Group #27536655 — subgroups

Subgroups — #27536655

Order 13568.

1006 conjugacy classes of subgroups; only the 347 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 347 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	C₂×#16811019	C₂	1	2	yes	maximal
3	#20283111	C₂	1	2	yes	maximal
4	#20274870	C₂	1	8	yes	maximal
5	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133217	C₂²	1	2	yes	—
8	C₂×#13133215	—	2	4	no	—
9	#16811021	—	2	4	no	—
10	#16811019	—	2	8	no	—
11	C₂×#13133149	C₂²	1	4	yes	—
12	C₂×#13133148	C₂²	1	8	yes	—
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
15	#16810492	—	2	4	no	—
16	#16810492	C₂²	1	8	yes	—
17	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#498	—	2	4	no	—
20	C₂×#106×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₀₆	—	2	2	no	—
23	C₂²×#88322	C₂³	1	2	yes	—
24	#13133217	—	2	4	no	—
25	#13133217	—	2	4	no	—
26	#13133215	—	4	16	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
29	C₂×#11719919	—	2	16	no	—
30	C₂×C₄×Dic₅₃	D₄	1	4	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
32	#13133149	—	2	8	no	—
33	#13133148	—	2	16	no	—
34	C₄×#88322	D₄	1	4	yes	—
35	C₈×Dic₅₃	Q₈	1	16	yes	complemented
36	C₂²×Q₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₅₃	—	2	4	no	—
39	C₄⋊Q₈×C₅₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#106×C₅₃	—	2	8	no	—
44	C₄×C₈×C₅₃	D₄	1	4	yes	—
45	C₂×Q₈×C₅₃	—	4	4	no	—
46	C₂×Q₈×C₅₃	—	4	8	no	—
47	C₂²×C₄×C₅₃	—	2	4	no	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂²×Dic₅₃	—	2	4	no	—
51	C₂²×Dic₅₃	C₂×D₄	1	4	yes	—
52	C₂×Dic₁₀₆	—	4	4	no	—
53	C₂×Dic₁₀₆	—	4	8	no	—
54	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
55	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	4	4	no	—
59	C₄²×C₅₃	C₂×D₄	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	2	yes	—
61	C₄⋊Dic₅₃	—	2	4	no	—
62	C₄⋊Dic₅₃	—	2	4	no	—
63	C₄⋊Dic₅₃	—	4	4	no	—
64	#11719919	—	4	32	no	—
65	C₄×Dic₅₃	SD₁₆	1	16	yes	—
66	C₄×Dic₅₃	C₂×Q₈	1	8	yes	—
67	C₂×#88322	C₂×D₄	1	4	yes	—
68	C₂×#88322	C₂×Q₈	1	8	yes	—
69	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
70	Q₈×C₅₃	—	8	8	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	4	8	no	—
73	C₂×C₄×C₅₃	#127	1	4	yes	—
74	C₂×C₄×C₅₃	C₂×SD₁₆	1	8	yes	—
75	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
78	C₂×Dic₅₃	—	4	4	no	—
79	C₂×Dic₅₃	—	4	8	no	—
80	C₂×Dic₅₃	C₄⋊Q₈	1	8	yes	—
81	C₂×Dic₅₃	C₂×SD₁₆	1	16	yes	—
82	Dic₁₀₆	—	8	8	no	—
83	C₈×C₅₃	C₄⋊Q₈	1	8	yes	complemented
84	#88322	C₄⋊Q₈	1	8	yes	complemented
85	#33310	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
88	C₂²×C₅₃	C₂²×SD₁₆	1	4	yes	—
89	C₄×C₅₃	—	8	4	no	—
90	C₄×C₅₃	#492	1	8	yes	—
91	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
92	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
93	Dic₅₃	—	8	4	no	—
94	Dic₅₃	#498	1	16	yes	complemented
95	C₂×#498	—	53	1	no	—
96	C₂×#498	—	53	2	no	—
97	C₂×#498	D₅₃	1	1	yes	characteristic
98	#2868	—	53	2	no	—
99	#1844	—	53	8	no	—
100	C₄²×C₈	—	53	1	no	—
101	C₂×C₅₃	#2868	1	1	yes	characteristic
102	C₂×C₅₃	C₂×#492	1	2	yes	—
103	C₂×C₅₃	C₂×#498	1	4	yes	—
104	C₂×C₄⋊Q₈	—	53	2	no	—
105	C₂×C₄⋊Q₈	—	106	4	no	—
106	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
107	#498	—	2	4	no	—
108	#498	—	106	8	no	—
109	#498	—	106	4	no	—
110	C₂×#106	—	53	4	no	—
111	C₂×#106	—	53	8	no	—
112	C₂×#106	C₂×D₅₃	1	4	yes	—
113	C₂×C₄×C₈	—	53	4	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#384	—	53	8	no	—
117	#384	—	106	4	no	—
118	C₄³	—	53	1	no	—
119	C₅₃	#33310	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	2	2	no	—
121	C₂²×Q₈	—	106	2	no	—
122	C₂²×C₈	—	53	2	no	—
123	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	2	4	no	—
126	C₄⋊Q₈	—	106	4	no	—
127	C₄⋊Q₈	—	106	4	no	—
128	C₄⋊Q₈	—	212	16	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	53	4	no	—
131	C₂×C₄⋊C₄	—	106	16	no	—
132	C₂×C₄⋊C₄	—	106	2	no	—
133	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
134	C₂×C₄²	—	53	4	no	—
135	C₂×C₄²	—	53	2	no	—
136	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	#106	—	2	8	no	—
138	#106	—	106	16	no	—
139	#106	—	106	8	no	—
140	C₄×C₈	—	53	16	no	—
141	C₄×C₈	—	53	4	no	—
142	C₄×C₈	C₂²⋊D₅₃	1	4	yes	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	4	8	no	—
145	C₂×Q₈	—	212	4	no	—
146	C₂×Q₈	—	212	8	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	53	4	no	—
149	C₂²×C₄	—	106	4	no	—
150	C₂²×C₄	D₄×D₅₃	1	2	yes	—
151	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
152	C₂×C₈	—	53	4	no	—
153	C₂×C₈	—	53	8	no	—
154	C₂×C₈	Q₈×D₅₃	1	8	yes	—
155	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	106	4	no	—
160	C₄⋊C₄	—	106	4	no	—
161	C₄⋊C₄	—	212	32	no	—
162	C₄⋊C₄	—	212	4	no	—
163	C₄²	—	53	16	no	—
164	C₄²	—	53	8	no	—
165	C₄²	D₄×D₅₃	1	2	yes	—
166	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
167	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
168	Q₈	—	8	8	no	—
169	Q₈	—	424	8	no	—
170	C₂×C₄	—	4	8	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	53	16	no	—
173	C₂×C₄	—	53	8	no	—
174	C₂×C₄	—	212	8	no	—
175	C₂×C₄	—	212	4	no	—
176	C₂×C₄	#13133213	1	4	yes	—
177	C₂×C₄	SD₁₆×D₅₃	1	8	yes	—
178	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₈×D₅₃	1	4	yes	—
181	C₈	—	53	8	no	—
182	C₈	#13133215	1	8	yes	complemented
183	C₂²	C₂×#13133213	1	1	yes	characteristic
184	C₂²	C₄⋊Q₈×D₅₃	1	2	yes	—
185	C₂²	C₂×SD₁₆×D₅₃	1	4	yes	—
186	C₄	—	8	4	no	—
187	C₄	—	53	16	no	—
188	C₄	—	424	4	no	—
189	C₄	#16810994	1	8	yes	—
190	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
191	C₄	C₂×#13133215	1	2	yes	—
192	C₂	#20283111	1	1	yes	characteristic
193	C₂	C₂×#16810994	1	2	yes	—
194	C₂	#498×D₅₃	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811021

C₂

yes

maximal, characteristic

C₂×#16811019

C₂

yes

maximal

#20283111

C₂

yes

maximal

#20274870

C₂

yes

maximal

C₄×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₅₃×#498

C₂

yes

maximal, characteristic

C₂×#13133217

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×#13133148

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

—

#16810492

—

#16810492

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#498

—

C₂×#106×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

C₂²×#88322

C₂³

yes

—

#13133217

—

#13133217

—

#13133215

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂³

yes

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

D₄

yes

—

C₂×C₄×Dic₅₃

C₂³

yes

—

#13133149

—

#13133148

—

C₄×#88322

D₄

yes

—

C₈×Dic₅₃

Q₈

yes

complemented

C₂²×Q₈×C₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂³

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#106×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

SD₁₆

yes

—

C₄×Dic₅₃

C₂×Q₈

yes

—

C₂×#88322

C₂×D₄

yes

—

C₂×#88322

C₂×Q₈

yes

—

C₂³×C₅₃

C₂²×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊Q₈

yes

—

C₂×Dic₅₃

C₂×SD₁₆

yes

—

Dic₁₀₆

—

C₈×C₅₃

C₄⋊Q₈

yes

complemented

#88322

C₄⋊Q₈

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₅₃

C₂²×SD₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

#492

yes

—

C₄×C₅₃

C₂×C₄⋊Q₈

yes

—

C₄×C₅₃

C₂×C₄⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

yes

—

—

D₅₃

yes

characteristic

#2868

—

#1844

—

100

C₄²×C₈

—

101

C₂×C₅₃

#2868

yes

characteristic

102

C₂×C₅₃

C₂×#492

yes

—

103

C₂×C₅₃

C₂×#498

yes

—

104

C₂×C₄⋊Q₈

—

105

C₂×C₄⋊Q₈

—

106

—

106

C₂×C₄⋊Q₈

C₂×D₅₃

yes

—

107

#498

—

108

#498

—

106

—

109

#498

—

106

—

110

C₂×#106

—

111

C₂×#106

—

112

C₂×#106

C₂×D₅₃

yes

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

116

#384

—

117

#384

—

106

—

118

C₄³

—

119

C₅₃

#33310

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

121

C₂²×Q₈

—

106

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²×D₅₃

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

106

—

127

C₄⋊Q₈

—

106

—

128

C₄⋊Q₈

—

212

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

106

—

132

C₂×C₄⋊C₄

—

106

—

133

C₂×C₄⋊C₄

C₂²×D₅₃

yes

—

134

C₂×C₄²

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²×D₅₃

yes

characteristic

137

#106

—

138

#106

—

106

—

139

#106

—

106

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

C₂²⋊D₅₃

yes

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

212

—

146

C₂×Q₈

—

212

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

106

—

150

C₂²×C₄

D₄×D₅₃

yes

—

151

C₂²×C₄

C₂³×D₅₃

yes

characteristic

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

Q₈×D₅₃

yes

—

155

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

106

—

160

C₄⋊C₄

—

106

—

161

C₄⋊C₄

—

212

—

162

C₄⋊C₄

—

212

—

163

C₄²

—

164

C₄²

—

165

C₄²

D₄×D₅₃

yes

—

166

C₄²

C₂×C₂²⋊D₅₃

yes

—

167

C₂³

C₂×D₄×D₅₃

yes

characteristic

168

Q₈

—

169

Q₈

—

424

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

212

—

175

C₂×C₄

—

212

—

176

C₂×C₄

#13133213

yes

—

177

C₂×C₄

SD₁₆×D₅₃

yes

—

178

C₂×C₄

C₂×D₄×D₅₃

yes

characteristic

179

C₂×C₄

C₂²×C₂²⋊D₅₃

yes

characteristic

180

C₂×C₄

C₂×Q₈×D₅₃

yes

—

181

C₈

—

182

C₈

#13133215

yes

complemented

183

C₂²

C₂×#13133213

yes

characteristic

184

C₂²

C₄⋊Q₈×D₅₃

yes

—

185

C₂²

C₂×SD₁₆×D₅₃

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

424

—

189

C₄

#16810994

yes

—

190

C₄

C₄⋊Q₈×D₅₃

yes

—

191

C₄

C₂×#13133215

yes

—

192

C₂

#20283111

yes

characteristic

193

C₂

C₂×#16810994

yes

—

194

C₂

#498×D₅₃

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#498	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂×#498	D₅₃
hypocenter	C₅₃	#33310
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

C₂×D₄×D₅₃

commutator, cosocle

C₂²×C₄×C₅₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₂×C₅₃×#498

C₂

socle, prime radical

C₂³×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
35. C₈×Dic₅₃	Q₈	semidirect
83. C₈×C₅₃	C₄⋊Q₈	semidirect
84. #88322	C₄⋊Q₈	semidirect
94. Dic₅₃	#498	semidirect
119. C₅₃	#33310	semidirect
182. C₈	#13133215	semidirect

H ≅ G/N

type

35. C₈×Dic₅₃

Q₈

semidirect

83. C₈×C₅₃

C₄⋊Q₈

semidirect

84. #88322

C₄⋊Q₈

semidirect

94. Dic₅₃

#498

semidirect

119. C₅₃

#33310

semidirect

182. C₈

#13133215

semidirect

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

subgroup	index
C₂×#16811021	2
C₂×#16811019	2
#20283111	2
#20274870	2
C₄×C₈×Dic₅₃	2
C₂×C₅₃×#498	2
#33310	53

subgroup

index

C₄×C₈×Dic₅₃

C₂×C₅₃×#498

#33310

Subgroups — #27536655

Conjugacy classes of subgroups

Complements

Maximal subgroups