Group #27541740 — subgroups

Subgroups — #27541740

Order 13568.

506 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 165 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	Q₈×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#20276942	C₂	1	3	yes	maximal, complemented
3	#20276857	C₂	1	3	yes	maximal, complemented
4	#20276757	C₂	1	3	yes	maximal
5	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13133160	C₂	1	3	yes	maximal
7	C₅₃×#3395	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×D₂₁₂	—	2	1	no	—
9	C₂×#13133160	C₂²	1	3	yes	—
10	#16811028	C₂²	1	3	yes	—
11	#16810904	—	2	3	no	—
12	C₄×D₄₂₄	C₂²	1	3	yes	—
13	#16810450	C₂²	1	3	yes	—
14	C₄⋊#11719908	C₂²	1	3	yes	—
15	#16810427	C₂²	1	6	yes	—
16	#16810427	C₂²	1	3	yes	—
17	#16810423	C₂²	1	1	yes	characteristic
18	C₄×#11719908	C₂²	1	3	yes	—
19	C₅₃×#557	—	2	1	no	—
20	C₅₃×#494	C₂²	1	3	yes	—
21	C₅₃×#462	—	2	3	no	—
22	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×Q₁₆×C₅₃	C₂²	1	3	yes	—
24	C₂×Q₈×D₅₃	—	4	1	no	—
25	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
26	#13133217	—	2	3	no	—
27	#13133197	—	4	3	no	—
28	C₄×D₂₁₂	—	2	3	no	—
29	#13133160	—	2	6	no	—
30	C₂×#11719908	C₂³	1	3	yes	—
31	C₂×#11719908	C₂³	1	1	yes	characteristic
32	#13133151	—	2	3	no	—
33	#13133151	C₂³	1	3	yes	—
34	#13133139	—	2	3	no	—
35	C₂×Q₁₆×C₅₃	—	2	3	no	—
36	C₂×Q₁₆×C₅₃	C₂³	1	3	yes	—
37	C₄⋊Q₈×C₅₃	—	2	3	no	—
38	#125×C₅₃	—	4	3	no	—
39	C₄×Q₈×C₅₃	—	2	3	no	—
40	C₄×Q₈×C₅₃	—	4	1	no	—
41	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
42	#107×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	D₄	1	3	yes	—
44	#103×C₅₃	—	2	3	no	—
45	C₄×C₈×C₅₃	C₂³	1	3	yes	—
46	C₂×Q₈×C₅₃	—	2	3	no	—
47	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
48	Q₈×D₅₃	—	8	4	no	—
49	C₂×D₂₁₂	—	2	1	no	—
50	C₂×C₄×D₅₃	—	4	3	no	—
51	C₂×Dic₁₀₆	—	4	3	no	—
52	Q₁₆×C₅₃	—	2	3	no	—
53	Q₁₆×C₅₃	—	2	3	no	—
54	Q₁₆×C₅₃	—	8	3	no	—
55	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
56	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₄⋊C₄×C₅₃	—	4	3	no	—
59	C₄⋊C₄×C₅₃	—	4	3	no	—
60	C₄⋊C₄×C₅₃	—	4	3	no	—
61	C₄⋊C₄×C₅₃	D₈	1	3	yes	—
62	C₄²×C₅₃	—	4	3	no	—
63	C₄²×C₅₃	C₂×D₄	1	3	yes	—
64	#11719921	—	4	3	no	—
65	C₄⋊Dic₅₃	—	2	3	no	—
66	C₄⋊Dic₅₃	—	4	3	no	—
67	D₄₂₄	—	2	1	no	—
68	D₄₂₄	#41	1	2	yes	—
69	#11719908	—	2	3	no	—
70	#11719908	—	2	1	no	—
71	C₂²×D₅₃	—	4	1	no	—
72	Q₈×C₅₃	—	4	3	no	—
73	Q₈×C₅₃	—	4	3	no	—
74	Q₈×C₅₃	SD₃₂	1	4	yes	complemented
75	C₂×C₄×C₅₃	—	4	3	no	—
76	C₂×C₄×C₅₃	—	4	1	no	—
77	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
79	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×Dic₅₃	—	4	3	no	—
81	D₂₁₂	—	2	2	no	—
82	D₂₁₂	—	4	1	no	—
83	C₄×D₅₃	—	8	6	no	—
84	Dic₁₀₆	—	8	6	no	—
85	C₈×C₅₃	—	2	3	no	—
86	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
87	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
88	#33926	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
90	C₂×D₅₃	—	4	2	no	—
91	C₂×D₅₃	—	8	1	no	—
92	C₂×D₅₃	—	8	1	no	—
93	C₄×C₅₃	—	2	3	no	—
94	C₄×C₅₃	—	8	3	no	—
95	C₄×C₅₃	—	8	1	no	—
96	C₄×C₅₃	C₂×SD₃₂	1	6	yes	—
97	C₄×C₅₃	#548	1	1	yes	characteristic
98	C₄×C₅₃	#577	1	1	yes	characteristic
99	Dic₅₃	—	8	3	no	—
100	#3395	D₅₃	1	1	yes	characteristic, complemented
101	Q₈×D₈	—	53	1	no	—
102	#2241	—	53	3	no	—
103	#2225	—	53	3	no	—
104	#2208	—	53	3	no	—
105	Q₈×C₁₆	—	53	1	no	—
106	C₄×SD₃₂	—	53	3	no	—
107	C₂×C₅₃	#3358	1	1	yes	characteristic
108	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
109	C₂×C₅₃	#3417	1	1	yes	characteristic
110	D₅₃	—	8	2	no	—
111	#557	—	2	1	no	—
112	D₄×Q₈	—	106	1	no	—
113	C₂×SD₃₂	—	53	3	no	—
114	#500	—	53	3	no	—
115	#494	C₂×D₅₃	1	3	yes	—
116	#474	—	106	3	no	—
117	#462	—	2	3	no	—
118	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
119	C₄×Q₁₆	C₂×D₅₃	1	3	yes	—
120	C₄×D₈	—	53	3	no	—
121	#367	—	53	3	no	—
122	C₄⋊C₁₆	—	53	3	no	—
123	#358	—	53	3	no	—
124	#358	—	53	6	no	—
125	#357	—	53	1	no	—
126	C₄×C₁₆	—	53	3	no	—
127	C₅₃	#33926	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	212	1	no	—
129	C₂×Q₁₆	—	2	3	no	—
130	C₂×Q₁₆	C₂²×D₅₃	1	3	yes	—
131	C₂×D₈	—	53	1	no	—
132	C₄⋊Q₈	—	2	3	no	—
133	C₄⋊Q₈	—	106	3	no	—
134	#125	—	4	3	no	—
135	C₂²⋊Q₈	—	212	3	no	—
136	C₄×Q₈	—	2	3	no	—
137	C₄×Q₈	—	4	1	no	—
138	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₄×D₄	—	106	3	no	—
140	SD₃₂	—	106	6	no	—
141	C₂×C₁₆	—	53	1	no	—
142	C₂×C₁₆	—	53	3	no	—
143	#107	—	53	3	no	—
144	#107	—	106	3	no	—
145	#107	C₂²×D₅₃	1	1	yes	characteristic
146	C₄⋊C₈	C₂²⋊D₅₃	1	3	yes	—
147	#103	—	2	3	no	—
148	#102	—	106	3	no	—
149	C₄×C₈	C₂²×D₅₃	1	3	yes	—
150	C₂×Q₈	—	2	3	no	—
151	C₂×Q₈	—	424	4	no	—
152	C₂×Q₈	—	212	3	no	—
153	C₂×Q₈	#11719922	1	1	yes	characteristic
154	C₂×D₄	—	106	1	no	—
155	C₂²×C₄	—	212	3	no	—
156	Q₁₆	—	2	3	no	—
157	Q₁₆	—	2	3	no	—
158	Q₁₆	—	8	3	no	—
159	D₈	—	53	2	no	—
160	D₈	—	106	1	no	—
161	C₂×C₈	C₂×C₂²⋊D₅₃	1	3	yes	—
162	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	4	3	no	—
165	C₄⋊C₄	—	4	3	no	—
166	C₄⋊C₄	—	4	3	no	—
167	C₄⋊C₄	—	106	3	no	—
168	C₄⋊C₄	—	212	3	no	—
169	C₄⋊C₄	#11719922	1	3	yes	—
170	C₂²⋊C₄	—	212	3	no	—
171	C₄²	—	4	3	no	—
172	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
173	C₁₆	—	106	3	no	—
174	C₁₆	—	106	1	no	—
175	C₂³	—	212	1	no	—
176	Q₈	—	4	3	no	—
177	Q₈	—	4	3	no	—
178	Q₈	—	424	6	no	—
179	Q₈	#13133160	1	4	yes	complemented
180	D₄	—	106	2	no	—
181	D₄	—	212	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	3	no	—
184	C₂×C₄	—	424	6	no	—
185	C₂×C₄	—	212	3	no	—
186	C₂×C₄	C₂×#11719922	1	3	yes	—
187	C₂×C₄	C₂×#11719922	1	3	yes	—
188	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
189	C₈	—	2	3	no	—
190	C₈	#41×D₅₃	1	1	yes	characteristic
191	C₈	#13133264	1	1	yes	characteristic
192	C₂²	—	424	1	no	—
193	C₂²	—	424	1	no	—
194	C₂²	—	212	2	no	—
195	C₂²	C₂²×#11719922	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	8	3	no	—
198	C₄	—	8	1	no	—
199	C₄	—	424	3	no	—
200	C₄	C₂×#13133160	1	6	yes	—
201	C₄	#16811280	1	1	yes	characteristic
202	C₄	#16811434	1	1	yes	characteristic
203	C₂	—	424	2	no	—
204	C₂	#20287346	1	1	yes	characteristic
205	C₂	C₂²×#13133160	1	1	yes	characteristic
206	C₂	#20287866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₄₂₄

C₂

yes

maximal, characteristic

#20276942

C₂

yes

maximal, complemented

#20276857

C₂

yes

maximal, complemented

#20276757

C₂

yes

maximal

Q₈×#11719908

C₂

yes

maximal, characteristic, complemented

C₄×#13133160

C₂

yes

maximal

C₅₃×#3395

C₂

yes

maximal, characteristic, complemented

Q₈×D₂₁₂

—

C₂×#13133160

C₂²

yes

—

#16811028

C₂²

yes

—

#16810904

—

C₄×D₄₂₄

C₂²

yes

—

#16810450

C₂²

yes

—

C₄⋊#11719908

C₂²

yes

—

#16810427

C₂²

yes

—

#16810427

C₂²

yes

—

#16810423

C₂²

yes

characteristic

C₄×#11719908

C₂²

yes

—

C₅₃×#557

—

C₅₃×#494

C₂²

yes

—

C₅₃×#462

—

C₈×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₁₆×C₅₃

C₂²

yes

—

C₂×Q₈×D₅₃

—

C₂×D₄₂₄

C₂³

yes

characteristic

#13133217

—

#13133197

—

C₄×D₂₁₂

—

#13133160

—

C₂×#11719908

C₂³

yes

—

C₂×#11719908

C₂³

yes

characteristic

#13133151

—

#13133151

C₂³

yes

—

#13133139

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂³

yes

—

C₄⋊Q₈×C₅₃

—

#125×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

D₄

yes

characteristic

#107×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

D₄

yes

—

#103×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

characteristic

Q₈×D₅₃

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₈

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

D₄₂₄

—

D₄₂₄

#41

yes

—

#11719908

—

#11719908

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

SD₃₂

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×#41

yes

characteristic

C₈×C₅₃

2₋¹⁺⁴

yes

characteristic

#33926

—

maximal, Sylow

C₂²×C₅₃

C₂²×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₃₂

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

100

#3395

D₅₃

yes

characteristic, complemented

101

Q₈×D₈

—

102

#2241

—

103

#2225

—

104

#2208

—

105

Q₈×C₁₆

—

106

C₄×SD₃₂

—

107

C₂×C₅₃

#3358

yes

characteristic

108

C₂×C₅₃

C₂²×SD₃₂

yes

characteristic

109

C₂×C₅₃

#3417

yes

characteristic

110

D₅₃

—

111

#557

—

112

D₄×Q₈

—

106

—

113

C₂×SD₃₂

—

114

#500

—

115

#494

C₂×D₅₃

yes

—

116

#474

—

106

—

117

#462

—

118

C₈×Q₈

C₂×D₅₃

yes

characteristic

119

C₄×Q₁₆

C₂×D₅₃

yes

—

120

C₄×D₈

—

121

#367

—

122

C₄⋊C₁₆

—

123

#358

—

124

#358

—

125

#357

—

126

C₄×C₁₆

—

127

C₅₃

#33926

yes

characteristic, complemented, Sylow

128

C₂²×Q₈

—

212

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

C₂²×D₅₃

yes

—

131

C₂×D₈

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

106

—

134

#125

—

135

C₂²⋊Q₈

—

212

—

136

C₄×Q₈

—

137

C₄×Q₈

—

138

C₄×Q₈

C₂²⋊D₅₃

yes

characteristic

139

C₄×D₄

—

106

—

140

SD₃₂

—

106

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

#107

—

144

#107

—

106

—

145

#107

C₂²×D₅₃

yes

characteristic

146

C₄⋊C₈

C₂²⋊D₅₃

yes

—

147

#103

—

148

#102

—

106

—

149

C₄×C₈

C₂²×D₅₃

yes

—

150

C₂×Q₈

—

151

C₂×Q₈

—

424

—

152

C₂×Q₈

—

212

—

153

C₂×Q₈

#11719922

yes

characteristic

154

C₂×D₄

—

106

—

155

C₂²×C₄

—

212

—

156

Q₁₆

—

157

Q₁₆

—

158

Q₁₆

—

159

D₈

—

160

D₈

—

106

—

161

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

162

C₂×C₈

C₂³×D₅₃

yes

characteristic

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

106

—

168

C₄⋊C₄

—

212

—

169

C₄⋊C₄

#11719922

yes

—

170

C₂²⋊C₄

—

212

—

171

C₄²

—

172

C₄²

C₂×C₂²⋊D₅₃

yes

—

173

C₁₆

—

106

—

174

C₁₆

—

106

—

175

C₂³

—

212

—

176

Q₈

—

177

Q₈

—

178

Q₈

—

424

—

179

Q₈

#13133160

yes

complemented

180

D₄

—

106

—

181

D₄

—

212

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

424

—

185

C₂×C₄

—

212

—

186

C₂×C₄

C₂×#11719922

yes

—

187

C₂×C₄

C₂×#11719922

yes

—

188

C₂×C₄

C₂²×C₂²⋊D₅₃

yes

characteristic

189

C₈

—

190

C₈

#41×D₅₃

yes

characteristic

191

C₈

#13133264

yes

characteristic

192

C₂²

—

424

—

193

C₂²

—

424

—

194

C₂²

—

212

—

195

C₂²

C₂²×#11719922

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

424

—

200

C₄

C₂×#13133160

yes

—

201

C₄

#16811280

yes

characteristic

202

C₄

#16811434

yes

characteristic

203

C₂

—

424

—

204

C₂

#20287346

yes

characteristic

205

C₂

C₂²×#13133160

yes

characteristic

206

C₂

#20287866

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3395	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄₂₄	C₂³
hypercenter	#3395	D₅₃
hypocenter	C₅₃	#33926
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

subgroup

G/H

center

C₂²

C₂²×#11719922

commutator, cosocle

C₂×C₈×C₅₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₃

Fitting

C₅₃×#3395

C₂

socle

C₂²×C₅₃

C₂²×D₈

prime radical

C₂×D₄₂₄

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

N	H ≅ G/N	type
2. #20276942	C₂	semidirect
3. #20276857	C₂	semidirect
5. Q₈×#11719908	C₂	semidirect
7. C₅₃×#3395	C₂	semidirect
74. Q₈×C₅₃	SD₃₂	semidirect
100. #3395	D₅₃	semidirect
127. C₅₃	#33926	semidirect
179. Q₈	#13133160	semidirect

H ≅ G/N

type

2. #20276942

C₂

semidirect

3. #20276857

C₂

semidirect

5. Q₈×#11719908

C₂

semidirect

7. C₅₃×#3395

C₂

semidirect

74. Q₈×C₅₃

SD₃₂

semidirect

100. #3395

D₅₃

semidirect

127. C₅₃

#33926

semidirect

179. Q₈

#13133160

semidirect

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

subgroup	index
Q₈×D₄₂₄	2
#20276942	2
#20276857	2
#20276757	2
Q₈×#11719908	2
C₄×#13133160	2
C₅₃×#3395	2
#33926	53

subgroup

index

Q₈×D₄₂₄

C₅₃×#3395

Subgroups — #27541740

Conjugacy classes of subgroups

Complements

Maximal subgroups