Group #27302673 — subgroups

Subgroups — #27302673

Order 13568.

366 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	2	yes	maximal
4	#20272806	C₂	1	2	yes	maximal
5	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133209	—	2	1	no	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	#16811025	—	2	2	no	—
9	#16811025	C₄	1	4	yes	complemented
10	C₂×#13133151	C₂²	1	1	yes	characteristic
11	C₂×#13133151	C₂²	1	1	yes	characteristic
12	C₂×#13133149	—	2	1	no	—
13	C₄⋊#11719908	—	2	2	no	—
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₅₃×#340	—	2	1	no	—
18	#13133209	—	2	4	no	—
19	#13133209	—	4	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	1	no	—
21	C₂×C₄⋊Dic₅₃	—	2	1	no	—
22	C₂×#11719919	—	4	1	no	—
23	C₂×#11719919	—	4	1	no	—
24	C₂×#11719908	—	2	2	no	—
25	C₂×#11719908	—	2	4	no	—
26	#13133151	—	2	1	no	—
27	#13133151	—	2	1	no	—
28	#13133151	C₂×C₄	1	2	yes	—
29	#13133151	C₂×C₄	1	2	yes	—
30	#13133149	—	2	4	no	—
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	—	4	1	no	—
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	#107×C₅₃	—	2	2	no	—
37	#107×C₅₃	Q₈	1	4	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	#95×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	1	no	—
46	C₂²×Dic₅₃	—	4	1	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₄²×C₅₃	—	8	2	no	—
56	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₅₃	SD₁₆	1	2	yes	—
58	C₄⋊Dic₅₃	—	2	2	no	—
59	C₄⋊Dic₅₃	—	2	2	no	—
60	C₄⋊Dic₅₃	—	4	1	no	—
61	C₄⋊Dic₅₃	—	4	1	no	—
62	#11719919	—	4	4	no	—
63	#11719919	—	4	4	no	—
64	#11719908	—	4	2	no	—
65	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	—	8	2	no	—
71	C₂×C₄×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	#102	1	2	yes	—
73	C₂×C₄×C₅₃	#102	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#137	1	2	yes	—
78	C₂×Dic₅₃	—	4	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₂×Dic₅₃	—	8	2	no	—
81	C₂×Dic₅₃	—	8	1	no	—
82	C₂×Dic₅₃	—	8	1	no	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	C₄×Q₈	1	2	yes	—
85	C₈×C₅₃	#124	1	1	yes	characteristic
86	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
87	#13277	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
89	C₂²×C₅₃	#474	1	2	yes	—
90	C₂²×C₅₃	#508	1	2	yes	—
91	C₂²×C₅₃	#509	1	1	yes	characteristic
92	C₂²×C₅₃	#510	1	1	yes	characteristic
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	8	4	no	—
96	C₄×C₅₃	#391	1	1	yes	characteristic
97	C₄×C₅₃	#419	1	1	yes	characteristic
98	C₄×C₅₃	#475	1	2	yes	—
99	Dic₅₃	—	8	2	no	—
100	Dic₅₃	—	8	2	no	—
101	C₂×#499	—	53	1	no	—
102	C₂×C₄⋊C₁₆	—	53	1	no	—
103	C₄×#107	D₅₃	1	1	yes	characteristic
104	#1380	—	53	2	no	—
105	#1380	—	53	2	no	—
106	C₂×C₅₃	#1988	1	1	yes	characteristic
107	C₂×C₅₃	#2138	1	1	yes	characteristic
108	C₂×C₅₃	#2139	1	1	yes	characteristic
109	C₂×C₅₃	#2228	1	2	yes	—
110	C₂×C₅₃	#2229	1	2	yes	—
111	C₂×#125	—	106	1	no	—
112	C₂²×C₁₆	—	53	2	no	—
113	#499	—	53	4	no	—
114	#499	—	106	2	no	—
115	C₂×#107	—	53	1	no	—
116	C₂×#107	—	53	1	no	—
117	C₂×#107	C₂×D₅₃	1	2	yes	—
118	C₂×#106	—	106	1	no	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₁₆	—	106	2	no	—
122	#340	—	2	1	no	—
123	C₅₃	#13277	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
126	#125	—	106	4	no	—
127	#125	—	212	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	106	1	no	—
130	C₂×C₄⋊C₄	—	106	1	no	—
131	C₂×C₄⋊C₄	—	212	1	no	—
132	C₂×C₄⋊C₄	—	212	1	no	—
133	C₂×C₄²	—	4	1	no	—
134	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂×C₁₆	—	106	2	no	—
136	C₂×C₁₆	—	106	4	no	—
137	#107	—	2	2	no	—
138	#107	—	53	2	no	—
139	#107	—	53	2	no	—
140	#107	—	106	1	no	—
141	#107	—	106	1	no	—
142	#107	Dic₁₀₆	1	4	yes	—
143	#106	—	106	4	no	—
144	C₄×C₈	C₄×D₅₃	1	2	yes	—
145	C₄×C₈	D₂₁₂	1	2	yes	—
146	#95	—	4	1	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	212	1	no	—
149	C₂²×C₄	—	212	1	no	—
150	C₂²×C₄	#11719922	1	1	yes	characteristic
151	C₂²×C₄	#11719922	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	#11719921	1	2	yes	—
155	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
157	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
158	C₂×C₈	#11719945	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	106	2	no	—
162	C₄⋊C₄	—	106	2	no	—
163	C₄⋊C₄	—	212	4	no	—
164	C₄⋊C₄	—	212	4	no	—
165	C₄⋊C₄	—	212	1	no	—
166	C₄⋊C₄	—	212	1	no	—
167	C₄²	—	8	2	no	—
168	C₄²	#11719921	1	2	yes	—
169	C₄²	#11719924	1	2	yes	—
170	C₁₆	—	212	2	no	—
171	C₂³	C₂×#11719922	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	424	2	no	—
179	C₂×C₄	—	424	1	no	—
180	C₂×C₄	—	424	1	no	—
181	C₂×C₄	—	212	2	no	—
182	C₂×C₄	—	212	2	no	—
183	C₂×C₄	#13133137	1	2	yes	—
184	C₂×C₄	#13133137	1	2	yes	—
185	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
186	C₂×C₄	#13133196	1	2	yes	—
187	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
188	C₂×C₄	#13133247	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	C₄×Dic₁₀₆	1	2	yes	—
191	C₈	#13133205	1	1	yes	characteristic
192	C₈	#13133217	1	1	yes	characteristic
193	C₂²	C₂×#13133137	1	1	yes	characteristic
194	C₂²	#16810903	1	2	yes	—
195	C₂²	#16811053	1	2	yes	—
196	C₂²	#16811059	1	1	yes	characteristic
197	C₂²	#16811064	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	8	4	no	—
201	C₄	—	424	2	no	—
202	C₄	—	424	2	no	—
203	C₄	#16810529	1	1	yes	characteristic
204	C₄	#16810631	1	1	yes	characteristic
205	C₄	#16810908	1	2	yes	—
206	C₂	#20275667	1	1	yes	characteristic
207	C₂	#20276420	1	1	yes	characteristic
208	C₂	#20276425	1	1	yes	characteristic
209	C₂	#20276872	1	2	yes	—
210	C₂	#20276879	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811025

C₂

yes

maximal, characteristic

C₂×C₄⋊#11719908

C₂

yes

maximal, characteristic

#20272806

C₂

yes

maximal

#20272806

C₂

yes

maximal

C₄×#107×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133209

—

C₂²×#11719908

C₂²

yes

—

#16811025

—

#16811025

C₄

yes

complemented

C₂×#13133151

C₂²

yes

characteristic

C₂×#13133151

C₂²

yes

characteristic

C₂×#13133149

—

C₄⋊#11719908

—

C₂×#107×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₅₃

—

C₅₃×#340

—

#13133209

—

#13133209

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

—

—

C₂×C₄

yes

—

#13133151

C₂×C₄

yes

—

#13133149

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#107×C₅₃

—

#107×C₅₃

Q₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

#11719908

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×Q₈

yes

—

C₈×C₅₃

#124

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#508

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

100

Dic₅₃

—

101

C₂×#499

—

102

C₂×C₄⋊C₁₆

—

103

C₄×#107

D₅₃

yes

characteristic

104

#1380

—

105

#1380

—

106

C₂×C₅₃

#1988

yes

characteristic

107

C₂×C₅₃

#2138

yes

characteristic

108

C₂×C₅₃

#2139

yes

characteristic

109

C₂×C₅₃

#2228

yes

—

110

C₂×C₅₃

#2229

yes

—

111

C₂×#125

—

106

—

112

C₂²×C₁₆

—

113

#499

—

114

#499

—

106

—

115

C₂×#107

—

116

C₂×#107

—

117

C₂×#107

C₂×D₅₃

yes

—

118

C₂×#106

—

106

—

119

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

120

C₄×C₄⋊C₄

—

121

C₄⋊C₁₆

—

106

—

122

#340

—

123

C₅₃

#13277

yes

characteristic, complemented, Sylow

124

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

125

C₂²×C₈

C₂²×D₅₃

yes

characteristic

126

#125

—

106

—

127

#125

—

212

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

106

—

130

C₂×C₄⋊C₄

—

106

—

131

C₂×C₄⋊C₄

—

212

—

132

C₂×C₄⋊C₄

—

212

—

133

C₂×C₄²

—

134

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

135

C₂×C₁₆

—

106

—

136

C₂×C₁₆

—

106

—

137

#107

—

138

#107

—

139

#107

—

140

#107

—

106

—

141

#107

—

106

—

142

#107

Dic₁₀₆

yes

—

143

#106

—

106

—

144

C₄×C₈

C₄×D₅₃

yes

—

145

C₄×C₈

D₂₁₂

yes

—

146

#95

—

147

C₂²×C₄

—

148

C₂²×C₄

—

212

—

149

C₂²×C₄

—

212

—

150

C₂²×C₄

#11719922

yes

characteristic

151

C₂²×C₄

#11719922

yes

characteristic

152

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

153

C₂×C₈

—

154

C₂×C₈

#11719921

yes

—

155

C₂×C₈

C₂×Dic₁₀₆

yes

—

156

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

157

C₂×C₈

C₂×D₂₁₂

yes

characteristic

158

C₂×C₈

#11719945

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

106

—

162

C₄⋊C₄

—

106

—

163

C₄⋊C₄

—

212

—

164

C₄⋊C₄

—

212

—

165

C₄⋊C₄

—

212

—

166

C₄⋊C₄

—

212

—

167

C₄²

—

168

C₄²

#11719921

yes

—

169

C₄²

#11719924

yes

—

170

C₁₆

—

212

—

171

C₂³

C₂×#11719922

yes

characteristic

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

424

—

179

C₂×C₄

—

424

—

180

C₂×C₄

—

424

—

181

C₂×C₄

—

212

—

182

C₂×C₄

—

212

—

183

C₂×C₄

#13133137

yes

—

184

C₂×C₄

#13133137

yes

—

185

C₂×C₄

C₂×#11719921

yes

characteristic

186

C₂×C₄

#13133196

yes

—

187

C₂×C₄

C₂×#11719924

yes

characteristic

188

C₂×C₄

#13133247

yes

—

189

C₈

—

190

C₈

C₄×Dic₁₀₆

yes

—

191

C₈

#13133205

yes

characteristic

192

C₈

#13133217

yes

characteristic

193

C₂²

C₂×#13133137

yes

characteristic

194

C₂²

#16810903

yes

—

195

C₂²

#16811053

yes

—

196

C₂²

#16811059

yes

characteristic

197

C₂²

#16811064

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

424

—

202

C₄

—

424

—

203

C₄

#16810529

yes

characteristic

204

C₄

#16810631

yes

characteristic

205

C₄

#16810908

yes

—

206

C₂

#20275667

yes

characteristic

207

C₂

#20276420

yes

characteristic

208

C₂

#20276425

yes

characteristic

209

C₂

#20276872

yes

—

210

C₂

#20276879

yes

—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×#107×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×#107	D₅₃
hypocenter	C₅₃	#13277
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

subgroup

G/H

center

C₂³

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₄×#107×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

C₂×D₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

N	H ≅ G/N	type
9. #16811025	C₄	semidirect
123. C₅₃	#13277	semidirect

H ≅ G/N

type

9. #16811025

C₄

semidirect

123. C₅₃

#13277

semidirect

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

subgroup	index
C₂×#16811025	2
C₂×C₄⋊#11719908	2
#20272806	2
C₄×#107×C₅₃	2
#13277	53

subgroup

index

C₄×#107×C₅₃

Subgroups — #27302673

Conjugacy classes of subgroups

Complements

Maximal subgroups