Group #27283854 — subgroups

Subgroups — #27283854

Order 13568.

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

normal subgroups only

This group has 726 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×#16811000	C₂	1	1	yes	maximal, characteristic
2	#20275520	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₂₄	—	2	2	no	—
6	C₂×#13133214	C₂²	1	2	yes	—
7	#16811000	—	2	2	no	—
8	#16811000	C₄	1	4	yes	complemented
9	C₂×#13133137	—	2	8	no	—
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	C₈×#88322	—	2	2	no	—
12	C₂×#107×C₅₃	—	2	2	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	C₂²	1	2	yes	—
15	C₂²×D₂₁₂	—	2	4	no	—
16	C₂²×D₂₁₂	—	2	2	no	—
17	C₂×D₄₂₄	—	2	8	no	—
18	C₂×D₄₂₄	—	4	4	no	—
19	C₂×D₄₂₄	—	4	4	no	—
20	C₂²×#88322	D₄	1	4	yes	—
21	#13133214	—	2	2	no	—
22	#13133214	—	2	4	no	—
23	#13133214	C₂×C₄	1	4	yes	—
24	#13133137	—	4	16	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#107×C₅₃	—	4	4	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	C₂³×D₅₃	—	4	2	no	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
38	C₂×D₂₁₂	—	2	4	no	—
39	C₂×D₂₁₂	—	2	8	no	—
40	C₂×D₂₁₂	—	4	8	no	—
41	C₂×D₂₁₂	—	4	4	no	—
42	C₂×D₂₁₂	—	4	4	no	—
43	C₂×D₂₁₂	—	4	8	no	—
44	C₂×D₂₁₂	—	4	4	no	—
45	C₂×D₂₁₂	—	4	2	no	—
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
47	C₂×C₈×C₅₃	D₈	1	4	yes	—
48	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
49	C₄⋊C₄×C₅₃	—	4	4	no	—
50	C₄⋊C₄×C₅₃	—	4	8	no	—
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	2	yes	—
54	C₂×#88322	—	2	4	no	—
55	C₂×#88322	—	2	4	no	—
56	C₂×#88322	D₈	1	8	yes	complemented
57	D₄₂₄	—	4	16	no	—
58	C₂³×C₅₃	#127	1	1	yes	characteristic
59	C₂²×D₅₃	—	4	4	no	—
60	C₂²×D₅₃	—	8	2	no	—
61	C₂²×D₅₃	—	8	4	no	—
62	C₂²×D₅₃	—	8	2	no	—
63	C₂²×D₅₃	—	8	4	no	—
64	C₂×C₄×C₅₃	—	8	4	no	—
65	C₂×C₄×C₅₃	—	8	2	no	—
66	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₅₃	#121	1	2	yes	—
69	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
71	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₅₃	#135	1	4	yes	—
74	D₂₁₂	—	4	8	no	—
75	D₂₁₂	—	4	16	no	—
76	D₂₁₂	—	8	8	no	—
77	D₂₁₂	—	8	4	no	—
78	C₈×C₅₃	#102	1	8	yes	complemented
79	#88322	—	2	8	no	—
80	#9824	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	#390	1	1	yes	characteristic
82	C₂²×C₅₃	#492	1	1	yes	characteristic
83	C₂²×C₅₃	#493	1	2	yes	—
84	C₂²×C₅₃	#493	1	1	yes	characteristic
85	C₂²×C₅₃	#495	1	2	yes	—
86	C₂×D₅₃	—	8	4	no	—
87	C₂×D₅₃	—	8	8	no	—
88	C₂×D₅₃	—	16	4	no	—
89	C₂×D₅₃	—	16	2	no	—
90	C₂×D₅₃	—	16	2	no	—
91	C₄×C₅₃	—	16	2	no	—
92	C₄×C₅₃	C₂×#102	1	2	yes	—
93	C₄×C₅₃	C₄×D₈	1	4	yes	—
94	C₄×C₅₃	#466	1	4	yes	—
95	C₄×C₅₃	#486	1	2	yes	—
96	C₂×#493	—	53	1	no	—
97	#1957	—	53	4	no	—
98	#1845	D₅₃	1	1	yes	characteristic, complemented
99	C₂×C₈²	—	53	1	no	—
100	C₂×C₅₃	#1706	1	2	yes	—
101	C₂×C₅₃	#1709	1	1	yes	characteristic
102	C₂×C₅₃	#1712	1	1	yes	characteristic
103	C₂×C₅₃	#1957	1	1	yes	characteristic
104	C₂×C₅₃	#1970	1	2	yes	—
105	D₅₃	—	16	4	no	—
106	C₂²×D₈	—	106	2	no	—
107	C₂×#127	—	53	2	no	—
108	#493	—	53	4	no	—
109	#493	—	106	2	no	—
110	C₂×#107	—	2	2	no	—
111	C₂×#102	—	106	8	no	—
112	C₂×C₄×C₈	—	53	2	no	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	#384	C₂×D₅₃	1	2	yes	—
115	C₈²	—	106	2	no	—
116	C₅₃	#9824	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₄	—	106	4	no	—
118	C₂²×D₄	—	106	2	no	—
119	C₂×D₈	—	106	8	no	—
120	C₂×D₈	—	212	4	no	—
121	C₂×D₈	—	212	4	no	—
122	C₂²×C₈	—	53	4	no	—
123	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
124	#127	—	53	4	no	—
125	#127	—	106	2	no	—
126	#127	—	106	4	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	#107	—	4	4	no	—
131	#102	—	212	16	no	—
132	C₄×C₈	—	106	2	no	—
133	C₄×C₈	—	106	2	no	—
134	C₄×C₈	C₄×D₅₃	1	2	yes	—
135	C₄×C₈	D₂₁₂	1	2	yes	—
136	C₂⁴	—	212	2	no	—
137	C₂×D₄	—	106	4	no	—
138	C₂×D₄	—	106	8	no	—
139	C₂×D₄	—	212	4	no	—
140	C₂×D₄	—	212	2	no	—
141	C₂×D₄	—	212	8	no	—
142	C₂×D₄	—	212	4	no	—
143	C₂×D₄	—	212	4	no	—
144	C₂×D₄	—	212	8	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	D₄×D₅₃	1	2	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	D₈	—	212	16	no	—
149	C₂×C₈	—	53	8	no	—
150	C₂×C₈	—	106	4	no	—
151	C₂×C₈	—	106	4	no	—
152	C₂×C₈	#11719921	1	4	yes	—
153	C₂×C₈	#11719922	1	4	yes	—
154	C₂×C₈	#11719924	1	4	yes	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
158	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
159	C₄²	#11719949	1	2	yes	—
160	C₂³	—	424	4	no	—
161	C₂³	—	424	2	no	—
162	C₂³	—	424	4	no	—
163	C₂³	—	424	2	no	—
164	C₂³	—	212	4	no	—
165	C₂³	#13133213	1	1	yes	characteristic
166	D₄	—	424	8	no	—
167	D₄	—	424	4	no	—
168	D₄	—	212	8	no	—
169	D₄	—	212	16	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
173	C₂×C₄	#13133179	1	2	yes	—
174	C₂×C₄	#13133192	1	2	yes	—
175	C₂×C₄	#13133206	1	1	yes	characteristic
176	C₂×C₄	D₈×D₅₃	1	4	yes	—
177	C₂×C₄	C₂×#11719922	1	2	yes	—
178	C₂×C₄	C₂×#11719924	1	2	yes	—
179	C₂×C₄	#13133240	1	4	yes	—
180	C₈	—	106	8	no	—
181	C₈	#13133137	1	8	yes	complemented
182	C₂²	—	848	4	no	—
183	C₂²	—	848	2	no	—
184	C₂²	—	848	2	no	—
185	C₂²	—	424	4	no	—
186	C₂²	—	424	8	no	—
187	C₂²	#16810524	1	1	yes	characteristic
188	C₂²	#16810997	1	1	yes	characteristic
189	C₂²	#16810998	1	2	yes	—
190	C₂²	#16810999	1	1	yes	characteristic
191	C₂²	#16811008	1	2	yes	—
192	C₄	—	16	2	no	—
193	C₄	C₂×#13133137	1	2	yes	—
194	C₄	#16810705	1	4	yes	—
195	C₄	#16810860	1	4	yes	—
196	C₄	#16810971	1	2	yes	—
197	C₂	—	848	4	no	—
198	C₂	#20274308	1	2	yes	—
199	C₂	#20274322	1	1	yes	characteristic
200	C₂	#20274329	1	1	yes	characteristic
201	C₂	#20275520	1	1	yes	characteristic
202	C₂	#20275579	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811000

C₂

yes

maximal, characteristic

#20275520

C₂

yes

maximal, complemented

C₂×C₈×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#1845

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₂₄

—

C₂×#13133214

C₂²

yes

—

#16811000

—

#16811000

C₄

yes

complemented

C₂×#13133137

—

C₂×C₄×#88322

C₂²

yes

—

C₈×#88322

—

C₂×#107×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

—

C₂²×D₂₁₂

—

C₂²×D₂₁₂

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂²×#88322

D₄

yes

—

#13133214

—

#13133214

—

#13133214

C₂×C₄

yes

—

#13133137

—

C₄×#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₈

yes

complemented

D₄₂₄

—

C₂³×C₅₃

#127

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

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×D₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#493

yes

characteristic

C₂²×C₅₃

#495

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#102

yes

—

C₄×C₅₃

C₄×D₈

yes

—

C₄×C₅₃

#466

yes

—

C₄×C₅₃

#486

yes

—

C₂×#493

—

#1957

—

#1845

D₅₃

yes

characteristic, complemented

C₂×C₈²

—

100

C₂×C₅₃

#1706

yes

—

101

C₂×C₅₃

#1709

yes

characteristic

102

C₂×C₅₃

#1712

yes

characteristic

103

C₂×C₅₃

#1957

yes

characteristic

104

C₂×C₅₃

#1970

yes

—

105

D₅₃

—

106

C₂²×D₈

—

106

—

107

C₂×#127

—

108

#493

—

109

#493

—

106

—

110

C₂×#107

—

111

C₂×#102

—

106

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

114

#384

C₂×D₅₃

yes

—

115

C₈²

—

106

—

116

C₅₃

#9824

yes

characteristic, complemented, Sylow

117

C₂²×D₄

—

106

—

118

C₂²×D₄

—

106

—

119

C₂×D₈

—

106

—

120

C₂×D₈

—

212

—

121

C₂×D₈

—

212

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₅₃

yes

—

124

#127

—

125

#127

—

106

—

126

#127

—

106

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄²

C₂²×D₅₃

yes

characteristic

130

#107

—

131

#102

—

212

—

132

C₄×C₈

—

106

—

133

C₄×C₈

—

106

—

134

C₄×C₈

C₄×D₅₃

yes

—

135

C₄×C₈

D₂₁₂

yes

—

136

C₂⁴

—

212

—

137

C₂×D₄

—

106

—

138

C₂×D₄

—

106

—

139

C₂×D₄

—

212

—

140

C₂×D₄

—

212

—

141

C₂×D₄

—

212

—

142

C₂×D₄

—

212

—

143

C₂×D₄

—

212

—

144

C₂×D₄

—

212

—

145

C₂²×C₄

—

146

C₂²×C₄

D₄×D₅₃

yes

—

147

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

148

D₈

—

212

—

149

C₂×C₈

—

150

C₂×C₈

—

106

—

151

C₂×C₈

—

106

—

152

C₂×C₈

#11719921

yes

—

153

C₂×C₈

#11719922

yes

—

154

C₂×C₈

#11719924

yes

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄²

C₂×C₄×D₅₃

yes

characteristic

158

C₄²

C₂×D₂₁₂

yes

characteristic

159

C₄²

#11719949

yes

—

160

C₂³

—

424

—

161

C₂³

—

424

—

162

C₂³

—

424

—

163

C₂³

—

424

—

164

C₂³

—

212

—

165

C₂³

#13133213

yes

characteristic

166

D₄

—

424

—

167

D₄

—

424

—

168

D₄

—

212

—

169

D₄

—

212

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

C₂×#11719921

yes

characteristic

173

C₂×C₄

#13133179

yes

—

174

C₂×C₄

#13133192

yes

—

175

C₂×C₄

#13133206

yes

characteristic

176

C₂×C₄

D₈×D₅₃

yes

—

177

C₂×C₄

C₂×#11719922

yes

—

178

C₂×C₄

C₂×#11719924

yes

—

179

C₂×C₄

#13133240

yes

—

180

C₈

—

106

—

181

C₈

#13133137

yes

complemented

182

C₂²

—

848

—

183

C₂²

—

848

—

184

C₂²

—

848

—

185

C₂²

—

424

—

186

C₂²

—

424

—

187

C₂²

#16810524

yes

characteristic

188

C₂²

#16810997

yes

characteristic

189

C₂²

#16810998

yes

—

190

C₂²

#16810999

yes

characteristic

191

C₂²

#16811008

yes

—

192

C₄

—

193

C₄

C₂×#13133137

yes

—

194

C₄

#16810705

yes

—

195

C₄

#16810860

yes

—

196

C₄

#16810971

yes

—

197

C₂

—

848

—

198

C₂

#20274308

yes

—

199

C₂

#20274322

yes

characteristic

200

C₂

#20274329

yes

characteristic

201

C₂

#20275520

yes

characteristic

202

C₂

#20275579

yes

—

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1845	C₂
socle	C₂³×C₅₃	#127
prime radical	C₂×#16811000	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#9824
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1845

C₂

C₂³×C₅₃

C₂

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
2. #20275520	C₂	semidirect
3. C₂×C₈×#88322	C₂	semidirect
4. C₅₃×#1845	C₂	semidirect
8. #16811000	C₄	semidirect
56. C₂×#88322	D₈	semidirect
78. C₈×C₅₃	#102	semidirect
98. #1845	D₅₃	semidirect
116. C₅₃	#9824	semidirect
181. C₈	#13133137	semidirect

H ≅ G/N

type

2. #20275520

C₂

semidirect

3. C₂×C₈×#88322

C₂

semidirect

4. C₅₃×#1845

C₂

semidirect

8. #16811000

C₄

semidirect

56. C₂×#88322

D₈

semidirect

78. C₈×C₅₃

#102

semidirect

98. #1845

D₅₃

semidirect

116. C₅₃

#9824

semidirect

181. C₈

#13133137

semidirect

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

subgroup	index
C₂×#16811000	2
#20275520	2
C₂×C₈×#88322	2
C₅₃×#1845	2
#9824	53

subgroup

index

C₂×#16811000

#20275520

C₂×C₈×#88322

C₅₃×#1845

#9824

Subgroups — #27283854

Conjugacy classes of subgroups

Complements

Maximal subgroups