Group #28302175 — subgroups

Subgroups — #28302175

Order 13568.

358 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 136 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	#20289917	C₂	1	2	yes	maximal
3	#20289909	C₂	1	1	yes	maximal, characteristic
4	#20289546	C₂	1	2	yes	maximal
5	#20273462	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133287	C₄	1	2	yes	—
7	C₄×#11719936	C₂²	1	2	yes	—
8	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
10	#16811499	C₄	1	2	yes	—
11	#88324⋊C₈	—	2	2	no	—
12	#16811464	C₄	1	2	yes	—
13	#16811464	C₄	1	2	yes	—
14	C₈×#88324	—	2	2	no	—
15	#16810675	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810376	C₄	1	1	yes	characteristic
18	#16810373	C₄	1	2	yes	—
19	C₅₃×#335	C₄	1	1	yes	characteristic
20	C₂×#11719936	C₂×C₄	1	4	yes	—
21	C₂×D₅₃⋊C₈	—	2	1	no	—
22	C₂×D₅₃⋊C₈	Q₈	1	2	yes	—
23	#13133287	C₈	1	8	yes	complemented
24	#13133272	—	2	2	no	—
25	C₄×#88324	—	2	2	no	—
26	C₄×#88324	—	2	1	no	—
27	C₄×#88324	C₂×C₄	1	2	yes	—
28	C₄×#88324	C₂×C₄	1	2	yes	—
29	C₄×#88324	D₄	1	2	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×D₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊#88322	C₂×C₄	1	2	yes	—
34	#13133129	—	2	2	no	—
35	C₄×#88322	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₅₃	—	2	2	no	—
37	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
44	C₂×#88324	—	2	4	no	—
45	C₂×#88324	—	2	2	no	—
46	C₂×#88324	C₄⋊C₄	1	4	yes	—
47	#11719936	—	2	4	no	—
48	#11719936	C₂×C₈	1	8	yes	—
49	D₅₃⋊C₈	—	4	2	no	—
50	D₅₃⋊C₈	—	4	1	no	—
51	D₅₃⋊C₈	—	4	1	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄×Dic₅₃	#41	1	1	yes	characteristic
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	C₄²	1	1	yes	characteristic
61	C₂×#88322	C₄²	1	1	yes	characteristic
62	C₈×D₅₃	—	2	2	no	—
63	C₈×D₅₃	C₂×C₈	1	4	yes	—
64	C₂²×D₅₃	C₄×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×Dic₅₃	#117	1	1	yes	characteristic
70	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
71	C₄×D₅₃	—	2	1	no	—
72	C₄×D₅₃	—	2	1	no	—
73	C₄×D₅₃	—	2	2	no	—
74	C₄×D₅₃	C₂²×C₈	1	2	yes	—
75	C₄×D₅₃	#131	1	2	yes	—
76	#88324	—	2	4	no	—
77	#88324	—	2	8	no	—
78	#88324	—	4	2	no	—
79	C₈×C₅₃	—	2	1	no	—
80	C₈×C₅₃	—	4	2	no	—
81	C₈×C₅₃	C₄×C₈	1	2	yes	—
82	#88322	—	2	1	no	—
83	#88322	—	4	2	no	—
84	#88322	C₄×C₈	1	2	yes	—
85	#8294	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂×D₅₃	—	2	1	no	—
88	C₂×D₅₃	—	2	1	no	—
89	C₂×D₅₃	C₈×Q₈	1	2	yes	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₅₃	#405	1	1	yes	characteristic
93	C₄×C₅₃	#444	1	1	yes	characteristic
94	C₄×C₅₃	#444	1	1	yes	characteristic
95	Dic₅₃	—	2	1	no	—
96	Dic₅₃	#430	1	2	yes	—
97	Dic₅₃	C₈×D₄	1	2	yes	—
98	C₄×#108	—	53	1	no	—
99	#1564	—	53	1	no	—
100	C₈×M₁₆	—	53	2	no	—
101	C₂×#322	—	53	1	no	—
102	#1278	—	53	2	no	—
103	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₅₃	#1773	1	1	yes	characteristic
105	C₂×C₅₃	#1778	1	1	yes	characteristic
106	D₅₃	—	2	2	no	—
107	#431	—	53	2	no	—
108	C₂×#108	—	53	2	no	—
109	C₄×M₁₆	—	53	2	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	1	no	—
113	#341	—	53	2	no	—
114	#335	—	53	1	no	—
115	#335	#6157	1	1	yes	characteristic
116	#334	—	53	2	no	—
117	#322	—	53	2	no	—
118	#322	—	53	2	no	—
119	#322	—	106	2	no	—
120	C₈²	—	106	2	no	—
121	C₅₃	#8294	1	1	yes	characteristic, complemented, Sylow
122	C₂×M₁₆	—	53	4	no	—
123	C₂²×C₈	—	53	1	no	—
124	C₂²×C₈	—	53	2	no	—
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	—	106	1	no	—
127	C₂×C₄²	—	53	1	no	—
128	#108	—	53	8	no	—
129	C₄⋊C₈	—	53	2	no	—
130	C₄⋊C₈	C₂×#6157	1	2	yes	—
131	C₂²⋊C₈	—	106	2	no	—
132	#97	—	106	2	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	53	1	no	—
136	C₄×C₈	—	53	1	no	—
137	C₄×C₈	—	53	2	no	—
138	C₄×C₈	—	53	1	no	—
139	C₄×C₈	—	106	2	no	—
140	C₄×C₈	—	106	2	no	—
141	C₄×C₈	—	106	1	no	—
142	C₄×C₈	C₂×#6157	1	1	yes	characteristic
143	C₂²×C₄	—	53	1	no	—
144	C₂²×C₄	—	53	1	no	—
145	C₂²×C₄	—	53	1	no	—
146	M₁₆	—	53	8	no	—
147	M₁₆	—	106	4	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	53	1	no	—
150	C₂×C₈	—	53	4	no	—
151	C₂×C₈	—	53	4	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	106	2	no	—
154	C₂×C₈	—	106	4	no	—
155	C₂×C₈	—	106	2	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	212	2	no	—
158	C₂×C₈	—	212	1	no	—
159	C₂×C₈	—	212	1	no	—
160	C₂×C₈	C₄×#6157	1	1	yes	characteristic
161	C₂×C₈	C₄×#6157	1	1	yes	characteristic
162	C₄²	—	53	1	no	—
163	C₄²	—	53	1	no	—
164	C₄²	—	53	1	no	—
165	C₄²	C₂²×#6157	1	1	yes	characteristic
166	C₂³	—	53	1	no	—
167	C₂×C₄	—	53	1	no	—
168	C₂×C₄	—	53	1	no	—
169	C₂×C₄	—	53	2	no	—
170	C₂×C₄	—	53	2	no	—
171	C₂×C₄	—	53	1	no	—
172	C₂×C₄	—	106	2	no	—
173	C₂×C₄	—	106	1	no	—
174	C₂×C₄	—	106	1	no	—
175	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
176	C₂×C₄	D₄×#6157	1	1	yes	characteristic
177	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	4	2	no	—
180	C₈	—	53	2	no	—
181	C₈	—	106	4	no	—
182	C₈	—	106	8	no	—
183	C₈	—	106	1	no	—
184	C₈	—	212	2	no	—
185	C₈	—	212	2	no	—
186	C₈	C₈×#6157	1	2	yes	—
187	C₂²	—	53	2	no	—
188	C₂²	—	106	1	no	—
189	C₂²	—	106	1	no	—
190	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	53	2	no	—
193	C₄	—	53	2	no	—
194	C₄	—	106	1	no	—
195	C₄	C₂×C₈×#6157	1	1	yes	characteristic
196	C₄	#16811589	1	1	yes	characteristic
197	C₄	#16811669	1	1	yes	characteristic
198	C₄	#16811670	1	1	yes	characteristic
199	C₂	—	106	2	no	—
200	C₂	#20290508	1	1	yes	characteristic
201	C₂	#20290522	1	1	yes	characteristic
202	C₂	#20290536	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133287

C₂

yes

maximal, characteristic

#20289917

C₂

yes

maximal

#20289909

C₂

yes

maximal, characteristic

#20289546

C₂

yes

maximal

#20273462

C₂

yes

maximal, characteristic

C₂×#13133287

C₄

yes

—

C₄×#11719936

C₂²

yes

—

C₄×D₅₃⋊C₈

C₂²

yes

characteristic

C₄×D₅₃⋊C₈

C₂²

yes

characteristic

#16811499

C₄

yes

—

#88324⋊C₈

—

#16811464

C₄

yes

—

#16811464

C₄

yes

—

C₈×#88324

—

#16810675

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic

#16810373

C₄

yes

—

C₅₃×#335

C₄

yes

characteristic

C₂×#11719936

C₂×C₄

yes

—

C₂×D₅₃⋊C₈

—

C₂×D₅₃⋊C₈

Q₈

yes

—

#13133287

C₈

yes

—

—

—

C₂×C₄

yes

—

C₄×#88324

C₂×C₄

yes

—

C₄×#88324

D₄

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

C₂×C₄×D₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

C₂×C₄×D₅₃

yes

—

—

C₄⋊C₄

yes

—

#11719936

—

#11719936

C₂×C₈

yes

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

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

C₄×Dic₅₃

yes

—

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

C₂×C₈

yes

—

C₂²×D₅₃

C₄×Q₈

yes

characteristic

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

#117

yes

characteristic

C₂×Dic₅₃

C₄×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂²×C₈

yes

—

C₄×D₅₃

#131

yes

—

#88324

—

#88324

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

—

#88322

—

#88322

—

#88322

C₄×C₈

yes

—

#8294

—

maximal, Sylow

C₂²×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₈×Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄×C₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

#430

yes

—

Dic₅₃

C₈×D₄

yes

—

C₄×#108

—

#1564

—

100

C₈×M₁₆

—

101

C₂×#322

—

102

#1278

—

103

C₂×C₅₃

C₈×C₄⋊C₄

yes

characteristic

104

C₂×C₅₃

#1773

yes

characteristic

105

C₂×C₅₃

#1778

yes

characteristic

106

D₅₃

—

107

#431

—

108

C₂×#108

—

109

C₄×M₁₆

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

#341

—

114

#335

—

115

yes

116

—

117

#322

—

118

#322

—

119

#322

—

106

—

120

C₈²

—

106

—

121

C₅₃

#8294

yes

characteristic, complemented, Sylow

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

106

—

127

C₂×C₄²

—

128

#108

—

129

C₄⋊C₈

—

130

C₄⋊C₈

C₂×#6157

yes

—

131

C₂²⋊C₈

—

106

—

132

#97

—

106

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

106

—

140

C₄×C₈

—

106

—

141

C₄×C₈

—

106

—

142

C₄×C₈

C₂×#6157

yes

characteristic

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

M₁₆

—

147

M₁₆

—

106

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

106

—

154

C₂×C₈

—

106

—

155

C₂×C₈

—

106

—

156

C₂×C₈

—

106

—

157

C₂×C₈

—

212

—

158

C₂×C₈

—

212

—

159

C₂×C₈

—

212

—

160

C₂×C₈

C₄×#6157

yes

characteristic

161

C₂×C₈

C₄×#6157

yes

characteristic

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

C₂²×#6157

yes

characteristic

166

C₂³

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

106

—

173

C₂×C₄

—

106

—

174

C₂×C₄

—

106

—

175

C₂×C₄

C₂×C₄×#6157

yes

characteristic

176

C₂×C₄

D₄×#6157

yes

characteristic

177

C₂×C₄

Q₈×#6157

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

106

—

182

C₈

—

106

—

183

C₈

—

106

—

184

C₈

—

212

—

185

C₈

—

212

—

186

C₈

C₈×#6157

yes

—

187

C₂²

—

188

C₂²

—

106

—

189

C₂²

—

106

—

190

C₂²

C₄⋊C₄×#6157

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

106

—

195

C₄

C₂×C₈×#6157

yes

characteristic

196

C₄

#16811589

yes

characteristic

197

C₄

#16811669

yes

characteristic

198

C₄

#16811670

yes

characteristic

199

C₂

—

106

—

200

C₂

#20290508

yes

characteristic

201

C₂

#20290522

yes

characteristic

202

C₂

#20290536

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂²×D₅₃	C₄×Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	#335	#6157
hypocenter	C₅₃	#8294
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#335

C₄

socle

C₂²×C₅₃

C₄×C₄⋊C₄

prime radical

C₂²×D₅₃

C₄×Q₈

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
23. #13133287	C₈	semidirect
121. C₅₃	#8294	semidirect

H ≅ G/N

type

23. #13133287

C₈

semidirect

121. C₅₃

#8294

semidirect

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

subgroup	index
C₄×#13133287	2
#20289917	2
#20289909	2
#20289546	2
#20273462	2
#8294	53

subgroup

index

Subgroups — #28302175

Conjugacy classes of subgroups

Complements

Maximal subgroups