Group #27282804 — subgroups

Subgroups — #27282804

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×#16810374	C₂	1	1	yes	maximal, characteristic
3	#20273004	C₂	1	1	yes	maximal, characteristic
4	#20272610	C₂	1	2	yes	maximal
5	C₅₃×#1567	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	#16810374	C₄	1	2	yes	—
12	#16810374	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×#335	—	2	1	no	—
18	C₅₃×#335	—	2	1	no	—
19	C₂×#11719917	D₄	1	1	yes	characteristic
20	C₂×#11719917	D₄	1	1	yes	characteristic
21	C₂²×#88322	—	2	2	no	—
22	#13133152	—	4	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	1	no	—
25	C₄⋊#88322	C₂×C₄	1	4	yes	—
26	C₂²⋊#88322	Q₈	1	2	yes	—
27	#13133128	—	2	1	no	—
28	C₄×#88322	—	2	1	no	—
29	C₄×#88322	—	2	1	no	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₅₃	—	4	2	no	—
44	M₁₆×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	4	1	no	—
47	C₂×C₈×C₅₃	—	4	2	no	—
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	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	1	yes	characteristic
55	C₄²×C₅₃	#41	1	1	yes	characteristic
56	#11719917	—	2	2	no	—
57	#11719917	—	2	2	no	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	—	2	1	no	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	4	1	no	—
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×C₄×C₅₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
74	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₅₃	#137	1	2	yes	—
76	C₈×C₅₃	—	2	1	no	—
77	C₈×C₅₃	—	2	1	no	—
78	C₈×C₅₃	—	2	1	no	—
79	C₈×C₅₃	—	8	2	no	—
80	C₈×C₅₃	#108	1	2	yes	complemented
81	#88322	—	4	2	no	—
82	#88322	—	4	4	no	—
83	#88322	—	4	1	no	—
84	#88322	—	4	1	no	—
85	#9626	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	#389	1	1	yes	characteristic
89	C₂²×C₅₃	#501	1	2	yes	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
94	C₄×C₅₃	#430	1	1	yes	characteristic
95	C₄×C₅₃	#440	1	2	yes	—
96	C₄×C₅₃	#469	1	1	yes	characteristic
97	C₄×C₅₃	#470	1	1	yes	characteristic
98	#1950	—	53	2	no	—
99	#1567	D₅₃	1	1	yes	characteristic
100	C₂×#334	—	53	1	no	—
101	#1451	—	53	1	no	—
102	#1301	—	53	2	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	#1944	1	1	yes	characteristic
105	C₂×C₅₃	#1950	1	1	yes	characteristic
106	C₂×C₅₃	#1953	1	1	yes	characteristic
107	#431	—	53	1	no	—
108	C₂×#108	—	106	2	no	—
109	#423	C₂×D₅₃	1	2	yes	—
110	C₂×C₄⋊C₈	—	53	2	no	—
111	C₄×M₁₆	—	53	1	no	—
112	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	#341	—	106	2	no	—
114	#335	—	2	1	no	—
115	#335	—	2	1	no	—
116	#334	—	53	2	no	—
117	#334	—	53	2	no	—
118	#322	—	106	1	no	—
119	#322	—	106	1	no	—
120	C₅₃	#9626	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	—	53	1	no	—
123	C₂×M₁₆	—	53	1	no	—
124	C₂²×C₈	—	106	2	no	—
125	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
126	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	#108	—	212	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	53	4	no	—
132	C₄⋊C₈	—	106	2	no	—
133	C₄⋊C₈	—	106	1	no	—
134	C₂²⋊C₈	—	53	2	no	—
135	#97	—	106	1	no	—
136	C₄×C₈	—	106	1	no	—
137	C₄×C₈	—	106	1	no	—
138	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
139	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
144	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
145	M₁₆	—	4	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	106	2	no	—
148	M₁₆	—	106	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	106	4	no	—
153	C₂×C₈	—	106	1	no	—
154	C₂×C₈	—	106	1	no	—
155	C₂×C₈	—	106	2	no	—
156	C₂×C₈	—	212	2	no	—
157	C₂×C₈	—	212	2	no	—
158	C₂×C₈	#11719919	1	1	yes	characteristic
159	C₂×C₈	#11719919	1	1	yes	characteristic
160	C₂×C₈	#11719919	1	1	yes	characteristic
161	C₂×C₈	#11719919	1	1	yes	characteristic
162	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
163	C₄²	#11719947	1	1	yes	characteristic
164	C₄²	#11719949	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂³	#13133216	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
171	C₂×C₄	#13133179	1	1	yes	characteristic
172	C₂×C₄	#13133182	1	1	yes	characteristic
173	C₂×C₄	#13133188	1	1	yes	characteristic
174	C₂×C₄	#13133193	1	1	yes	characteristic
175	C₂×C₄	#13133208	1	1	yes	characteristic
176	C₂×C₄	#13133241	1	2	yes	—
177	C₂×C₄	#13133249	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	8	2	no	—
182	C₈	—	212	2	no	—
183	C₈	—	212	1	no	—
184	C₈	—	212	1	no	—
185	C₈	—	212	4	no	—
186	C₈	#13133152	1	2	yes	complemented
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	#16810519	1	1	yes	characteristic
190	C₂²	#16811031	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#13133152	1	1	yes	characteristic
195	C₄	#16810671	1	1	yes	characteristic
196	C₄	#16810722	1	2	yes	—
197	C₄	#16810873	1	1	yes	characteristic
198	C₄	#16810878	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	#20275456	1	1	yes	characteristic
201	C₂	#20275488	1	1	yes	characteristic
202	C₂	#20275503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275488

C₂

yes

maximal

C₂×#16810374

C₂

yes

maximal, characteristic

#20273004

C₂

yes

maximal, characteristic

#20272610

C₂

yes

maximal

C₅₃×#1567

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

characteristic

C₂×#13133152

—

C₂×C₄⋊#88322

C₂²

yes

—

C₄×#11719917

C₂²

yes

characteristic

#16810387

—

#16810374

C₄

yes

—

#16810374

C₄

yes

—

#16810345

—

#16810344

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₅₃×#335

—

C₂×#11719917

D₄

yes

characteristic

C₂×#11719917

D₄

yes

characteristic

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

—

D₅₃

yes

characteristic

100

C₂×#334

—

101

#1451

—

102

#1301

—

103

C₂×C₅₃

—

104

C₂×C₅₃

#1944

yes

characteristic

105

C₂×C₅₃

#1950

yes

characteristic

106

C₂×C₅₃

#1953

yes

characteristic

107

#431

—

108

C₂×#108

—

106

—

109

#423

C₂×D₅₃

yes

—

110

C₂×C₄⋊C₈

—

111

C₄×M₁₆

—

112

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

113

#341

—

106

—

114

#335

—

115

#335

—

116

#334

—

117

#334

—

118

#322

—

106

—

119

#322

—

106

—

120

C₅₃

#9626

yes

characteristic, complemented, Sylow

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

124

C₂²×C₈

—

106

—

125

C₂²×C₈

Dic₁₀₆

yes

characteristic

126

C₂²×C₈

Dic₁₀₆

yes

characteristic

127

C₂×C₄²

C₂²×D₅₃

yes

characteristic

128

#108

—

212

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

106

—

133

C₄⋊C₈

—

106

—

134

C₂²⋊C₈

—

135

#97

—

106

—

136

C₄×C₈

—

106

—

137

C₄×C₈

—

106

—

138

C₄×C₈

C₄×D₅₃

yes

characteristic

139

C₄×C₈

C₄×D₅₃

yes

characteristic

140

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

141

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

142

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

143

C₂²×C₄

D₄×D₅₃

yes

characteristic

144

C₂²×C₄

Q₈×D₅₃

yes

characteristic

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

106

—

148

M₁₆

—

106

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

106

—

153

C₂×C₈

—

106

—

154

C₂×C₈

—

106

—

155

C₂×C₈

—

106

—

156

C₂×C₈

—

212

—

157

C₂×C₈

—

212

—

158

C₂×C₈

#11719919

yes

characteristic

159

C₂×C₈

#11719919

yes

characteristic

160

C₂×C₈

#11719919

yes

characteristic

161

C₂×C₈

#11719919

yes

characteristic

162

C₄²

C₂×C₄×D₅₃

yes

characteristic

163

C₄²

#11719947

yes

characteristic

164

C₄²

#11719949

yes

characteristic

165

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

166

C₂³

#13133216

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

C₂×#11719919

yes

characteristic

171

C₂×C₄

#13133179

yes

characteristic

172

C₂×C₄

#13133182

yes

characteristic

173

C₂×C₄

#13133188

yes

characteristic

174

C₂×C₄

#13133193

yes

characteristic

175

C₂×C₄

#13133208

yes

characteristic

176

C₂×C₄

#13133241

yes

—

177

C₂×C₄

#13133249

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

212

—

183

C₈

—

212

—

184

C₈

—

212

—

185

C₈

—

212

—

186

C₈

#13133152

yes

complemented

187

C₂²

—

188

C₂²

—

189

C₂²

#16810519

yes

characteristic

190

C₂²

#16811031

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

C₂×#13133152

yes

characteristic

195

C₄

#16810671

yes

characteristic

196

C₄

#16810722

yes

—

197

C₄

#16810873

yes

characteristic

198

C₄

#16810878

yes

characteristic

199

C₂

—

200

C₂

#20275456

yes

characteristic

201

C₂

#20275488

yes

characteristic

202

C₂

#20275503

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1567	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1567	D₅₃
hypocenter	C₅₃	#9626
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#13133188

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1567

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 ◁ #1567

N	H ≅ G/N	type
80. C₈×C₅₃	#108	semidirect
120. C₅₃	#9626	semidirect
186. C₈	#13133152	semidirect

H ≅ G/N

type

80. C₈×C₅₃

#108

semidirect

120. C₅₃

#9626

semidirect

186. 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×#16810374	2
#20273004	2
#20272610	2
C₅₃×#1567	2
#9626	53

subgroup

index

C₅₃×#1567

Subgroups — #27282804

Conjugacy classes of subgroups

Complements

Maximal subgroups