Group #27302260 — subgroups

Subgroups — #27302260

Order 13568.

442 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 151 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	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810407	C₂	1	4	yes	maximal, complemented
3	C₂×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₄⋊#11719908	—	2	2	no	—
8	#16810407	—	2	8	no	—
9	#16810399	—	2	2	no	—
10	C₂×#131×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#532	C₄	1	1	yes	characteristic
12	C₅₃×#436	—	2	2	no	—
13	C₈×D₄×C₅₃	—	2	2	no	—
14	C₅₃×#423	C₄	1	1	yes	characteristic
15	C₅₃×#408	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	#13133157	—	2	4	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	D₄	1	4	yes	complemented
22	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
23	#131×C₅₃	—	2	4	no	—
24	#131×C₅₃	—	2	4	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
31	#127×C₅₃	C₈	1	1	yes	characteristic
32	#124×C₅₃	C₈	1	2	yes	—
33	#121×C₅₃	—	2	2	no	—
34	C₄×D₄×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₂²⋊C₈×C₅₃	—	2	4	no	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	#41×C₅₃	—	2	4	no	—
41	#41×C₅₃	—	2	4	no	—
42	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×D₄×C₅₃	—	2	2	no	—
44	C₂×D₄×C₅₃	—	2	2	no	—
45	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₅₃	—	2	4	no	—
50	M₁₆×C₅₃	—	4	4	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	2	4	no	—
62	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
65	#11719908	—	2	4	no	—
66	#11719908	—	4	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	Q₈×C₅₃	—	2	4	no	—
70	D₄×C₅₃	—	2	4	no	—
71	D₄×C₅₃	—	4	4	no	—
72	D₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#131	1	2	yes	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	4	2	no	—
85	C₈×C₅₃	#121	1	2	yes	—
86	C₈×C₅₃	#124	1	1	yes	characteristic
87	C₈×C₅₃	#127	1	1	yes	characteristic
88	#13192	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	—	4	4	no	—
91	C₂²×C₅₃	—	4	2	no	—
92	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
93	C₂²×C₅₃	#436	1	2	yes	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	4	2	no	—
97	C₄×C₅₃	#390	1	1	yes	characteristic
98	C₄×C₅₃	#408	1	1	yes	characteristic
99	C₄×C₅₃	C₈×D₄	1	2	yes	—
100	#2931	D₅₃	1	1	yes	characteristic
101	#2150	—	53	1	no	—
102	C₂×#350	—	53	4	no	—
103	C₂×#346	—	53	1	no	—
104	C₂×C₅₃	—	2	2	no	—
105	C₂×C₅₃	—	8	2	no	—
106	C₂×C₅₃	#1954	1	1	yes	characteristic
107	C₂×C₅₃	#2117	1	2	yes	—
108	C₂×#131	C₂×D₅₃	1	2	yes	—
109	#532	Dic₅₃	1	1	yes	characteristic
110	C₂×M₃₂	—	53	2	no	—
111	C₂²×C₁₆	—	53	2	no	—
112	#436	—	2	2	no	—
113	C₈×D₄	—	2	2	no	—
114	#423	Dic₅₃	1	1	yes	characteristic
115	#408	Dic₅₃	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₄⋊C₁₆	—	106	2	no	—
118	#350	—	106	8	no	—
119	#346	—	106	2	no	—
120	C₅₃	#13192	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	C₂×Dic₅₃	1	2	yes	—
122	#131	—	2	4	no	—
123	#131	—	2	4	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
129	C₄⋊Q₈	#88322	1	1	yes	characteristic
130	#127	#88322	1	1	yes	characteristic
131	#124	#88322	1	2	yes	—
132	#121	—	2	2	no	—
133	C₄×D₄	—	2	2	no	—
134	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
135	M₃₂	—	106	4	no	—
136	C₂×C₁₆	—	53	4	no	—
137	C₂×C₁₆	—	106	2	no	—
138	C₂×C₁₆	—	106	2	no	—
139	C₂×C₁₆	—	106	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	4	no	—
142	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
143	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
144	#41	—	2	4	no	—
145	#41	—	2	4	no	—
146	C₂×Q₈	C₂×#88322	1	2	yes	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	C₂×#88322	1	2	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	#11719917	1	2	yes	—
152	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
153	M₁₆	—	2	4	no	—
154	M₁₆	—	4	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
160	C₂×C₈	D₄×D₅₃	1	2	yes	—
161	C₂×C₈	#11719947	1	2	yes	—
162	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
163	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	4	no	—
166	C₄²	C₂×#88322	1	1	yes	characteristic
167	C₄²	C₂×#88322	1	1	yes	characteristic
168	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
169	C₁₆	—	106	4	no	—
170	C₁₆	—	212	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₂×#11719917	1	1	yes	characteristic
173	Q₈	—	2	4	no	—
174	D₄	—	2	4	no	—
175	D₄	—	4	4	no	—
176	D₄	—	4	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂²⋊#88322	1	2	yes	—
182	C₂×C₄	C₂²⋊#88322	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
185	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
186	C₂×C₄	#13133224	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	#13133188	1	2	yes	—
190	C₈	#13133204	1	1	yes	characteristic
191	C₈	#13133213	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	—	4	4	no	—
195	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
196	C₂²	#16810701	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	#16810526	1	1	yes	characteristic
201	C₄	#16810590	1	1	yes	characteristic
202	C₄	D₄×#88322	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	8	2	no	—
205	C₂	#20275510	1	1	yes	characteristic
206	C₂	#20276343	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276471

C₂

yes

maximal, characteristic, complemented

C₂×#16810407

C₂

yes

maximal, complemented

C₂×#16810399

C₂

yes

maximal, characteristic, complemented

C₅₃×#2931

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂²×#11719908

C₂²

yes

—

C₄⋊#11719908

—

#16810407

—

#16810399

—

C₂×#131×C₅₃

C₂²

yes

—

C₅₃×#532

C₄

yes

characteristic

C₅₃×#436

—

C₈×D₄×C₅₃

—

C₅₃×#423

C₄

yes

characteristic

C₅₃×#408

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

—

D₄

yes

complemented

C₂×#41×C₅₃

C₂×C₄

yes

—

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊Q₈×C₅₃

C₈

yes

characteristic

#127×C₅₃

C₈

yes

characteristic

#124×C₅₃

C₈

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₈

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

#11719908

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#121

yes

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

#436

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₈×D₄

yes

—

100

#2931

D₅₃

yes

characteristic

101

#2150

—

102

C₂×#350

—

103

C₂×#346

—

104

C₂×C₅₃

—

105

C₂×C₅₃

—

106

C₂×C₅₃

#1954

yes

characteristic

107

C₂×C₅₃

#2117

yes

—

108

C₂×#131

C₂×D₅₃

yes

—

109

#532

Dic₅₃

yes

characteristic

110

C₂×M₃₂

—

111

C₂²×C₁₆

—

112

#436

—

113

C₈×D₄

—

114

#423

Dic₅₃

yes

characteristic

115

#408

Dic₅₃

yes

—

116

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

117

C₄⋊C₁₆

—

106

—

118

#350

—

106

—

119

#346

—

106

—

120

C₅₃

#13192

yes

characteristic, complemented, Sylow

121

C₂×#41

C₂×Dic₅₃

yes

—

122

#131

—

123

#131

—

124

C₂×M₁₆

—

125

C₂×M₁₆

C₂×Dic₅₃

yes

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

128

C₂²×C₈

C₂²×D₅₃

yes

characteristic

129

C₄⋊Q₈

yes

130

yes

131

yes

—

132

#121

—

133

C₄×D₄

—

134

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

135

M₃₂

—

106

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

106

—

138

C₂×C₁₆

—

106

—

139

C₂×C₁₆

—

106

—

140

C₄⋊C₈

—

141

C₂²⋊C₈

—

142

C₄×C₈

C₂²⋊D₅₃

yes

—

143

C₄×C₈

C₂²⋊D₅₃

yes

—

144

#41

—

145

#41

—

146

C₂×Q₈

C₂×#88322

yes

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

C₂×#88322

yes

—

150

C₂²×C₄

—

151

C₂²×C₄

#11719917

yes

—

152

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

153

M₁₆

—

154

M₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₂²⋊Dic₅₃

yes

—

160

C₂×C₈

D₄×D₅₃

yes

—

161

C₂×C₈

#11719947

yes

—

162

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

163

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

164

C₄⋊C₄

—

165

C₂²⋊C₄

—

166

C₄²

C₂×#88322

yes

characteristic

167

C₄²

C₂×#88322

yes

characteristic

168

C₄²

C₂²⋊Dic₅₃

yes

—

169

C₁₆

—

106

—

170

C₁₆

—

212

—

171

C₂³

—

172

C₂³

C₂×#11719917

yes

characteristic

173

Q₈

—

174

D₄

—

175

D₄

—

176

D₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

C₂²⋊#88322

yes

—

182

C₂×C₄

C₂²⋊#88322

yes

—

183

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

184

C₂×C₄

D₄×Dic₅₃

yes

—

185

C₂×C₄

C₂²×#88322

yes

characteristic

186

C₂×C₄

#13133224

yes

—

187

C₈

—

188

C₈

—

189

C₈

#13133188

yes

—

190

C₈

#13133204

yes

characteristic

191

C₈

#13133213

yes

characteristic

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

C₂×C₂²⋊#88322

yes

characteristic

196

C₂²

#16810701

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

#16810526

yes

characteristic

201

C₄

#16810590

yes

characteristic

202

C₄

D₄×#88322

yes

—

203

C₂

—

204

C₂

—

205

C₂

#20275510

yes

characteristic

206

C₂

#20276343

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2931	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₅₃×#532	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2931	D₅₃
hypocenter	C₅₃	#13192
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#2931

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

C₅₃×#532

C₄

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

N	H ≅ G/N	type
1. #20276471	C₂	semidirect
2. C₂×#16810407	C₂	semidirect
3. C₂×#16810399	C₂	semidirect
21. C₂×#11719908	D₄	semidirect
120. C₅₃	#13192	semidirect

H ≅ G/N

type

1. #20276471

C₂

semidirect

2. C₂×#16810407

C₂

semidirect

3. C₂×#16810399

C₂

semidirect

21. C₂×#11719908

D₄

semidirect

120. C₅₃

#13192

semidirect

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

subgroup	index
#20276471	2
C₂×#16810407	2
C₂×#16810399	2
C₅₃×#2931	2
#13192	53

subgroup

index

C₅₃×#2931

Subgroups — #27302260

Conjugacy classes of subgroups

Complements

Maximal subgroups