Group #27277969 — subgroups

Subgroups — #27277969

Order 13568.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20275652	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133136	C₂	1	1	yes	maximal, characteristic
3	#20273468	C₂	1	2	yes	maximal
4	#20272551	C₂	1	2	yes	maximal
5	C₅₃×#2983	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
7	C₂×#13133136	C₄	1	2	yes	—
8	C₄×#11719917	C₂²	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810392	C₄	1	1	yes	characteristic
11	#16810392	C₄	1	1	yes	characteristic
12	#16810378	—	2	2	no	—
13	#16810375	—	2	2	no	—
14	#16810373	—	2	2	no	—
15	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#446	—	2	2	no	—
17	C₅₃×#446	—	2	2	no	—
18	C₅₃×#432	—	2	2	no	—
19	C₅₃×#431	C₂²	1	2	yes	—
20	C₂×#11719917	C₂×C₄	1	2	yes	—
21	C₂×#11719917	C₂×C₄	1	2	yes	—
22	C₂²×#88322	—	2	1	no	—
23	C₂²×#88322	—	2	1	no	—
24	C₄⋊#88322	—	2	2	no	—
25	#13133136	—	2	4	no	—
26	#13133128	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	—	2	2	no	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
39	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
40	C₂²⋊C₈×C₅₃	—	2	2	no	—
41	#97×C₅₃	—	2	4	no	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₂×Q₈×C₅₃	—	2	2	no	—
44	C₂×Q₈×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	C₂×D₄	1	2	yes	—
59	C₄²×C₅₃	#41	1	2	yes	—
60	#11719917	—	2	2	no	—
61	#11719917	—	2	2	no	—
62	#11719917	—	2	2	no	—
63	#11719917	M₁₆	1	4	yes	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	—	4	1	no	—
68	C₂×#88322	—	4	1	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	—	4	1	no	—
71	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	Q₈×C₅₃	—	4	2	no	—
73	Q₈×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	4	1	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
89	C₈×C₅₃	—	4	2	no	—
90	C₈×C₅₃	—	4	2	no	—
91	#88322	—	2	4	no	—
92	#88322	—	4	2	no	—
93	#88322	—	4	2	no	—
94	#88322	—	8	1	no	—
95	#8838	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	#385	1	1	yes	characteristic
99	C₂²×C₅₃	#423	1	1	yes	characteristic
100	C₂²×C₅₃	#424	1	1	yes	characteristic
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	4	2	no	—
104	C₄×C₅₃	—	4	2	no	—
105	C₄×C₅₃	#435	1	2	yes	—
106	C₄×C₅₃	#436	1	2	yes	—
107	C₄×C₅₃	#454	1	1	yes	characteristic
108	C₄×C₅₃	#456	1	1	yes	characteristic
109	#2983	D₅₃	1	1	yes	characteristic
110	#1984	—	53	1	no	—
111	C₄×#101	—	53	1	no	—
112	#1565	—	53	2	no	—
113	#1278	—	53	2	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	—	2	1	no	—
116	C₂×C₅₃	#1851	1	1	yes	characteristic
117	C₂×C₅₃	#1857	1	1	yes	characteristic
118	C₂×C₅₃	#1858	1	1	yes	characteristic
119	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
120	#446	—	2	2	no	—
121	#446	—	2	2	no	—
122	#432	—	2	2	no	—
123	#431	C₂×D₅₃	1	2	yes	—
124	C₂×C₄⋊C₈	—	53	1	no	—
125	C₂×#101	—	53	2	no	—
126	C₄×M₁₆	—	53	2	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	#343	—	53	1	no	—
129	#343	—	53	1	no	—
130	#336	—	106	2	no	—
131	#335	—	106	2	no	—
132	#334	—	106	2	no	—
133	C₅₃	#8838	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
135	C₂×M₁₆	—	53	2	no	—
136	C₂×M₁₆	—	53	2	no	—
137	C₂²×C₈	—	106	1	no	—
138	C₂²×C₈	—	106	1	no	—
139	C₄×Q₈	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
142	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
144	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	106	2	no	—
148	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
149	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
150	#101	—	106	4	no	—
151	C₂²⋊C₈	—	2	2	no	—
152	#97	—	2	4	no	—
153	#97	—	106	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	106	2	no	—
156	C₄×C₈	—	106	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂²×C₄	#11719919	1	2	yes	—
162	C₂²×C₄	#11719919	1	2	yes	—
163	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
166	M₁₆	—	53	4	no	—
167	M₁₆	—	106	2	no	—
168	M₁₆	—	106	2	no	—
169	M₁₆	—	106	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	106	2	no	—
173	C₂×C₈	—	106	2	no	—
174	C₂×C₈	—	106	2	no	—
175	C₂×C₈	—	212	2	no	—
176	C₂×C₈	—	212	1	no	—
177	C₂×C₈	—	212	1	no	—
178	C₂×C₈	—	212	1	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	#11719945	1	2	yes	—
184	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
185	C₂³	C₂×#11719919	1	1	yes	characteristic
186	Q₈	—	4	2	no	—
187	Q₈	—	4	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
197	C₂×C₄	#13133186	1	1	yes	characteristic
198	C₂×C₄	#13133196	1	2	yes	—
199	C₂×C₄	#13133200	1	1	yes	characteristic
200	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
201	C₂×C₄	#13133225	1	1	yes	characteristic
202	C₂×C₄	#13133225	1	1	yes	characteristic
203	C₈	—	4	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	106	4	no	—
206	C₈	—	424	1	no	—
207	C₈	—	212	2	no	—
208	C₈	—	212	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#16810496	1	1	yes	characteristic
212	C₂²	#16810646	1	1	yes	characteristic
213	C₂²	#16810651	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	#16810692	1	2	yes	—
219	C₄	#16810699	1	2	yes	—
220	C₄	#16810796	1	1	yes	characteristic
221	C₄	#16810806	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#20274898	1	1	yes	characteristic
225	C₂	#20274930	1	1	yes	characteristic
226	C₂	#20274935	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275652

C₂

yes

maximal, characteristic

C₄×#13133136

C₂

yes

maximal, characteristic

#20273468

C₂

yes

maximal

#20272551

C₂

yes

maximal

C₅₃×#2983

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

characteristic

C₂×#13133136

C₄

yes

—

C₄×#11719917

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

#16810392

C₄

yes

characteristic

#16810392

C₄

yes

—

—

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×#446

—

C₅₃×#446

—

C₅₃×#432

—

C₅₃×#431

C₂²

yes

—

C₂×#11719917

C₂×C₄

yes

—

C₂×#11719917

C₂×C₄

yes

—

C₂²×#88322

—

C₂²×#88322

—

—

—

—

—

—

C₂²×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

C₄⋊C₈×C₅₃

D₄

yes

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

M₁₆

yes

—

—

—

—

—

—

—

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

100

C₂²×C₅₃

#424

yes

characteristic

101

C₄×C₅₃

—

102

C₄×C₅₃

—

103

C₄×C₅₃

—

104

C₄×C₅₃

—

105

C₄×C₅₃

#435

yes

—

106

C₄×C₅₃

#436

yes

—

107

C₄×C₅₃

#454

yes

characteristic

108

C₄×C₅₃

#456

yes

characteristic

109

#2983

D₅₃

yes

characteristic

110

#1984

—

111

C₄×#101

—

112

#1565

—

113

#1278

—

114

C₂×C₅₃

—

115

C₂×C₅₃

—

116

C₂×C₅₃

#1851

yes

characteristic

117

C₂×C₅₃

#1857

yes

characteristic

118

C₂×C₅₃

#1858

yes

characteristic

119

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

120

#446

—

121

#446

—

122

#432

—

123

#431

C₂×D₅₃

yes

—

124

C₂×C₄⋊C₈

—

125

C₂×#101

—

126

C₄×M₁₆

—

127

C₂×C₄×C₈

—

128

#343

—

129

#343

—

130

#336

—

106

—

131

#335

—

106

—

132

#334

—

106

—

133

C₅₃

#8838

yes

characteristic, complemented, Sylow

134

C₂²×Q₈

C₄×D₅₃

yes

characteristic

135

C₂×M₁₆

—

136

C₂×M₁₆

—

137

C₂²×C₈

—

106

—

138

C₂²×C₈

—

106

—

139

C₄×Q₈

—

140

C₄×Q₈

—

141

C₂×C₄⋊C₄

Dic₁₀₆

yes

—

142

C₂×C₄⋊C₄

C₄×D₅₃

yes

characteristic

143

C₂×C₄²

C₂²⋊D₅₃

yes

—

144

C₂×C₄²

C₂²×D₅₃

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

106

—

148

C₄⋊C₈

C₂²⋊D₅₃

yes

—

149

C₄⋊C₈

C₂²⋊D₅₃

yes

—

150

#101

—

106

—

151

C₂²⋊C₈

—

152

#97

—

153

#97

—

106

—

154

C₄×C₈

—

155

C₄×C₈

—

106

—

156

C₄×C₈

—

106

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂²×C₄

#11719919

yes

—

162

C₂²×C₄

#11719919

yes

—

163

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

164

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

165

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

166

M₁₆

—

167

M₁₆

—

106

—

168

M₁₆

—

106

—

169

M₁₆

—

106

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

106

—

173

C₂×C₈

—

106

—

174

C₂×C₈

—

106

—

175

C₂×C₈

—

212

—

176

C₂×C₈

—

212

—

177

C₂×C₈

—

212

—

178

C₂×C₈

—

212

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄²

—

182

C₄²

—

183

C₄²

#11719945

yes

—

184

C₄²

C₂×C₂²⋊D₅₃

yes

—

185

C₂³

C₂×#11719919

yes

characteristic

186

Q₈

—

187

Q₈

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

C₄×C₂²⋊D₅₃

yes

—

197

C₂×C₄

#13133186

yes

characteristic

198

C₂×C₄

#13133196

yes

—

199

C₂×C₄

#13133200

yes

characteristic

200

C₂×C₄

M₁₆×D₅₃

yes

—

201

C₂×C₄

#13133225

yes

characteristic

202

C₂×C₄

#13133225

yes

characteristic

203

C₈

—

204

C₈

—

205

C₈

—

106

—

206

C₈

—

424

—

207

C₈

—

212

—

208

C₈

—

212

—

209

C₂²

—

210

C₂²

—

211

C₂²

#16810496

yes

characteristic

212

C₂²

#16810646

yes

characteristic

213

C₂²

#16810651

yes

characteristic

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

#16810692

yes

—

219

C₄

#16810699

yes

—

220

C₄

#16810796

yes

characteristic

221

C₄

#16810806

yes

characteristic

222

C₂

—

223

C₂

—

224

C₂

#20274898

yes

characteristic

225

C₂

#20274930

yes

characteristic

226

C₂

#20274935

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133186
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2983	C₂
socle	C₂²×C₅₃	#385
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2983	D₅₃
hypocenter	C₅₃	#8838
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2983

subgroup

G/H

center

C₂×C₄

#13133186

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2983

C₂

socle

C₂²×C₅₃

#385

prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2983

N	H ≅ G/N	type
133. C₅₃	#8838	semidirect

H ≅ G/N

type

133. C₅₃

#8838

semidirect

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

subgroup	index
#20275652	2
C₄×#13133136	2
#20273468	2
#20272551	2
C₅₃×#2983	2
#8838	53

subgroup

index

C₅₃×#2983

Subgroups — #27277969

Conjugacy classes of subgroups

Complements

Maximal subgroups