Group #28305657 — subgroups

Subgroups — #28305657

Order 13568.

290 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 92 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	#20290829	C₂	1	2	yes	maximal
2	#20290273	C₂	1	1	yes	maximal, characteristic
3	#20289925	C₂	1	1	yes	maximal, characteristic
4	#20289607	C₂	1	2	yes	maximal
5	#334×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811642	C₂²	1	1	yes	characteristic
7	C₂×#13133288	—	2	2	no	—
8	#16811627	C₂²	1	2	yes	—
9	C₄×#11719936	C₂²	1	1	yes	characteristic
10	#16811499	—	2	2	no	—
11	#16811486	—	2	1	no	—
12	#16811486	—	2	1	no	—
13	#88324⋊C₈	—	2	1	no	—
14	#16811464	—	2	1	no	—
15	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810374	C₄	1	1	yes	characteristic
18	#16810373	C₄	1	2	yes	—
19	C₅₃×#334	C₄	1	1	yes	characteristic
20	C₂×#11719936	—	2	2	no	—
21	C₂×#11719936	D₄	1	1	yes	characteristic
22	C₂×#11719936	D₄	1	1	yes	characteristic
23	#13133288	—	4	4	no	—
24	C₄⋊#88324	—	2	1	no	—
25	C₄⋊#88324	—	2	2	no	—
26	C₄⋊#88324	—	2	2	no	—
27	#13133274	Q₈	1	2	yes	—
28	#13133272	—	2	1	no	—
29	C₄×#88324	—	2	1	no	—
30	C₄×#88324	—	2	1	no	—
31	C₂×C₈×D₅₃	—	2	2	no	—
32	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
34	C₄²×D₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊#88322	C₂×C₄	1	2	yes	—
36	#13133146	—	2	2	no	—
37	C₄×#88322	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	D₄	1	1	yes	characteristic
39	C₈×Dic₅₃	D₄	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×#88324	—	2	1	no	—
46	C₂×#88324	—	2	1	no	—
47	C₂×#88324	—	2	2	no	—
48	C₂×#88324	—	4	2	no	—
49	#11719936	—	2	2	no	—
50	#11719936	—	2	2	no	—
51	#11719936	—	4	2	no	—
52	#11719936	—	4	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄×Dic₅₃	#41	1	1	yes	characteristic
59	C₄×Dic₅₃	#41	1	1	yes	characteristic
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
62	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
63	C₈×D₅₃	—	2	2	no	—
64	C₈×D₅₃	—	4	1	no	—
65	C₈×D₅₃	—	4	2	no	—
66	C₈×D₅₃	—	4	2	no	—
67	C₂²×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×Dic₅₃	#121	1	1	yes	characteristic
72	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×Dic₅₃	#125	1	1	yes	characteristic
74	C₄×D₅₃	—	2	1	no	—
75	C₄×D₅₃	—	2	1	no	—
76	C₄×D₅₃	—	4	1	no	—
77	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₄×D₅₃	#137	1	2	yes	—
79	#88324	—	4	2	no	—
80	#88324	—	4	1	no	—
81	#88324	—	4	1	no	—
82	#88324	—	8	2	no	—
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	4	2	no	—
85	C₈×C₅₃	#108	1	2	yes	complemented
86	#88322	—	2	1	no	—
87	#88322	—	2	1	no	—
88	#88322	—	4	2	no	—
89	#9626	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	#389	1	1	yes	characteristic
91	C₂×D₅₃	—	2	1	no	—
92	C₂×D₅₃	—	2	1	no	—
93	C₂×D₅₃	#501	1	2	yes	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
96	C₄×C₅₃	#430	1	1	yes	characteristic
97	C₄×C₅₃	#440	1	2	yes	—
98	Dic₅₃	—	2	1	no	—
99	Dic₅₃	—	2	1	no	—
100	Dic₅₃	#469	1	1	yes	characteristic
101	Dic₅₃	#470	1	1	yes	characteristic
102	#1950	—	53	2	no	—
103	#1567	—	53	1	no	—
104	C₂×#334	—	53	1	no	—
105	#1451	—	53	1	no	—
106	#1301	—	53	2	no	—
107	C₂×C₅₃	#1944	1	1	yes	characteristic
108	C₂×C₅₃	#1950	1	1	yes	characteristic
109	C₂×C₅₃	#1953	1	1	yes	characteristic
110	D₅₃	—	2	2	no	—
111	#431	—	53	1	no	—
112	C₂×#108	—	106	2	no	—
113	#423	—	53	2	no	—
114	C₂×C₄⋊C₈	—	53	2	no	—
115	C₄×M₁₆	—	53	1	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	#341	—	106	2	no	—
118	#335	—	106	1	no	—
119	#335	—	106	1	no	—
120	#334	—	53	2	no	—
121	#334	—	53	1	no	—
122	#334	#6157	1	1	yes	characteristic
123	#322	—	106	1	no	—
124	#322	—	106	1	no	—
125	C₅₃	#9626	1	1	yes	characteristic, complemented, Sylow
126	C₂×M₁₆	—	53	1	no	—
127	C₂×M₁₆	—	53	1	no	—
128	C₂×M₁₆	—	106	2	no	—
129	C₂²×C₈	—	53	1	no	—
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	—	106	2	no	—
132	C₂×C₄²	—	53	1	no	—
133	#108	—	212	4	no	—
134	C₄⋊C₈	—	53	2	no	—
135	C₄⋊C₈	—	106	2	no	—
136	C₄⋊C₈	—	106	1	no	—
137	C₄⋊C₈	—	106	2	no	—
138	C₄⋊C₈	—	106	2	no	—
139	C₄⋊C₈	C₂×#6157	1	2	yes	—
140	C₂²⋊C₈	—	53	2	no	—
141	#97	—	106	1	no	—
142	C₄×C₈	—	53	1	no	—
143	C₄×C₈	—	53	1	no	—
144	C₄×C₈	—	53	1	no	—
145	C₄×C₈	—	106	1	no	—
146	C₄×C₈	—	106	1	no	—
147	C₄×C₈	C₂×#6157	1	1	yes	characteristic
148	C₂²×C₄	—	53	1	no	—
149	C₂²×C₄	—	53	1	no	—
150	C₂²×C₄	—	53	1	no	—
151	M₁₆	—	106	2	no	—
152	M₁₆	—	106	2	no	—
153	M₁₆	—	212	2	no	—
154	M₁₆	—	212	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	53	1	no	—
157	C₂×C₈	—	53	1	no	—
158	C₂×C₈	—	106	2	no	—
159	C₂×C₈	—	106	1	no	—
160	C₂×C₈	—	106	1	no	—
161	C₂×C₈	—	106	2	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	212	1	no	—
164	C₂×C₈	—	212	2	no	—
165	C₂×C₈	—	212	2	no	—
166	C₂×C₈	—	212	2	no	—
167	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
168	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
169	C₄²	—	53	1	no	—
170	C₄²	—	53	1	no	—
171	C₄²	—	53	1	no	—
172	C₄²	C₂²×#6157	1	1	yes	characteristic
173	C₂³	—	53	1	no	—
174	C₂×C₄	—	53	1	no	—
175	C₂×C₄	—	53	2	no	—
176	C₂×C₄	—	53	2	no	—
177	C₂×C₄	—	53	1	no	—
178	C₂×C₄	—	53	1	no	—
179	C₂×C₄	—	106	1	no	—
180	C₂×C₄	—	106	1	no	—
181	C₂×C₄	—	212	1	no	—
182	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
183	C₂×C₄	D₄×#6157	1	1	yes	characteristic
184	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	4	2	no	—
187	C₈	—	106	1	no	—
188	C₈	—	106	1	no	—
189	C₈	—	424	2	no	—
190	C₈	—	212	1	no	—
191	C₈	—	212	1	no	—
192	C₈	—	212	2	no	—
193	C₈	—	212	2	no	—
194	C₈	#13133288	1	2	yes	complemented
195	C₂²	—	53	2	no	—
196	C₂²	—	106	1	no	—
197	C₂²	—	106	1	no	—
198	C₂²	#16811574	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	53	1	no	—
201	C₄	—	53	1	no	—
202	C₄	—	106	1	no	—
203	C₄	—	106	1	no	—
204	C₄	C₂×#13133288	1	1	yes	characteristic
205	C₄	#16811638	1	1	yes	characteristic
206	C₄	#16811662	1	2	yes	—
207	C₂	—	106	2	no	—
208	C₂	#20290821	1	1	yes	characteristic
209	C₂	#20290829	1	1	yes	characteristic
210	C₂	#20290835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290829

C₂

yes

maximal

#20290273

C₂

yes

maximal, characteristic

#20289925

C₂

yes

maximal, characteristic

#20289607

C₂

yes

maximal

#334×D₅₃

C₂

yes

maximal, characteristic

#16811642

C₂²

yes

characteristic

C₂×#13133288

—

#16811627

C₂²

yes

—

C₄×#11719936

C₂²

yes

—

—

—

—

—

C₄⋊C₈×D₅₃

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810374

C₄

yes

characteristic

#16810373

C₄

yes

—

C₅₃×#334

C₄

yes

characteristic

C₂×#11719936

—

C₂×#11719936

D₄

yes

characteristic

C₂×#11719936

D₄

yes

—

—

—

—

Q₈

yes

—

#13133272

—

C₄×#88324

—

C₄×#88324

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

C₂³

yes

characteristic

C₄⋊#88322

C₂×C₄

yes

—

#13133146

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

—

—

—

—

—

—

—

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

C₄×Dic₅₃

yes

C₄×Dic₅₃

yes

—

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₂²×D₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

characteristic

C₂×Dic₅₃

#121

yes

characteristic

C₂×Dic₅₃

C₂²⋊Q₈

yes

characteristic

C₂×Dic₅₃

#125

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

AGL(1,ℤ/8)

yes

—

C₄×D₅₃

#137

yes

—

—

—

—

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#501

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

—

100

Dic₅₃

#469

yes

characteristic

101

Dic₅₃

#470

yes

characteristic

102

#1950

—

103

#1567

—

104

C₂×#334

—

105

#1451

—

106

#1301

—

107

C₂×C₅₃

#1944

yes

characteristic

108

C₂×C₅₃

#1950

yes

characteristic

109

C₂×C₅₃

#1953

yes

characteristic

110

D₅₃

—

111

#431

—

112

C₂×#108

—

106

—

113

#423

—

114

C₂×C₄⋊C₈

—

115

C₄×M₁₆

—

116

C₂×C₄×C₈

—

117

#341

—

106

—

118

#335

—

106

—

119

#335

—

106

—

120

#334

—

121

#334

—

122

yes

123

—

106

—

124

#322

—

106

—

125

C₅₃

#9626

yes

characteristic, complemented, Sylow

126

C₂×M₁₆

—

127

C₂×M₁₆

—

128

C₂×M₁₆

—

106

—

129

C₂²×C₈

—

130

C₂²×C₈

—

131

C₂²×C₈

—

106

—

132

C₂×C₄²

—

133

#108

—

212

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

106

—

136

C₄⋊C₈

—

106

—

137

C₄⋊C₈

—

106

—

138

C₄⋊C₈

—

106

—

139

C₄⋊C₈

C₂×#6157

yes

—

140

C₂²⋊C₈

—

141

#97

—

106

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

106

—

146

C₄×C₈

—

106

—

147

C₄×C₈

C₂×#6157

yes

characteristic

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

M₁₆

—

106

—

152

M₁₆

—

106

—

153

M₁₆

—

212

—

154

M₁₆

—

212

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

106

—

159

C₂×C₈

—

106

—

160

C₂×C₈

—

106

—

161

C₂×C₈

—

106

—

162

C₂×C₈

—

106

—

163

C₂×C₈

—

212

—

164

C₂×C₈

—

212

—

165

C₂×C₈

—

212

—

166

C₂×C₈

—

212

—

167

C₂×C₈

C₄⋊#6157

yes

characteristic

168

C₂×C₈

C₄⋊#6157

yes

characteristic

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

C₂²×#6157

yes

characteristic

173

C₂³

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

106

—

180

C₂×C₄

—

106

—

181

C₂×C₄

—

212

—

182

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

183

C₂×C₄

D₄×#6157

yes

characteristic

184

C₂×C₄

Q₈×#6157

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

—

106

—

188

C₈

—

106

—

189

C₈

—

424

—

190

C₈

—

212

—

191

C₈

—

212

—

192

C₈

—

212

—

193

C₈

—

212

—

194

C₈

#13133288

yes

complemented

195

C₂²

—

196

C₂²

—

106

—

197

C₂²

—

106

—

198

C₂²

#16811574

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

106

—

203

C₄

—

106

—

204

C₄

C₂×#13133288

yes

characteristic

205

C₄

#16811638

yes

characteristic

206

C₄

#16811662

yes

—

207

C₂

—

106

—

208

C₂

#20290821

yes

characteristic

209

C₂

#20290829

yes

characteristic

210

C₂

#20290835

yes

characteristic

	subgroup	G/H
center	C₂²	#16811574
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#334	C₄
socle	C₂²×C₅₃	#389
prime radical	C₂²×D₅₃	C₄⋊Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	#334	#6157
hypocenter	C₅₃	#9626
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#16811574

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#334

C₄

socle

C₂²×C₅₃

#389

prime radical

C₂²×D₅₃

C₄⋊Q₈

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
85. C₈×C₅₃	#108	semidirect
125. C₅₃	#9626	semidirect
194. C₈	#13133288	semidirect

H ≅ G/N

type

85. C₈×C₅₃

#108

semidirect

125. C₅₃

#9626

semidirect

194. C₈

#13133288

semidirect

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

subgroup	index
#20290829	2
#20290273	2
#20289925	2
#20289607	2
#334×D₅₃	2
#9626	53

subgroup

index

#334×D₅₃

Subgroups — #28305657

Conjugacy classes of subgroups

Complements

Maximal subgroups