Group #27541744 — subgroups

Subgroups — #27541744

Order 13568.

506 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 167 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	#20287721	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276944	C₂	1	3	yes	maximal, complemented
3	#20276854	C₂	1	3	yes	maximal, complemented
4	#20276760	C₂	1	3	yes	maximal
5	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13133161	C₂	1	3	yes	maximal
7	Q₈×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16811352	—	2	1	no	—
9	C₂×#13133161	C₂²	1	3	yes	—
10	#16811003	C₂²	1	3	yes	—
11	#16810834	—	2	3	no	—
12	C₄×Dic₂₁₂	C₂²	1	3	yes	—
13	#16810450	C₂²	1	3	yes	—
14	C₄⋊#11719908	C₂²	1	3	yes	—
15	#16810426	C₂²	1	6	yes	—
16	#16810426	C₂²	1	3	yes	—
17	#16810424	C₂²	1	1	yes	characteristic
18	C₄×#11719908	C₂²	1	3	yes	—
19	D₄×Q₈×C₅₃	—	2	1	no	—
20	C₅₃×#500	C₂²	1	3	yes	—
21	C₅₃×#474	—	2	3	no	—
22	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×D₈×C₅₃	C₂²	1	3	yes	—
24	C₂×Dic₂₁₂	—	2	3	no	—
25	C₂×Dic₂₁₂	C₂³	1	3	yes	—
26	#13133217	—	2	3	no	—
27	#13133208	—	4	3	no	—
28	Q₈×Dic₅₃	—	4	1	no	—
29	C₄×Dic₁₀₆	—	2	3	no	—
30	#13133161	—	2	6	no	—
31	C₂×#11719908	C₂³	1	3	yes	—
32	C₂×#11719908	C₂³	1	1	yes	characteristic
33	#13133151	C₂³	1	1	yes	characteristic
34	#13133142	—	2	3	no	—
35	C₂²×Q₈×C₅₃	—	4	1	no	—
36	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₅₃	—	2	3	no	—
38	C₂²⋊Q₈×C₅₃	—	4	3	no	—
39	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×D₄×C₅₃	—	2	3	no	—
41	#107×C₅₃	—	2	3	no	—
42	#107×C₅₃	C₂³	1	3	yes	—
43	C₄⋊C₈×C₅₃	D₄	1	3	yes	—
44	#102×C₅₃	—	2	3	no	—
45	C₄×C₈×C₅₃	C₂³	1	3	yes	—
46	C₂×Q₈×C₅₃	—	4	3	no	—
47	C₂×Q₈×C₅₃	—	8	4	no	—
48	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
49	C₂×D₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	4	3	no	—
51	C₂×Dic₁₀₆	—	2	3	no	—
52	D₈×C₅₃	—	2	1	no	—
53	D₈×C₅₃	#41	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	—	2	3	no	—
57	C₄⋊C₄×C₅₃	—	4	3	no	—
58	C₄⋊C₄×C₅₃	D₈	1	3	yes	—
59	C₂²⋊C₄×C₅₃	—	4	3	no	—
60	C₄²×C₅₃	C₂×D₄	1	3	yes	—
61	C₄⋊Dic₅₃	—	2	1	no	—
62	C₄⋊Dic₅₃	—	4	3	no	—
63	C₄⋊Dic₅₃	—	4	3	no	—
64	#11719919	—	4	3	no	—
65	C₄×Dic₅₃	—	4	3	no	—
66	Dic₂₁₂	—	2	3	no	—
67	Dic₂₁₂	—	2	3	no	—
68	Dic₂₁₂	—	8	3	no	—
69	#11719908	—	2	3	no	—
70	#11719908	—	2	1	no	—
71	C₂³×C₅₃	—	4	1	no	—
72	Q₈×C₅₃	—	8	6	no	—
73	Q₈×C₅₃	SD₃₂	1	4	yes	complemented
74	D₄×C₅₃	—	2	2	no	—
75	D₄×C₅₃	—	4	1	no	—
76	C₂×C₄×C₅₃	—	4	3	no	—
77	C₂×C₄×C₅₃	—	8	6	no	—
78	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
79	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
80	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₅₃	—	4	3	no	—
82	C₂×Dic₅₃	—	4	1	no	—
83	Dic₁₀₆	—	4	3	no	—
84	Dic₁₀₆	—	4	3	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	—	4	2	no	—
90	C₂²×C₅₃	—	8	1	no	—
91	C₂²×C₅₃	—	8	1	no	—
92	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
93	C₄×C₅₃	—	2	3	no	—
94	C₄×C₅₃	—	8	3	no	—
95	C₄×C₅₃	C₂×SD₃₂	1	6	yes	—
96	C₄×C₅₃	#548	1	1	yes	characteristic
97	C₄×C₅₃	#577	1	1	yes	characteristic
98	Dic₅₃	—	8	3	no	—
99	Dic₅₃	—	8	1	no	—
100	#3395	—	53	1	no	—
101	Q₈×D₈	D₅₃	1	1	yes	characteristic
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	—	8	2	no	—
108	C₂×C₅₃	#3358	1	1	yes	characteristic
109	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
110	C₂×C₅₃	#3417	1	1	yes	characteristic
111	#557	—	106	1	no	—
112	D₄×Q₈	—	2	1	no	—
113	C₂×SD₃₂	—	53	3	no	—
114	#500	C₂×D₅₃	1	3	yes	—
115	#494	—	53	3	no	—
116	#474	—	2	3	no	—
117	#462	—	106	3	no	—
118	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
119	C₄×Q₁₆	—	53	3	no	—
120	C₄×D₈	C₂×D₅₃	1	3	yes	—
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	—	4	1	no	—
129	C₂×Q₁₆	—	53	3	no	—
130	C₂×Q₁₆	—	106	3	no	—
131	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	3	no	—
133	C₄⋊Q₈	—	106	3	no	—
134	#125	—	212	3	no	—
135	C₂²⋊Q₈	—	4	3	no	—
136	C₄×Q₈	—	106	3	no	—
137	C₄×Q₈	—	212	1	no	—
138	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₄×D₄	—	2	3	no	—
140	SD₃₂	—	106	6	no	—
141	C₂×C₁₆	—	53	1	no	—
142	C₂×C₁₆	—	53	3	no	—
143	#107	—	2	3	no	—
144	#107	—	53	1	no	—
145	#107	C₂²×D₅₃	1	3	yes	—
146	C₄⋊C₈	C₂²⋊D₅₃	1	3	yes	—
147	#103	—	106	3	no	—
148	#102	—	2	3	no	—
149	C₄×C₈	C₂²×D₅₃	1	3	yes	—
150	C₂×Q₈	—	4	3	no	—
151	C₂×Q₈	—	8	4	no	—
152	C₂×Q₈	—	106	3	no	—
153	C₂×Q₈	#11719922	1	1	yes	characteristic
154	C₂×D₄	—	2	1	no	—
155	C₂²×C₄	—	4	3	no	—
156	Q₁₆	—	106	3	no	—
157	Q₁₆	—	106	3	no	—
158	Q₁₆	—	424	3	no	—
159	D₈	—	2	1	no	—
160	D₈	#11719949	1	2	yes	—
161	C₂×C₈	C₂×C₂²⋊D₅₃	1	3	yes	—
162	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
163	C₄⋊C₄	—	2	3	no	—
164	C₄⋊C₄	—	4	3	no	—
165	C₄⋊C₄	—	106	1	no	—
166	C₄⋊C₄	—	212	3	no	—
167	C₄⋊C₄	—	212	3	no	—
168	C₄⋊C₄	—	212	3	no	—
169	C₄⋊C₄	#11719922	1	3	yes	—
170	C₂²⋊C₄	—	4	3	no	—
171	C₄²	—	212	3	no	—
172	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
173	C₁₆	—	106	3	no	—
174	C₁₆	—	106	1	no	—
175	C₂³	—	4	1	no	—
176	Q₈	—	8	6	no	—
177	Q₈	—	212	3	no	—
178	Q₈	—	212	3	no	—
179	Q₈	#13133161	1	4	yes	complemented
180	D₄	—	2	2	no	—
181	D₄	—	4	1	no	—
182	C₂×C₄	—	4	3	no	—
183	C₂×C₄	—	8	6	no	—
184	C₂×C₄	—	212	1	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	C₂×#11719949	1	1	yes	characteristic
191	C₈	#13133265	1	1	yes	characteristic
192	C₂²	—	4	2	no	—
193	C₂²	—	8	1	no	—
194	C₂²	—	8	1	no	—
195	C₂²	C₂²×#11719922	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	8	3	no	—
198	C₄	—	424	3	no	—
199	C₄	—	424	1	no	—
200	C₄	C₂×#13133161	1	6	yes	—
201	C₄	#16811284	1	1	yes	characteristic
202	C₄	#16811431	1	1	yes	characteristic
203	C₂	—	8	2	no	—
204	C₂	#20287350	1	1	yes	characteristic
205	C₂	C₂²×#13133161	1	1	yes	characteristic
206	C₂	#20287866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20287721

C₂

yes

maximal, characteristic, complemented

#20276944

C₂

yes

maximal, complemented

#20276854

C₂

yes

maximal, complemented

#20276760

C₂

yes

maximal

Q₈×#11719908

C₂

yes

maximal, characteristic, complemented

C₄×#13133161

C₂

yes

maximal

Q₈×D₈×C₅₃

C₂

yes

maximal, characteristic

#16811352

—

C₂×#13133161

C₂²

yes

—

#16811003

C₂²

yes

—

#16810834

—

C₄×Dic₂₁₂

C₂²

yes

—

#16810450

C₂²

yes

—

C₄⋊#11719908

C₂²

yes

—

#16810426

C₂²

yes

—

#16810426

C₂²

yes

—

#16810424

C₂²

yes

characteristic

C₄×#11719908

C₂²

yes

—

D₄×Q₈×C₅₃

—

C₅₃×#500

C₂²

yes

—

C₅₃×#474

—

C₈×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₈×C₅₃

C₂²

yes

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂³

yes

—

#13133217

—

#13133208

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

—

#13133161

—

C₂×#11719908

C₂³

yes

—

C₂×#11719908

C₂³

yes

characteristic

#13133151

C₂³

yes

characteristic

#13133142

—

C₂²×Q₈×C₅₃

—

C₂×D₈×C₅₃

C₂³

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

D₄

yes

characteristic

C₄×D₄×C₅₃

—

#107×C₅₃

—

#107×C₅₃

C₂³

yes

—

C₄⋊C₈×C₅₃

D₄

yes

—

#102×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

characteristic

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₈

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

#11719908

—

#11719908

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

SD₃₂

yes

complemented

D₄×C₅₃

—

D₄×C₅₃

—

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

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×#41

yes

characteristic

C₈×C₅₃

2₋¹⁺⁴

yes

characteristic

#33926

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₃₂

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

100

#3395

—

101

Q₈×D₈

D₅₃

yes

characteristic

102

#2241

—

103

#2225

—

104

#2208

—

105

Q₈×C₁₆

—

106

C₄×SD₃₂

—

107

C₂×C₅₃

—

108

C₂×C₅₃

#3358

yes

characteristic

109

C₂×C₅₃

C₂²×SD₃₂

yes

characteristic

110

C₂×C₅₃

#3417

yes

characteristic

111

#557

—

106

—

112

D₄×Q₈

—

113

C₂×SD₃₂

—

114

#500

C₂×D₅₃

yes

—

115

#494

—

116

#474

—

117

#462

—

106

—

118

C₈×Q₈

C₂×D₅₃

yes

characteristic

119

C₄×Q₁₆

—

120

C₄×D₈

C₂×D₅₃

yes

—

121

#367

—

122

C₄⋊C₁₆

—

123

#358

—

124

#358

—

125

#357

—

126

C₄×C₁₆

—

127

C₅₃

#33926

yes

characteristic, complemented, Sylow

128

C₂²×Q₈

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

106

—

131

C₂×D₈

C₂²×D₅₃

yes

characteristic

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

106

—

134

#125

—

212

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

106

—

137

C₄×Q₈

—

212

—

138

C₄×Q₈

C₂²⋊D₅₃

yes

characteristic

139

C₄×D₄

—

140

SD₃₂

—

106

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

#107

—

144

#107

—

145

#107

C₂²×D₅₃

yes

—

146

C₄⋊C₈

C₂²⋊D₅₃

yes

—

147

#103

—

106

—

148

#102

—

149

C₄×C₈

C₂²×D₅₃

yes

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

106

—

153

C₂×Q₈

#11719922

yes

characteristic

154

C₂×D₄

—

155

C₂²×C₄

—

156

Q₁₆

—

106

—

157

Q₁₆

—

106

—

158

Q₁₆

—

424

—

159

D₈

—

160

D₈

#11719949

yes

—

161

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

162

C₂×C₈

C₂³×D₅₃

yes

characteristic

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

106

—

166

C₄⋊C₄

—

212

—

167

C₄⋊C₄

—

212

—

168

C₄⋊C₄

—

212

—

169

C₄⋊C₄

#11719922

yes

—

170

C₂²⋊C₄

—

171

C₄²

—

212

—

172

C₄²

C₂×C₂²⋊D₅₃

yes

—

173

C₁₆

—

106

—

174

C₁₆

—

106

—

175

C₂³

—

176

Q₈

—

177

Q₈

—

212

—

178

Q₈

—

212

—

179

Q₈

#13133161

yes

complemented

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

212

—

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

C₂×#11719949

yes

characteristic

191

C₈

#13133265

yes

characteristic

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

C₂²×#11719922

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

424

—

199

C₄

—

424

—

200

C₄

C₂×#13133161

yes

—

201

C₄

#16811284

yes

characteristic

202

C₄

#16811431

yes

characteristic

203

C₂

—

204

C₂

#20287350

yes

characteristic

205

C₂

C₂²×#13133161

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	Q₈×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₈×C₅₃	C₂³
hypercenter	Q₈×D₈	D₅₃
hypocenter	C₅₃	#33926
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×D₈

subgroup

G/H

center

C₂²

C₂²×#11719922

commutator, cosocle

C₂×C₈×C₅₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₃

Fitting

Q₈×D₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₈

prime radical

C₂×D₈×C₅₃

C₂³

hypercenter

Q₈×D₈

D₅₃

hypocenter

C₅₃

#33926

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×D₈

N	H ≅ G/N	type
1. #20287721	C₂	semidirect
2. #20276944	C₂	semidirect
3. #20276854	C₂	semidirect
5. Q₈×#11719908	C₂	semidirect
73. Q₈×C₅₃	SD₃₂	semidirect
127. C₅₃	#33926	semidirect
179. Q₈	#13133161	semidirect

H ≅ G/N

type

1. #20287721

C₂

semidirect

2. #20276944

C₂

semidirect

3. #20276854

C₂

semidirect

5. Q₈×#11719908

C₂

semidirect

73. Q₈×C₅₃

SD₃₂

semidirect

127. C₅₃

#33926

semidirect

179. Q₈

#13133161

semidirect

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

subgroup	index
#20287721	2
#20276944	2
#20276854	2
#20276760	2
Q₈×#11719908	2
C₄×#13133161	2
Q₈×D₈×C₅₃	2
#33926	53

subgroup

index

Q₈×D₈×C₅₃

Subgroups — #27541744

Conjugacy classes of subgroups

Complements

Maximal subgroups