Group #27282799 — subgroups

Subgroups — #27282799

Order 13568.

290 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20275488	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	#20273004	C₂	1	1	yes	maximal, characteristic
4	#20272610	C₂	1	2	yes	maximal
5	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	C₂×#13133152	—	2	2	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₄×#11719917	C₂²	1	1	yes	characteristic
10	#16810387	—	2	2	no	—
11	#16810376	C₄	1	2	yes	—
12	#16810376	C₄	1	2	yes	—
13	#16810345	—	2	1	no	—
14	#16810344	—	2	1	no	—
15	C₅₃×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#334	—	2	1	no	—
18	C₅₃×#334	—	2	1	no	—
19	C₂×#11719917	D₄	1	2	yes	—
20	C₂²×#88322	—	2	2	no	—
21	#13133152	—	4	4	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	1	no	—
24	C₄⋊#88322	C₂×C₄	1	4	yes	—
25	C₂²⋊#88322	Q₈	1	2	yes	—
26	#13133128	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	—	2	1	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₅₃	—	4	2	no	—
43	M₁₆×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	4	1	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	1	yes	characteristic
55	#11719917	—	2	2	no	—
56	#11719917	—	2	2	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	—	2	2	no	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	4	2	no	—
61	C₂×#88322	—	4	2	no	—
62	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	4	1	no	—
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
72	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₅₃	#137	1	2	yes	—
74	C₈×C₅₃	—	2	1	no	—
75	C₈×C₅₃	—	2	1	no	—
76	C₈×C₅₃	—	2	1	no	—
77	C₈×C₅₃	—	8	2	no	—
78	C₈×C₅₃	#108	1	2	yes	complemented
79	#88322	—	4	2	no	—
80	#88322	—	4	4	no	—
81	#88322	—	4	2	no	—
82	#9625	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	#389	1	1	yes	characteristic
86	C₂²×C₅₃	#501	1	2	yes	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
91	C₄×C₅₃	#430	1	1	yes	characteristic
92	C₄×C₅₃	#441	1	1	yes	characteristic
93	C₄×C₅₃	#442	1	1	yes	characteristic
94	C₄×C₅₃	#468	1	2	yes	—
95	#1950	—	53	2	no	—
96	#1566	D₅₃	1	1	yes	characteristic
97	C₂×#335	—	53	1	no	—
98	#1451	—	53	1	no	—
99	#1301	—	53	2	no	—
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	#1943	1	1	yes	characteristic
102	C₂×C₅₃	#1950	1	1	yes	characteristic
103	C₂×C₅₃	#1953	1	1	yes	characteristic
104	#431	—	53	1	no	—
105	C₂×#108	—	106	2	no	—
106	#423	C₂×D₅₃	1	2	yes	—
107	C₂×C₄⋊C₈	—	53	2	no	—
108	C₄×M₁₆	—	53	1	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	#341	—	106	2	no	—
111	#335	—	53	2	no	—
112	#335	—	53	2	no	—
113	#334	—	2	1	no	—
114	#334	—	2	1	no	—
115	#322	—	106	1	no	—
116	#322	—	106	1	no	—
117	C₅₃	#9625	1	1	yes	characteristic, complemented, Sylow
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	53	2	no	—
120	C₂²×C₈	—	106	2	no	—
121	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
122	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	#108	—	212	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	53	4	no	—
128	C₄⋊C₈	—	106	2	no	—
129	C₄⋊C₈	—	106	1	no	—
130	C₂²⋊C₈	—	53	2	no	—
131	#97	—	106	1	no	—
132	C₄×C₈	—	106	1	no	—
133	C₄×C₈	—	106	1	no	—
134	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
135	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
139	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
141	M₁₆	—	4	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	106	2	no	—
144	M₁₆	—	106	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	106	4	no	—
149	C₂×C₈	—	106	2	no	—
150	C₂×C₈	—	106	2	no	—
151	C₂×C₈	—	212	2	no	—
152	C₂×C₈	—	212	2	no	—
153	C₂×C₈	#11719919	1	1	yes	characteristic
154	C₂×C₈	#11719919	1	1	yes	characteristic
155	C₂×C₈	#11719919	1	1	yes	characteristic
156	C₂×C₈	#11719919	1	1	yes	characteristic
157	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
158	C₄²	#11719947	1	1	yes	characteristic
159	C₄²	#11719949	1	1	yes	characteristic
160	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
161	C₂³	#13133216	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
166	C₂×C₄	#13133179	1	1	yes	characteristic
167	C₂×C₄	#13133182	1	1	yes	characteristic
168	C₂×C₄	#13133188	1	1	yes	characteristic
169	C₂×C₄	#13133193	1	1	yes	characteristic
170	C₂×C₄	#13133208	1	1	yes	characteristic
171	C₂×C₄	#13133244	1	2	yes	—
172	C₂×C₄	#13133246	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	8	2	no	—
177	C₈	—	212	2	no	—
178	C₈	—	212	2	no	—
179	C₈	—	212	4	no	—
180	C₈	#13133152	1	2	yes	complemented
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#16810519	1	1	yes	characteristic
184	C₂²	#16811030	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	C₂×#13133152	1	1	yes	characteristic
189	C₄	#16810671	1	1	yes	characteristic
190	C₄	#16810729	1	1	yes	characteristic
191	C₄	#16810734	1	1	yes	characteristic
192	C₄	#16810866	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#20275451	1	1	yes	characteristic
195	C₂	#20275488	1	1	yes	characteristic
196	C₂	#20275503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275488

C₂

yes

maximal

C₂×#16810376

C₂

yes

maximal, characteristic

#20273004

C₂

yes

maximal, characteristic

#20272610

C₂

yes

maximal

C₅₃×#1566

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

characteristic

C₂×#13133152

—

C₂×C₄⋊#88322

C₂²

yes

—

C₄×#11719917

C₂²

yes

characteristic

#16810387

—

#16810376

C₄

yes

—

#16810376

C₄

yes

—

#16810345

—

#16810344

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₅₃×#334

—

C₂×#11719917

D₄

yes

—

C₂²×#88322

—

—

—

—

C₂×C₄

yes

—

C₂²⋊#88322

Q₈

yes

—

#13133128

—

C₄×#88322

—

C₄×#88322

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

—

—

—

—

—

—

C₂³×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#389

yes

characteristic

C₂²×C₅₃

#501

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1950

—

#1566

D₅₃

yes

—

—

—

100

C₂×C₅₃

—

101

C₂×C₅₃

#1943

yes

characteristic

102

C₂×C₅₃

#1950

yes

characteristic

103

C₂×C₅₃

#1953

yes

characteristic

104

#431

—

105

C₂×#108

—

106

—

106

#423

C₂×D₅₃

yes

—

107

C₂×C₄⋊C₈

—

108

C₄×M₁₆

—

109

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

110

#341

—

106

—

111

#335

—

112

#335

—

113

#334

—

114

#334

—

115

#322

—

106

—

116

#322

—

106

—

117

C₅₃

#9625

yes

characteristic, complemented, Sylow

118

C₂×M₁₆

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

106

—

121

C₂²×C₈

Dic₁₀₆

yes

characteristic

122

C₂²×C₈

Dic₁₀₆

yes

characteristic

123

C₂×C₄²

C₂²×D₅₃

yes

characteristic

124

#108

—

212

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

106

—

129

C₄⋊C₈

—

106

—

130

C₂²⋊C₈

—

131

#97

—

106

—

132

C₄×C₈

—

106

—

133

C₄×C₈

—

106

—

134

C₄×C₈

C₄×D₅₃

yes

characteristic

135

C₄×C₈

C₄×D₅₃

yes

characteristic

136

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

137

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

138

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

139

C₂²×C₄

D₄×D₅₃

yes

characteristic

140

C₂²×C₄

Q₈×D₅₃

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

106

—

144

M₁₆

—

106

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

106

—

149

C₂×C₈

—

106

—

150

C₂×C₈

—

106

—

151

C₂×C₈

—

212

—

152

C₂×C₈

—

212

—

153

C₂×C₈

#11719919

yes

characteristic

154

C₂×C₈

#11719919

yes

characteristic

155

C₂×C₈

#11719919

yes

characteristic

156

C₂×C₈

#11719919

yes

characteristic

157

C₄²

C₂×C₄×D₅₃

yes

characteristic

158

C₄²

#11719947

yes

characteristic

159

C₄²

#11719949

yes

characteristic

160

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

161

C₂³

#13133216

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

C₂×#11719919

yes

characteristic

166

C₂×C₄

#13133179

yes

characteristic

167

C₂×C₄

#13133182

yes

characteristic

168

C₂×C₄

#13133188

yes

characteristic

169

C₂×C₄

#13133193

yes

characteristic

170

C₂×C₄

#13133208

yes

characteristic

171

C₂×C₄

#13133244

yes

—

172

C₂×C₄

#13133246

yes

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

212

—

178

C₈

—

212

—

179

C₈

—

212

—

180

C₈

#13133152

yes

complemented

181

C₂²

—

182

C₂²

—

183

C₂²

#16810519

yes

characteristic

184

C₂²

#16811030

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

C₂×#13133152

yes

characteristic

189

C₄

#16810671

yes

characteristic

190

C₄

#16810729

yes

characteristic

191

C₄

#16810734

yes

characteristic

192

C₄

#16810866

yes

—

193

C₂

—

194

C₂

#20275451

yes

characteristic

195

C₂

#20275488

yes

characteristic

196

C₂

#20275503

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1566	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1566	D₅₃
hypocenter	C₅₃	#9625
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#13133188

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1566

C₂

socle

C₂²×C₅₃

#389

prime radical

C₂³×C₅₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
78. C₈×C₅₃	#108	semidirect
117. C₅₃	#9625	semidirect
180. C₈	#13133152	semidirect

H ≅ G/N

type

78. C₈×C₅₃

#108

semidirect

117. C₅₃

#9625

semidirect

180. C₈

#13133152

semidirect

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

subgroup	index
#20275488	2
C₂×#16810376	2
#20273004	2
#20272610	2
C₅₃×#1566	2
#9625	53

subgroup

index

C₅₃×#1566

Subgroups — #27282799

Conjugacy classes of subgroups

Complements

Maximal subgroups