
83	#88322	—	2	1	no	—
84	#88322	—	2	2	no	—
85	#88322	—	4	1	no	—
86	#88322	#97	1	2	yes	—
87	#8295	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂×D₅₃	—	2	1	no	—
90	C₂×D₅₃	—	2	1	no	—
91	C₂×D₅₃	#446	1	2	yes	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
94	C₄×C₅₃	#405	1	1	yes	characteristic
95	C₄×C₅₃	#430	1	2	yes	—
96	Dic₅₃	—	2	1	no	—
97	Dic₅₃	#436	1	2	yes	—
98	Dic₅₃	#444	1	2	yes	—
99	C₄×#108	—	53	1	no	—
100	#1564	—	53	1	no	—
101	#1451	—	53	2	no	—
102	C₂×#322	—	53	1	no	—
103	#1278	—	53	2	no	—
104	C₂×C₅₃	#1770	1	1	yes	characteristic
105	C₂×C₅₃	#1773	1	1	yes	characteristic
106	C₂×C₅₃	#1778	1	1	yes	characteristic
107	D₅₃	—	2	2	no	—
108	#431	—	53	2	no	—
109	C₂×#108	—	53	2	no	—
110	C₄×M₁₆	—	53	2	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	1	no	—
113	C₂×C₄×C₈	—	53	1	no	—
114	#341	—	53	2	no	—
115	#335	—	53	2	no	—
116	#334	—	53	2	no	—
117	#322	—	53	1	no	—
118	#322	—	53	1	no	—
119	#322	—	53	1	no	—
120	#322	—	106	2	no	—
121	#322	—	106	2	no	—
122	#322	#6157	1	1	yes	characteristic
123	C₅₃	#8295	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₁₆	—	53	4	no	—
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	—	53	1	no	—
127	C₂²×C₈	—	53	2	no	—
128	C₂²×C₈	—	106	1	no	—
129	C₂×C₄²	—	53	1	no	—
130	#108	—	106	4	no	—
131	C₄⋊C₈	—	53	4	no	—
132	C₂²⋊C₈	—	106	2	no	—
133	#97	—	106	2	no	—
134	C₄×C₈	—	53	1	no	—
135	C₄×C₈	—	53	1	no	—
136	C₄×C₈	—	53	2	no	—
137	C₄×C₈	—	53	1	no	—
138	C₄×C₈	—	53	1	no	—
139	C₄×C₈	—	53	1	no	—
140	C₄×C₈	—	106	1	no	—
141	C₄×C₈	—	106	2	no	—
142	C₄×C₈	—	106	2	no	—
143	C₄×C₈	C₂×#6157	1	1	yes	characteristic
144	C₄×C₈	C₂×#6157	1	1	yes	characteristic
145	C₄×C₈	C₂×#6157	1	1	yes	characteristic
146	C₂²×C₄	—	53	1	no	—
147	C₂²×C₄	—	53	1	no	—
148	C₂²×C₄	—	53	1	no	—
149	M₁₆	—	106	4	no	—
150	M₁₆	—	106	4	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	4	no	—
154	C₂×C₈	—	53	1	no	—
155	C₂×C₈	—	53	2	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	106	4	no	—
158	C₂×C₈	—	106	4	no	—
159	C₂×C₈	—	106	1	no	—
160	C₂×C₈	—	212	2	no	—
161	C₂×C₈	—	212	2	no	—
162	C₂×C₈	C₄×#6157	1	1	yes	characteristic
163	C₂×C₈	C₄×#6157	1	1	yes	characteristic
164	C₂×C₈	C₄⋊#6157	1	2	yes	—
165	C₄²	—	53	1	no	—
166	C₄²	—	53	1	no	—
167	C₄²	—	53	1	no	—
168	C₄²	C₂²×#6157	1	1	yes	characteristic
169	C₂³	—	53	1	no	—
170	C₂×C₄	—	53	1	no	—
171	C₂×C₄	—	53	1	no	—
172	C₂×C₄	—	53	2	no	—
173	C₂×C₄	—	53	2	no	—
174	C₂×C₄	—	53	1	no	—
175	C₂×C₄	—	106	2	no	—
176	C₂×C₄	—	106	2	no	—
177	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
179	C₂×C₄	#13133296	1	1	yes	characteristic
180	C₈	—	2	1	no	—
181	C₈	—	2	2	no	—
182	C₈	—	4	1	no	—
183	C₈	—	53	2	no	—
184	C₈	—	106	1	no	—
185	C₈	—	106	2	no	—
186	C₈	—	212	1	no	—
187	C₈	—	212	4	no	—
188	C₈	—	212	4	no	—
189	C₈	#13133271	1	2	yes	—
190	C₂²	—	53	2	no	—
191	C₂²	—	106	1	no	—
192	C₂²	—	106	1	no	—
193	C₂²	C₄×C₄⋊#6157	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	53	2	no	—
196	C₄	—	53	2	no	—
197	C₄	—	106	1	no	—
198	C₄	C₂×#13133271	1	1	yes	characteristic
199	C₄	#16811586	1	1	yes	characteristic
200	C₄	#16811638	1	2	yes	—
201	C₂	—	106	2	no	—
202	C₂	#20290509	1	1	yes	characteristic
203	C₂	#20290521	1	1	yes	characteristic
204	C₂	#20290535	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133287

C₂

yes

maximal, characteristic

#20290265

C₂

yes

maximal, characteristic

#20289925

C₂

yes

maximal

#20289547

C₂

yes

maximal

#322×D₅₃

C₂

yes

maximal, characteristic

#16811642

C₂²

yes

—

C₂×#13133287

C₄

yes

—

C₄×#11719936

C₂²

yes

—

#16811499

C₄

yes

—

#16811486

C₄

yes

—

#16811485

C₄

yes

—

#88324⋊C₈

—

#16811464

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810345

C₄

yes

characteristic

#16810345

C₄

yes

characteristic

#16810344

C₄

yes

characteristic

C₅₃×#322

C₄

yes

characteristic

C₂×#11719936

C₂×C₄

yes

—

#13133287

—

C₄⋊#88324

C₂×C₄

yes

—

—

—

—

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

—

C₄²×D₅₃

C₂³

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

—

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

C₂×C₄×D₅₃

yes

—

—

—

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

C₄×Dic₅₃

yes

—

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

M₁₆

yes

—

C₂²×D₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

characteristic

C₂×Dic₅₃

C₄×D₄

yes

characteristic

C₂×Dic₅₃

C₄×D₄

yes

characteristic

C₂×Dic₅₃

C₄×Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×M₁₆

yes

—

C₄×D₅₃

#131

yes

—

#88324

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#97

yes

—

—

—

—

yes

—

#8295

—

maximal, Sylow

C₂²×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#446

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#97

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

#436

yes

—

Dic₅₃

#444

yes

—

C₄×#108

—

100

#1564

—

101

#1451

—

102

C₂×#322

—

103

#1278

—

104

C₂×C₅₃

#1770

yes

characteristic

105

C₂×C₅₃

#1773

yes

characteristic

106

C₂×C₅₃

#1778

yes

characteristic

107

D₅₃

—

108

#431

—

109

C₂×#108

—

110

C₄×M₁₆

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

#341

—

115

#335

—

116

#334

—

117

#322

—

118

#322

—

119

#322

—

120

#322

—

106

—

121

#322

—

106

—

122

#322

#6157

yes

characteristic

123

C₅₃

#8295

yes

characteristic, complemented, Sylow

124

C₂×M₁₆

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

106

—

129

C₂×C₄²

—

130

#108

—

106

—

131

C₄⋊C₈

—

132

C₂²⋊C₈

—

106

—

133

#97

—

106

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

106

—

141

C₄×C₈

—

106

—

142

C₄×C₈

—

106

—

143

C₄×C₈

C₂×#6157

yes

characteristic

144

C₄×C₈

C₂×#6157

yes

characteristic

145

C₄×C₈

C₂×#6157

yes

characteristic

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

M₁₆

—

106

—

150

M₁₆

—

106

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

106

—

157

C₂×C₈

—

106

—

158

C₂×C₈

—

106

—

159

C₂×C₈

—

106

—

160

C₂×C₈

—

212

—

161

C₂×C₈

—

212

—

162

C₂×C₈

C₄×#6157

yes

characteristic

163

C₂×C₈

C₄×#6157

yes

characteristic

164

C₂×C₈

C₄⋊#6157

yes

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₂²×#6157

yes

characteristic

169

C₂³

—

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

C₂×C₄×#6157

yes

characteristic

178

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

179

C₂×C₄

#13133296

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

106

—

185

C₈

—

106

—

186

C₈

—

212

—

187

C₈

—

212

—

188

C₈

—

212

—

189

C₈

#13133271

yes

—

190

C₂²

—

191

C₂²

—

106

—

192

C₂²

—

106

—

193

C₂²

C₄×C₄⋊#6157

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

106

—

198

C₄

C₂×#13133271

yes

characteristic

199

C₄

#16811586

yes

characteristic

200

C₄

#16811638

yes

—

201

C₂

—

106

—

202

C₂

#20290509

yes

characteristic

203

C₂

#20290521

yes

characteristic

204

C₂

#20290535

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#322	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂²×D₅₃	C₄×Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	#322	#6157
hypocenter	C₅₃	#8295
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#322

C₄

socle

C₂²×C₅₃

C₄×C₄⋊C₄

prime radical

C₂²×D₅₃

C₄×Q₈

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
123. C₅₃	#8295	semidirect

H ≅ G/N

type

123. C₅₃

#8295

semidirect

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

subgroup	index
C₄×#13133287	2
#20290265	2
#20289925	2
#20289547	2
#322×D₅₃	2
#8295	53

subgroup

index

#322×D₅₃

Subgroups — #28302176

Conjugacy classes of subgroups

Complements

Maximal subgroups