Group #27284063 — subgroups

Subgroups — #27284063

Order 13568.

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

normal subgroups only

This group has 514 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	#20285143	C₂	1	1	yes	maximal, characteristic
2	#20273165	C₂	1	4	yes	maximal, complemented
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133239	—	2	2	no	—
6	#16811173	C₂²	1	2	yes	—
7	#16811007	—	2	2	no	—
8	#16811003	C₄	1	1	yes	characteristic
9	#16811000	C₄	1	1	yes	characteristic
10	#16810996	C₄	1	2	yes	—
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	#16810350	—	2	8	no	—
13	#16810344	—	2	2	no	—
14	C₂×#108×C₅₃	—	2	2	no	—
15	C₅₃×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719945	—	2	2	no	—
18	#13133239	—	4	4	no	—
19	#13133239	—	4	4	no	—
20	C₂×Dic₂₁₂	—	2	2	no	—
21	C₂×#11719913	—	2	4	no	—
22	C₂×D₄₂₄	—	2	2	no	—
23	C₂²×#88322	—	2	2	no	—
24	#13133217	C₂×C₄	1	2	yes	—
25	#13133214	C₂×C₄	1	2	yes	—
26	#13133205	—	2	2	no	—
27	#13133191	—	2	4	no	—
28	C₄×D₂₁₂	—	2	4	no	—
29	C₄×#88322	—	2	2	no	—
30	C₄×#88322	D₄	1	4	yes	—
31	C₂×M₁₆×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#108×C₅₃	—	4	4	no	—
35	C₄⋊C₈×C₅₃	—	2	4	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	2	yes	—
41	C₂×C₂²⋊D₅₃	—	4	2	no	—
42	#11719945	—	4	4	no	—
43	#11719945	—	4	4	no	—
44	C₂×D₂₁₂	—	2	2	no	—
45	C₂×D₂₁₂	—	2	4	no	—
46	C₂×C₄×D₅₃	—	4	2	no	—
47	C₂×Dic₁₀₆	—	2	2	no	—
48	M₁₆×C₅₃	—	4	4	no	—
49	C₂×C₈×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₃	D₈	1	4	yes	—
53	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
54	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×D₄	1	2	yes	—
57	#11719921	—	4	4	no	—
58	C₄⋊Dic₅₃	—	2	4	no	—
59	C₂×#88322	—	2	4	no	—
60	C₂×#88322	—	4	4	no	—
61	Dic₂₁₂	—	4	4	no	—
62	D₄₂₄	—	4	4	no	—
63	#11719913	—	4	8	no	—
64	C₂³×C₅₃	#124	1	1	yes	characteristic
65	C₂²×D₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₅₃	#121	1	2	yes	—
70	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
73	C₂²⋊D₅₃	—	8	4	no	—
74	C₂×Dic₅₃	—	4	2	no	—
75	D₂₁₂	—	4	4	no	—
76	D₂₁₂	—	4	8	no	—
77	C₄×D₅₃	—	8	2	no	—
78	C₄×D₅₃	—	8	2	no	—
79	Dic₁₀₆	—	4	4	no	—
80	C₈×C₅₃	—	8	2	no	—
81	C₈×C₅₃	#102	1	4	yes	—
82	C₈×C₅₃	#102	1	4	yes	—
83	#88322	—	4	4	no	—
84	#9862	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	#390	1	1	yes	characteristic
87	C₂²×C₅₃	#486	1	2	yes	—
88	C₂×D₅₃	—	8	2	no	—
89	C₂×D₅₃	—	8	4	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	C₂×#102	1	2	yes	—
92	C₄×C₅₃	#444	1	4	yes	—
93	C₄×C₅₃	#492	1	1	yes	characteristic
94	C₄×C₅₃	#493	1	1	yes	characteristic
95	Dic₅₃	—	8	2	no	—
96	#3147	—	53	1	no	—
97	#1848	D₅₃	1	1	yes	characteristic, complemented
98	#1492	—	53	4	no	—
99	C₂×#322	—	53	1	no	—
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	#1957	1	1	yes	characteristic
102	C₂×C₅₃	#1975	1	2	yes	—
103	D₅₃	—	16	2	no	—
104	C₂×#135	—	106	2	no	—
105	#532	—	53	2	no	—
106	#495	—	106	2	no	—
107	#494	—	53	1	no	—
108	#493	—	53	1	no	—
109	#492	—	53	2	no	—
110	C₂×#108	—	2	2	no	—
111	#423	C₂×D₅₃	1	2	yes	—
112	C₂×C₄×C₈	—	53	2	no	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	D₄⋊C₈	—	106	8	no	—
115	#322	—	106	2	no	—
116	C₅₃	#9862	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	—	106	2	no	—
118	#135	—	212	4	no	—
119	#135	—	212	4	no	—
120	C₂×Q₁₆	—	106	2	no	—
121	C₂×SD₁₆	—	106	4	no	—
122	C₂×D₈	—	106	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	—	106	2	no	—
125	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
126	C₄⋊Q₈	—	53	2	no	—
127	#127	—	53	2	no	—
128	#124	—	106	2	no	—
129	#121	—	106	4	no	—
130	C₄×D₄	—	106	4	no	—
131	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
132	#108	—	4	4	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄×C₈	—	53	4	no	—
135	C₄×C₈	—	106	2	no	—
136	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
137	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
138	C₄×C₈	D₂₁₂	1	2	yes	—
139	#41	—	212	4	no	—
140	#41	—	212	4	no	—
141	C₂×Q₈	—	106	2	no	—
142	C₂×D₄	—	106	2	no	—
143	C₂×D₄	—	106	4	no	—
144	C₂×D₄	—	212	2	no	—
145	C₂²×C₄	—	212	2	no	—
146	C₂²×C₄	#11719949	1	2	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	Q₁₆	—	212	4	no	—
149	SD₁₆	—	212	8	no	—
150	D₈	—	212	4	no	—
151	M₁₆	—	4	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	106	4	no	—
154	C₂×C₈	—	212	4	no	—
155	C₂×C₈	#11719921	1	2	yes	—
156	C₂×C₈	#11719921	1	2	yes	—
157	C₂×C₈	#11719922	1	4	yes	—
158	C₂×C₈	#11719924	1	4	yes	—
159	C₄⋊C₄	—	106	4	no	—
160	C₂²⋊C₄	—	212	4	no	—
161	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
162	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
163	C₄²	D₄×D₅₃	1	2	yes	—
164	C₂³	—	212	2	no	—
165	C₂³	#13133206	1	1	yes	characteristic
166	Q₈	—	212	4	no	—
167	D₄	—	424	4	no	—
168	D₄	—	212	4	no	—
169	D₄	—	212	8	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	424	2	no	—
172	C₂×C₄	—	424	2	no	—
173	C₂×C₄	—	212	2	no	—
174	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
175	C₂×C₄	#13133179	1	2	yes	—
176	C₂×C₄	#13133192	1	2	yes	—
177	C₂×C₄	#13133213	1	1	yes	characteristic
178	C₂×C₄	C₂×#11719922	1	2	yes	—
179	C₂×C₄	C₂×#11719924	1	2	yes	—
180	C₈	—	8	2	no	—
181	C₈	—	212	4	no	—
182	C₈	#13133137	1	4	yes	—
183	C₈	#13133137	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	424	2	no	—
186	C₂²	—	424	4	no	—
187	C₂²	#16810524	1	1	yes	characteristic
188	C₂²	#16810971	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	424	2	no	—
191	C₄	C₂×#13133137	1	2	yes	—
192	C₄	#16810744	1	4	yes	—
193	C₄	#16810997	1	1	yes	characteristic
194	C₄	#16810999	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	848	2	no	—
197	C₂	#20275520	1	1	yes	characteristic
198	C₂	#20275606	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20285143

C₂

yes

maximal, characteristic

#20273165

C₂

yes

maximal, complemented

C₂×#16810344

C₂

yes

maximal, characteristic, complemented

C₅₃×#1848

C₂

yes

maximal, characteristic, complemented

C₂×#13133239

—

#16811173

C₂²

yes

—

#16811007

—

#16811003

C₄

yes

characteristic

#16811000

C₄

yes

characteristic

#16810996

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810350

—

#16810344

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

C₂×Dic₂₁₂

—

C₂×#11719913

—

C₂×D₄₂₄

—

C₂²×#88322

—

#13133217

C₂×C₄

yes

—

#13133214

C₂×C₄

yes

—

#13133205

—

#13133191

—

C₄×D₂₁₂

—

C₄×#88322

—

C₄×#88322

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

Dic₂₁₂

—

D₄₂₄

—

#11719913

—

C₂³×C₅₃

#124

yes

characteristic

C₂²×D₅₃

—

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

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

#102

yes

—

#88322

—

#9862

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#486

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#102

yes

—

C₄×C₅₃

#444

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#3147

—

#1848

D₅₃

yes

characteristic, complemented

#1492

—

C₂×#322

—

100

C₂×C₅₃

—

101

C₂×C₅₃

#1957

yes

characteristic

102

C₂×C₅₃

#1975

yes

—

103

D₅₃

—

104

C₂×#135

—

106

—

105

#532

—

106

#495

—

106

—

107

#494

—

108

#493

—

109

#492

—

110

C₂×#108

—

111

#423

C₂×D₅₃

yes

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

114

D₄⋊C₈

—

106

—

115

#322

—

106

—

116

C₅₃

#9862

yes

characteristic, complemented, Sylow

117

C₂×#41

—

106

—

118

#135

—

212

—

119

#135

—

212

—

120

C₂×Q₁₆

—

106

—

121

C₂×SD₁₆

—

106

—

122

C₂×D₈

—

106

—

123

C₂×M₁₆

—

124

C₂²×C₈

—

106

—

125

C₂²×C₈

C₂²⋊D₅₃

yes

—

126

C₄⋊Q₈

—

127

#127

—

128

#124

—

106

—

129

#121

—

106

—

130

C₄×D₄

—

106

—

131

C₂×C₄²

C₂²×D₅₃

yes

characteristic

132

#108

—

133

C₄⋊C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

106

—

136

C₄×C₈

C₄×D₅₃

yes

characteristic

137

C₄×C₈

C₄×D₅₃

yes

characteristic

138

C₄×C₈

D₂₁₂

yes

—

139

#41

—

212

—

140

#41

—

212

—

141

C₂×Q₈

—

106

—

142

C₂×D₄

—

106

—

143

C₂×D₄

—

106

—

144

C₂×D₄

—

212

—

145

C₂²×C₄

—

212

—

146

C₂²×C₄

#11719949

yes

—

147

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

148

Q₁₆

—

212

—

149

SD₁₆

—

212

—

150

D₈

—

212

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

106

—

154

C₂×C₈

—

212

—

155

C₂×C₈

#11719921

yes

—

156

C₂×C₈

#11719921

yes

—

157

C₂×C₈

#11719922

yes

—

158

C₂×C₈

#11719924

yes

—

159

C₄⋊C₄

—

106

—

160

C₂²⋊C₄

—

212

—

161

C₄²

C₂×C₄×D₅₃

yes

characteristic

162

C₄²

C₂×D₂₁₂

yes

characteristic

163

C₄²

D₄×D₅₃

yes

—

164

C₂³

—

212

—

165

C₂³

#13133206

yes

characteristic

166

Q₈

—

212

—

167

D₄

—

424

—

168

D₄

—

212

—

169

D₄

—

212

—

170

C₂×C₄

—

171

C₂×C₄

—

424

—

172

C₂×C₄

—

424

—

173

C₂×C₄

—

212

—

174

C₂×C₄

C₂×#11719921

yes

characteristic

175

C₂×C₄

#13133179

yes

—

176

C₂×C₄

#13133192

yes

—

177

C₂×C₄

#13133213

yes

characteristic

178

C₂×C₄

C₂×#11719922

yes

—

179

C₂×C₄

C₂×#11719924

yes

—

180

C₈

—

181

C₈

—

212

—

182

C₈

#13133137

yes

—

183

C₈

#13133137

yes

—

184

C₂²

—

185

C₂²

—

424

—

186

C₂²

—

424

—

187

C₂²

#16810524

yes

characteristic

188

C₂²

#16810971

yes

—

189

C₄

—

190

C₄

—

424

—

191

C₄

C₂×#13133137

yes

—

192

C₄

#16810744

yes

—

193

C₄

#16810997

yes

characteristic

194

C₄

#16810999

yes

characteristic

195

C₂

—

196

C₂

—

848

—

197

C₂

#20275520

yes

characteristic

198

C₂

#20275606

yes

—

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	#390
prime radical	#20285143	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#9862
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1848

C₂

C₂²×C₅₃

C₂

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
2. #20273165	C₂	semidirect
3. C₂×#16810344	C₂	semidirect
4. C₅₃×#1848	C₂	semidirect
97. #1848	D₅₃	semidirect
116. C₅₃	#9862	semidirect

H ≅ G/N

type

2. #20273165

C₂

semidirect

3. C₂×#16810344

C₂

semidirect

4. C₅₃×#1848

C₂

semidirect

97. #1848

D₅₃

semidirect

116. C₅₃

#9862

semidirect

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

subgroup	index
#20285143	2
#20273165	2
C₂×#16810344	2
C₅₃×#1848	2
#9862	53

subgroup

index

C₅₃×#1848

Subgroups — #27284063

Conjugacy classes of subgroups

Complements

Maximal subgroups