Group #27277016 — subgroups

Subgroups — #27277016

Order 13568.

398 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20274711	C₂	1	2	yes	maximal
2	#20273471	C₂	1	2	yes	maximal
3	#20272590	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810647	C₂²	1	4	yes	—
6	C₂×#13133136	—	2	2	no	—
7	#16810392	—	2	2	no	—
8	#16810374	—	2	2	no	—
9	#16810374	—	2	2	no	—
10	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₅₃	—	2	2	no	—
12	C₈×Q₈×C₅₃	—	2	2	no	—
13	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719917	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	—	2	4	no	—
20	#13133136	—	4	4	no	—
21	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₅₃	—	2	2	no	—
26	C₄×Q₈×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	4	no	—
32	C₄⋊C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₂×Q₈×C₅₃	—	2	2	no	—
38	C₂×Q₈×C₅₃	—	2	2	no	—
39	C₂×Q₈×C₅₃	SD₁₆	1	4	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	4	1	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	C₂×D₄	1	2	yes	—
57	C₄²×C₅₃	#41	1	2	yes	—
58	#11719917	—	4	4	no	—
59	#11719917	—	4	4	no	—
60	C₂×#88322	—	4	4	no	—
61	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	Q₈×C₅₃	—	2	8	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	1	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	#106	1	4	yes	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#121	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	#135	1	2	yes	—
77	C₈×C₅₃	—	2	1	no	—
78	C₈×C₅₃	—	2	2	no	—
79	C₈×C₅₃	—	4	2	no	—
80	C₈×C₅₃	#101	1	2	yes	complemented
81	#88322	—	8	4	no	—
82	#8615	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	#380	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
87	C₂²×C₅₃	#427	1	1	yes	characteristic
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	—	2	4	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	4	2	no	—
92	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
93	C₄×C₅₃	#413	1	1	yes	characteristic
94	C₄×C₅₃	#465	1	4	yes	—
95	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
96	#1801	—	53	2	no	—
97	#1566	—	53	2	no	—
98	#1292	—	53	2	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	#1801	1	1	yes	characteristic
102	C₂×C₅₃	#1807	1	1	yes	characteristic
103	C₂×C₅₃	#1813	1	1	yes	characteristic
104	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
105	C₈×Q₈	—	2	2	no	—
106	C₈×Q₈	—	2	2	no	—
107	#423	—	53	4	no	—
108	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
109	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
110	C₂×#101	—	106	2	no	—
111	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	#343	—	106	2	no	—
114	#334	—	106	2	no	—
115	#334	—	106	2	no	—
116	C₅₃	#8615	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
118	C₂×M₁₆	—	106	4	no	—
119	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
120	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
121	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
122	C₄×Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
125	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
126	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	106	4	no	—
131	C₄⋊C₈	—	106	4	no	—
132	#101	—	212	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
136	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	#11719913	1	4	yes	—
140	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
141	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
142	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
145	M₁₆	—	212	4	no	—
146	M₁₆	—	212	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	212	4	no	—
152	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	#11719945	1	2	yes	—
160	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
161	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
162	Q₈	—	2	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#13133149	1	4	yes	—
171	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
172	C₂×C₄	#13133176	1	1	yes	characteristic
173	C₂×C₄	#13133191	1	2	yes	—
174	C₂×C₄	#13133196	1	2	yes	—
175	C₂×C₄	C₂×#11719913	1	2	yes	—
176	C₂×C₄	#13133239	1	2	yes	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	424	4	no	—
181	C₈	#13133136	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#16810483	1	1	yes	characteristic
185	C₂²	C₂×#13133149	1	1	yes	characteristic
186	C₂²	#16810661	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	C₂×#13133136	1	1	yes	characteristic
192	C₄	#16810607	1	1	yes	characteristic
193	C₄	#16810852	1	4	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#20274711	1	1	yes	characteristic
197	C₂	#20274740	1	1	yes	characteristic
198	C₂	#20274765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274711

C₂

yes

maximal

#20273471

C₂

yes

maximal

#20272590

C₂

yes

maximal

C₂×C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic

#16810647

C₂²

yes

—

—

—

—

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₈×Q₈×C₅₃

—

C₈×Q₈×C₅₃

—

C₂×C₄⋊C₈×C₅₃

C₄

yes

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

—

C₂²×Q₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

SD₁₆

yes

—

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

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

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

#106

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₈×Q₈

D₅₃

yes

—

—

—

C₂×C₅₃

—

100

C₂×C₅₃

—

101

C₂×C₅₃

#1801

yes

characteristic

102

C₂×C₅₃

#1807

yes

characteristic

103

C₂×C₅₃

#1813

yes

characteristic

104

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

105

C₈×Q₈

—

106

C₈×Q₈

—

107

#423

—

108

C₂×C₄⋊C₈

Dic₅₃

yes

—

109

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

110

C₂×#101

—

106

—

111

C₂×C₄×C₈

Dic₅₃

yes

—

112

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

113

#343

—

106

—

114

#334

—

106

—

115

#334

—

106

—

116

C₅₃

#8615

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

D₂₁₂

yes

characteristic

118

C₂×M₁₆

—

106

—

119

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

120

C₂²×C₈

C₂×Dic₅₃

yes

—

121

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

122

C₄×Q₈

—

123

C₄×Q₈

—

124

C₂×C₄⋊C₄

Dic₁₀₆

yes

—

125

C₂×C₄⋊C₄

D₂₁₂

yes

characteristic

126

C₂×C₄²

C₂×Dic₅₃

yes

—

127

C₂×C₄²

C₂²×D₅₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

106

—

131

C₄⋊C₈

—

106

—

132

#101

—

212

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

C₂²⋊D₅₃

yes

—

136

C₄×C₈

C₂²⋊D₅₃

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

#11719913

yes

—

140

C₂²×C₄

C₄⋊Dic₅₃

yes

—

141

C₂²×C₄

C₄⋊Dic₅₃

yes

—

142

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

143

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

144

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

145

M₁₆

—

212

—

146

M₁₆

—

212

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

212

—

152

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

153

C₂×C₈

C₂²⋊Dic₅₃

yes

—

154

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄²

—

158

C₄²

—

159

C₄²

#11719945

yes

—

160

C₄²

C₂×C₂²⋊D₅₃

yes

—

161

C₂³

C₂×C₄⋊Dic₅₃

yes

characteristic

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#13133149

yes

—

171

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

172

C₂×C₄

#13133176

yes

characteristic

173

C₂×C₄

#13133191

yes

—

174

C₂×C₄

#13133196

yes

—

175

C₂×C₄

C₂×#11719913

yes

—

176

C₂×C₄

#13133239

yes

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

424

—

181

C₈

#13133136

yes

complemented

182

C₂²

—

183

C₂²

—

184

C₂²

#16810483

yes

characteristic

185

C₂²

C₂×#13133149

yes

characteristic

186

C₂²

#16810661

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

C₂×#13133136

yes

characteristic

192

C₄

#16810607

yes

characteristic

193

C₄

#16810852

yes

—

194

C₂

—

195

C₂

—

196

C₂

#20274711

yes

characteristic

197

C₂

#20274740

yes

characteristic

198

C₂

#20274765

yes

characteristic

	subgroup	G/H
center	C₂²	#16810483
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#8615
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

subgroup

G/H

center

C₂²

#16810483

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

#380

prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

hypercenter

C₂×C₈×Q₈

D₅₃

hypocenter

C₅₃

#8615

LCS

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

N	H ≅ G/N	type
80. C₈×C₅₃	#101	semidirect
116. C₅₃	#8615	semidirect
181. C₈	#13133136	semidirect

H ≅ G/N

type

80. C₈×C₅₃

#101

semidirect

116. C₅₃

#8615

semidirect

181. C₈

#13133136

semidirect

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

subgroup	index
#20274711	2
#20273471	2
#20272590	2
C₂×C₈×Q₈×C₅₃	2
#8615	53

subgroup

index

C₂×C₈×Q₈×C₅₃

Subgroups — #27277016

Conjugacy classes of subgroups

Complements

Maximal subgroups