Group #27283825 — subgroups

Subgroups — #27283825

Order 13568.

526 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 199 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×#16811021	C₂	1	1	yes	maximal, characteristic
2	#20275675	C₂	1	2	yes	maximal
3	#20275516	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	2	yes	—
7	#16811021	—	2	2	no	—
8	#16811021	C₄	1	4	yes	complemented
9	C₂×#13133149	—	2	2	no	—
10	C₂×#13133140	—	2	4	no	—
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	#16810385	—	2	4	no	—
13	C₈×#88322	—	2	2	no	—
14	C₂×#106×C₅₃	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#384	C₂²	1	1	yes	characteristic
17	C₅₃×#384	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₀₆	—	2	2	no	—
19	C₂²×#88322	D₄	1	4	yes	—
20	#13133217	—	2	2	no	—
21	#13133217	—	2	2	no	—
22	#13133217	—	2	2	no	—
23	#13133217	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	#13133149	—	2	8	no	—
27	#13133140	—	4	8	no	—
28	C₄×#88322	—	2	2	no	—
29	C₄×#88322	—	2	2	no	—
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#106×C₅₃	—	4	4	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₂²×C₄×C₅₃	—	4	1	no	—
40	C₂²×C₄×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂²×Dic₅₃	—	4	2	no	—
44	C₂×Dic₁₀₆	—	2	4	no	—
45	C₂×Dic₁₀₆	—	4	4	no	—
46	C₂×Dic₁₀₆	—	4	4	no	—
47	C₂×Dic₁₀₆	—	4	2	no	—
48	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
49	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
50	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	—	4	4	no	—
54	C₄⋊C₄×C₅₃	—	4	4	no	—
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	#41	1	2	yes	—
58	C₄⋊Dic₅₃	—	2	4	no	—
59	C₄⋊Dic₅₃	—	2	4	no	—
60	C₄⋊Dic₅₃	—	4	2	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	2	4	no	—
64	C₂×#88322	SD₁₆	1	8	yes	—
65	C₂³×C₅₃	#127	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	8	2	no	—
67	C₂×C₄×C₅₃	—	8	1	no	—
68	C₂×C₄×C₅₃	—	8	1	no	—
69	C₂×C₄×C₅₃	—	8	2	no	—
70	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₅₃	#121	1	2	yes	—
73	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
75	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	#135	1	4	yes	—
78	C₂×Dic₅₃	—	4	4	no	—
79	C₂×Dic₅₃	—	8	2	no	—
80	C₂×Dic₅₃	—	8	2	no	—
81	Dic₁₀₆	—	4	8	no	—
82	Dic₁₀₆	—	8	4	no	—
83	C₈×C₅₃	#103	1	8	yes	complemented
84	#88322	—	2	8	no	—
85	#9819	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	#390	1	1	yes	characteristic
87	C₂²×C₅₃	#492	1	2	yes	—
88	C₂²×C₅₃	#492	1	1	yes	characteristic
89	C₂²×C₅₃	#494	1	1	yes	characteristic
90	C₂²×C₅₃	#495	1	2	yes	—
91	C₄×C₅₃	—	16	1	no	—
92	C₄×C₅₃	—	16	1	no	—
93	C₄×C₅₃	C₂×#103	1	2	yes	—
94	C₄×C₅₃	C₄×SD₁₆	1	4	yes	—
95	C₄×C₅₃	#465	1	4	yes	—
96	C₄×C₅₃	#487	1	2	yes	—
97	Dic₅₃	—	8	4	no	—
98	C₂×#498	—	53	1	no	—
99	#1990	—	53	2	no	—
100	#1956	—	53	2	no	—
101	#1844	D₅₃	1	1	yes	characteristic
102	C₂×C₈²	—	53	1	no	—
103	C₂×C₅₃	#1705	1	1	yes	characteristic
104	C₂×C₅₃	#1707	1	2	yes	—
105	C₂×C₅₃	#1708	1	1	yes	characteristic
106	C₂×C₅₃	#1956	1	1	yes	characteristic
107	C₂×C₅₃	#1968	1	2	yes	—
108	C₂×C₄⋊Q₈	—	53	2	no	—
109	#498	—	53	4	no	—
110	#498	—	106	2	no	—
111	C₂×#106	—	2	2	no	—
112	C₂×#106	—	106	2	no	—
113	C₂×#103	—	106	4	no	—
114	C₂×C₄×C₈	—	53	2	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#384	C₂×D₅₃	1	1	yes	characteristic
117	#384	C₂×D₅₃	1	1	yes	characteristic
118	#340	—	106	4	no	—
119	C₈²	—	106	2	no	—
120	C₅₃	#9819	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	106	2	no	—
122	C₂²×C₈	—	53	4	no	—
123	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
124	C₄⋊Q₈	—	53	4	no	—
125	C₄⋊Q₈	—	106	2	no	—
126	C₄⋊Q₈	—	106	2	no	—
127	C₄⋊Q₈	—	106	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	106	2	no	—
133	C₂×C₄⋊C₄	—	106	2	no	—
134	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
135	#106	—	4	4	no	—
136	#106	—	106	8	no	—
137	#103	—	212	8	no	—
138	C₄×C₈	—	106	2	no	—
139	C₄×C₈	—	106	2	no	—
140	C₄×C₈	C₄×D₅₃	1	2	yes	—
141	C₄×C₈	D₂₁₂	1	2	yes	—
142	C₂×Q₈	—	106	4	no	—
143	C₂×Q₈	—	212	4	no	—
144	C₂×Q₈	—	212	4	no	—
145	C₂×Q₈	—	212	2	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	212	2	no	—
149	C₂²×C₄	D₄×D₅₃	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	C₂×C₈	—	53	8	no	—
152	C₂×C₈	—	106	4	no	—
153	C₂×C₈	—	106	4	no	—
154	C₂×C₈	#11719921	1	4	yes	—
155	C₂×C₈	#11719923	1	4	yes	—
156	C₂×C₈	#11719925	1	4	yes	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	106	4	no	—
162	C₄⋊C₄	—	106	4	no	—
163	C₄⋊C₄	—	212	2	no	—
164	C₄⋊C₄	—	212	2	no	—
165	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
166	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
167	C₄²	#11719949	1	2	yes	—
168	C₂³	#13133213	1	1	yes	characteristic
169	Q₈	—	424	4	no	—
170	Q₈	—	212	8	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	424	2	no	—
176	C₂×C₄	—	424	2	no	—
177	C₂×C₄	—	212	4	no	—
178	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
179	C₂×C₄	#13133179	1	2	yes	—
180	C₂×C₄	#13133192	1	2	yes	—
181	C₂×C₄	#13133206	1	1	yes	characteristic
182	C₂×C₄	SD₁₆×D₅₃	1	4	yes	—
183	C₂×C₄	C₂×#11719923	1	2	yes	—
184	C₂×C₄	C₂×#11719925	1	2	yes	—
185	C₂×C₄	#13133238	1	4	yes	—
186	C₈	—	106	8	no	—
187	C₈	#13133140	1	8	yes	complemented
188	C₂²	#16810524	1	1	yes	characteristic
189	C₂²	#16810994	1	2	yes	—
190	C₂²	#16810995	1	1	yes	characteristic
191	C₂²	#16811002	1	1	yes	characteristic
192	C₂²	#16811004	1	2	yes	—
193	C₄	—	16	1	no	—
194	C₄	—	16	1	no	—
195	C₄	—	424	4	no	—
196	C₄	C₂×#13133140	1	2	yes	—
197	C₄	#16810710	1	4	yes	—
198	C₄	#16810853	1	4	yes	—
199	C₄	#16810975	1	2	yes	—
200	C₂	#20274301	1	1	yes	characteristic
201	C₂	#20274310	1	2	yes	—
202	C₂	#20274314	1	1	yes	characteristic
203	C₂	#20275516	1	1	yes	characteristic
204	C₂	#20275567	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811021

C₂

yes

maximal, characteristic

#20275675

C₂

yes

maximal

#20275516

C₂

yes

maximal

C₂×C₈×#88322

C₂

yes

maximal, characteristic

C₅₃×#1844

C₂

yes

maximal, characteristic

C₂×#13133217

C₂²

yes

—

#16811021

—

#16811021

C₄

yes

complemented

C₂×#13133149

—

C₂×#13133140

—

C₂×C₄×#88322

C₂²

yes

—

#16810385

—

C₈×#88322

—

C₂×#106×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

C₂²×#88322

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

SD₁₆

yes

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#103

yes

—

C₄×C₅₃

C₄×SD₁₆

yes

—

C₄×C₅₃

#465

yes

—

C₄×C₅₃

#487

yes

—

Dic₅₃

—

C₂×#498

—

#1990

—

100

#1956

—

101

#1844

D₅₃

yes

characteristic

102

C₂×C₈²

—

103

C₂×C₅₃

#1705

yes

characteristic

104

C₂×C₅₃

#1707

yes

—

105

C₂×C₅₃

#1708

yes

characteristic

106

C₂×C₅₃

#1956

yes

characteristic

107

C₂×C₅₃

#1968

yes

—

108

C₂×C₄⋊Q₈

—

109

#498

—

110

#498

—

106

—

111

C₂×#106

—

112

C₂×#106

—

106

—

113

C₂×#103

—

106

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

116

#384

C₂×D₅₃

yes

characteristic

117

#384

C₂×D₅₃

yes

characteristic

118

#340

—

106

—

119

C₈²

—

106

—

120

C₅₃

#9819

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

106

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₅₃

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

106

—

126

C₄⋊Q₈

—

106

—

127

C₄⋊Q₈

—

106

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

106

—

133

C₂×C₄⋊C₄

—

106

—

134

C₂×C₄²

C₂²×D₅₃

yes

characteristic

135

#106

—

136

#106

—

106

—

137

#103

—

212

—

138

C₄×C₈

—

106

—

139

C₄×C₈

—

106

—

140

C₄×C₈

C₄×D₅₃

yes

—

141

C₄×C₈

D₂₁₂

yes

—

142

C₂×Q₈

—

106

—

143

C₂×Q₈

—

212

—

144

C₂×Q₈

—

212

—

145

C₂×Q₈

—

212

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

212

—

149

C₂²×C₄

D₄×D₅₃

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

106

—

153

C₂×C₈

—

106

—

154

C₂×C₈

#11719921

yes

—

155

C₂×C₈

#11719923

yes

—

156

C₂×C₈

#11719925

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

106

—

162

C₄⋊C₄

—

106

—

163

C₄⋊C₄

—

212

—

164

C₄⋊C₄

—

212

—

165

C₄²

C₂×C₄×D₅₃

yes

characteristic

166

C₄²

C₂×D₂₁₂

yes

characteristic

167

C₄²

#11719949

yes

—

168

C₂³

#13133213

yes

characteristic

169

Q₈

—

424

—

170

Q₈

—

212

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

424

—

176

C₂×C₄

—

424

—

177

C₂×C₄

—

212

—

178

C₂×C₄

C₂×#11719921

yes

characteristic

179

C₂×C₄

#13133179

yes

—

180

C₂×C₄

#13133192

yes

—

181

C₂×C₄

#13133206

yes

characteristic

182

C₂×C₄

SD₁₆×D₅₃

yes

—

183

C₂×C₄

C₂×#11719923

yes

—

184

C₂×C₄

C₂×#11719925

yes

—

185

C₂×C₄

#13133238

yes

—

186

C₈

—

106

—

187

C₈

#13133140

yes

complemented

188

C₂²

#16810524

yes

characteristic

189

C₂²

#16810994

yes

—

190

C₂²

#16810995

yes

characteristic

191

C₂²

#16811002

yes

characteristic

192

C₂²

#16811004

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

424

—

196

C₄

C₂×#13133140

yes

—

197

C₄

#16810710

yes

—

198

C₄

#16810853

yes

—

199

C₄

#16810975

yes

—

200

C₂

#20274301

yes

characteristic

201

C₂

#20274310

yes

—

202

C₂

#20274314

yes

characteristic

203

C₂

#20275516

yes

characteristic

204

C₂

#20275567

yes

—

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1844	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1844	D₅₃
hypocenter	C₅₃	#9819
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1844

C₂

socle, prime radical

C₂³×C₅₃

#127

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
8. #16811021	C₄	semidirect
83. C₈×C₅₃	#103	semidirect
120. C₅₃	#9819	semidirect
187. C₈	#13133140	semidirect

H ≅ G/N

type

8. #16811021

C₄

semidirect

83. C₈×C₅₃

#103

semidirect

120. C₅₃

#9819

semidirect

187. C₈

#13133140

semidirect

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

subgroup	index
C₂×#16811021	2
#20275675	2
#20275516	2
C₂×C₈×#88322	2
C₅₃×#1844	2
#9819	53

subgroup

index

C₂×#16811021

#20275675

#20275516

C₂×C₈×#88322

C₅₃×#1844

#9819

Subgroups — #27283825

Conjugacy classes of subgroups

Complements

Maximal subgroups