Group #27284040 — subgroups

Subgroups — #27284040

Order 13568.

428 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20274853	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273170	C₂	1	2	yes	maximal, complemented
3	#20273166	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3131	C₂	1	1	yes	maximal, characteristic
6	#16810647	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	#16810388	—	2	2	no	—
9	Q₈⋊#88322	—	2	4	no	—
10	D₄⋊#88322	—	2	4	no	—
11	C₈×#88322	—	2	2	no	—
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₅₃×#488	—	2	2	no	—
15	C₅₃×#487	C₄	1	2	yes	—
16	C₅₃×#486	C₄	1	2	yes	—
17	C₅₃×#416	—	2	2	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719917	—	2	2	no	—
20	C₂²×#88322	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄×#88322	—	2	2	no	—
24	C₄×#88322	D₄	1	4	yes	—
25	C₂×#41×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	D₄	1	2	yes	—
27	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#125×C₅₃	—	2	1	no	—
31	#124×C₅₃	—	2	1	no	—
32	C₂²⋊Q₈×C₅₃	—	2	2	no	—
33	#121×C₅₃	—	2	2	no	—
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×D₄×C₅₃	—	2	2	no	—
36	#117×C₅₃	—	2	1	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#103×C₅₃	—	2	4	no	—
39	#102×C₅₃	—	2	4	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	#41×C₅₃	—	4	4	no	—
43	C₂×Q₈×C₅₃	—	2	1	no	—
44	C₂×Q₈×C₅₃	—	2	2	no	—
45	C₂×D₄×C₅₃	—	2	1	no	—
46	C₂×D₄×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	—	4	1	no	—
48	C₂²×C₄×C₅₃	—	4	1	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	#41	1	2	yes	—
51	C₂×C₈×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₂²⋊C₄×C₅₃	—	4	1	no	—
58	C₂²⋊C₄×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	—	4	1	no	—
60	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×D₄	1	2	yes	—
63	#11719917	—	4	4	no	—
64	C₂×#88322	—	2	8	no	—
65	C₂×#88322	—	2	4	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂³×C₅₃	—	4	1	no	—
68	C₂³×C₅₃	#124	1	1	yes	characteristic
69	Q₈×C₅₃	—	4	2	no	—
70	Q₈×C₅₃	—	4	4	no	—
71	D₄×C₅₃	—	4	2	no	—
72	D₄×C₅₃	—	4	4	no	—
73	D₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	4	1	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	8	2	no	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₅₃	#121	1	2	yes	—
81	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#135	1	4	yes	—
83	C₈×C₅₃	—	2	4	no	—
84	#88322	—	2	8	no	—
85	#88322	—	8	2	no	—
86	#9857	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	2	2	no	—
88	C₂²×C₅₃	—	8	1	no	—
89	C₂²×C₅₃	—	8	2	no	—
90	C₂²×C₅₃	#390	1	1	yes	characteristic
91	C₂²×C₅₃	#488	1	2	yes	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	8	1	no	—
94	C₄×C₅₃	—	8	2	no	—
95	C₄×C₅₃	#416	1	2	yes	—
96	C₄×C₅₃	#443	1	4	yes	—
97	C₄×C₅₃	#495	1	2	yes	—
98	#3131	D₅₃	1	1	yes	characteristic
99	#1837	—	53	1	no	—
100	#1493	—	53	2	no	—
101	#1492	—	53	2	no	—
102	C₂×C₈²	—	53	1	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	—	16	1	no	—
105	C₂×C₅₃	#1958	1	1	yes	characteristic
106	C₂×C₅₃	#1974	1	2	yes	—
107	#533	C₂×D₅₃	1	1	yes	characteristic
108	#532	C₂×D₅₃	1	1	yes	characteristic
109	#488	—	2	2	no	—
110	#487	Dic₅₃	1	2	yes	—
111	#486	Dic₅₃	1	2	yes	—
112	#423	—	53	2	no	—
113	#416	—	2	2	no	—
114	C₂×C₄×C₈	—	53	2	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#341	—	106	2	no	—
117	Q₈⋊C₈	—	106	4	no	—
118	D₄⋊C₈	—	106	4	no	—
119	C₈²	—	106	2	no	—
120	C₅₃	#9857	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	2	1	no	—
122	C₂×M₁₆	—	106	2	no	—
123	C₂²×C₈	—	106	2	no	—
124	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
125	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
126	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
127	#127	C₂×Dic₅₃	1	1	yes	characteristic
128	#125	—	2	1	no	—
129	#124	—	2	1	no	—
130	C₂²⋊Q₈	—	2	2	no	—
131	#121	—	2	2	no	—
132	C₄×Q₈	—	2	2	no	—
133	C₄×D₄	—	2	2	no	—
134	#117	—	2	1	no	—
135	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
136	C₄⋊C₈	—	106	2	no	—
137	C₄⋊C₈	—	106	2	no	—
138	#103	—	2	4	no	—
139	#102	—	2	4	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	106	2	no	—
142	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
143	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
144	#41	—	4	4	no	—
145	C₂×Q₈	—	2	1	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	4	1	no	—
150	C₂²×C₄	—	4	1	no	—
151	C₂²×C₄	#11719947	1	2	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	M₁₆	—	212	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	106	8	no	—
156	C₂×C₈	—	106	4	no	—
157	C₂×C₈	—	212	2	no	—
158	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₂²⋊C₄	—	4	2	no	—
165	C₄²	—	4	1	no	—
166	C₄²	D₄×D₅₃	1	2	yes	—
167	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
168	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
169	C₂³	—	4	1	no	—
170	C₂³	#13133204	1	1	yes	characteristic
171	Q₈	—	4	2	no	—
172	Q₈	—	4	4	no	—
173	D₄	—	4	2	no	—
174	D₄	—	4	4	no	—
175	D₄	—	8	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
182	C₂×C₄	#13133188	1	2	yes	—
183	C₂×C₄	#13133213	1	1	yes	characteristic
184	C₂×C₄	#13133237	1	4	yes	—
185	C₈	—	2	4	no	—
186	C₈	—	106	8	no	—
187	C₈	—	424	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	8	1	no	—
190	C₂²	—	8	2	no	—
191	C₂²	#16810526	1	1	yes	characteristic
192	C₂²	#16810977	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	8	1	no	—
195	C₄	—	8	2	no	—
196	C₄	#16810619	1	2	yes	—
197	C₄	#16810740	1	4	yes	—
198	C₄	#16811005	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	16	1	no	—
201	C₂	#20275527	1	1	yes	characteristic
202	C₂	#20275604	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274853

C₂

yes

maximal, characteristic, complemented

#20273170

C₂

yes

maximal, complemented

#20273166

C₂

yes

maximal

C₂×C₈×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#3131

C₂

yes

maximal, characteristic

#16810647

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

—

—

—

—

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#488

—

C₅₃×#487

C₄

yes

—

C₅₃×#486

C₄

yes

—

C₅₃×#416

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

C₂²×#88322

—

—

—

—

D₄

yes

—

C₂×#41×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

characteristic

#127×C₅₃

C₂×C₄

yes

characteristic

#125×C₅₃

—

#124×C₅₃

—

C₂²⋊Q₈×C₅₃

—

#121×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×D₄×C₅₃

—

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#103×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

#124

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

#127

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#488

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#416

yes

—

C₄×C₅₃

#443

yes

—

C₄×C₅₃

#495

yes

—

#3131

D₅₃

yes

characteristic

#1837

—

100

#1493

—

101

#1492

—

102

C₂×C₈²

—

103

C₂×C₅₃

—

104

C₂×C₅₃

—

105

C₂×C₅₃

#1958

yes

characteristic

106

C₂×C₅₃

#1974

yes

—

107

#533

C₂×D₅₃

yes

characteristic

108

#532

C₂×D₅₃

yes

characteristic

109

#488

—

110

#487

Dic₅₃

yes

—

111

#486

Dic₅₃

yes

—

112

#423

—

113

#416

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

116

#341

—

106

—

117

Q₈⋊C₈

—

106

—

118

D₄⋊C₈

—

106

—

119

C₈²

—

106

—

120

C₅₃

#9857

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

C₂×M₁₆

—

106

—

123

C₂²×C₈

—

106

—

124

C₂²×C₈

C₂²⋊D₅₃

yes

—

125

C₄⋊Q₈

C₂×Dic₅₃

yes

characteristic

126

C₄⋊Q₈

C₂×Dic₅₃

yes

—

127

#127

C₂×Dic₅₃

yes

characteristic

128

#125

—

129

#124

—

130

C₂²⋊Q₈

—

131

#121

—

132

C₄×Q₈

—

133

C₄×D₄

—

134

#117

—

135

C₂×C₄²

C₂²×D₅₃

yes

characteristic

136

C₄⋊C₈

—

106

—

137

C₄⋊C₈

—

106

—

138

#103

—

139

#102

—

140

C₄×C₈

—

141

C₄×C₈

—

106

—

142

C₄×C₈

C₂×Dic₅₃

yes

—

143

C₄×C₈

C₂²⋊D₅₃

yes

—

144

#41

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂²×C₄

—

151

C₂²×C₄

#11719947

yes

—

152

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

153

M₁₆

—

212

—

154

C₂×C₈

—

155

C₂×C₈

—

106

—

156

C₂×C₈

—

106

—

157

C₂×C₈

—

212

—

158

C₂×C₈

C₂²⋊Dic₅₃

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

D₄×D₅₃

yes

—

167

C₄²

C₂²×Dic₅₃

yes

characteristic

168

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

169

C₂³

—

170

C₂³

#13133204

yes

characteristic

171

Q₈

—

172

Q₈

—

173

D₄

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

181

C₂×C₄

D₄×Dic₅₃

yes

—

182

C₂×C₄

#13133188

yes

—

183

C₂×C₄

#13133213

yes

characteristic

184

C₂×C₄

#13133237

yes

—

185

C₈

—

186

C₈

—

106

—

187

C₈

—

424

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

#16810526

yes

characteristic

192

C₂²

#16810977

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

#16810619

yes

—

197

C₄

#16810740

yes

—

198

C₄

#16811005

yes

—

199

C₂

—

200

C₂

—

201

C₂

#20275527

yes

characteristic

202

C₂

#20275604

yes

—

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3131	C₂
socle	C₂²×C₅₃	#390
prime radical	C₅₃×#532	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3131	D₅₃
hypocenter	C₅₃	#9857
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#3131

C₂

socle

C₂²×C₅₃

#390

prime radical

C₅₃×#532

C₂²

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

N	H ≅ G/N	type
1. #20274853	C₂	semidirect
2. #20273170	C₂	semidirect
4. C₂×C₈×#88322	C₂	semidirect
120. C₅₃	#9857	semidirect

H ≅ G/N

type

1. #20274853

C₂

semidirect

2. #20273170

C₂

semidirect

4. C₂×C₈×#88322

C₂

semidirect

120. C₅₃

#9857

semidirect

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

subgroup	index
#20274853	2
#20273170	2
#20273166	2
C₂×C₈×#88322	2
C₅₃×#3131	2
#9857	53

subgroup

index

#20274853

#20273170

#20273166

C₂×C₈×#88322

C₅₃×#3131

#9857

Subgroups — #27284040

Conjugacy classes of subgroups

Complements

Maximal subgroups