Group #27277090 — subgroups

Subgroups — #27277090

Order 13568.

388 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20283427	C₂	1	1	yes	maximal, characteristic
2	#20273476	C₂	1	2	yes	maximal, complemented
3	#20273350	C₂	1	1	yes	maximal, characteristic
4	#20272625	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1536	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133224	C₄	1	2	yes	—
7	#16810701	—	2	2	no	—
8	D₄×#88322	—	2	2	no	—
9	#16810680	C₂²	1	2	yes	—
10	#16810647	C₂²	1	1	yes	characteristic
11	#16810590	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810375	—	2	2	no	—
14	#16810373	—	2	2	no	—
15	#16810363	—	2	2	no	—
16	C₅₃×#532	C₂²	1	1	yes	characteristic
17	C₅₃×#432	C₂²	1	2	yes	—
18	C₅₃×#329	—	2	2	no	—
19	#13133224	—	4	4	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	C₂×C₄	1	2	yes	—
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₂²⋊#88322	—	2	4	no	—
28	C₂²⋊#88322	—	2	2	no	—
29	#13133128	—	2	2	no	—
30	C₄×#88322	D₄	1	2	yes	—
31	C₄×#88322	D₄	1	2	yes	—
32	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₅₃	Q₈	1	1	yes	characteristic
34	#127×C₅₃	Q₈	1	1	yes	characteristic
35	#124×C₅₃	D₄	1	2	yes	—
36	#121×C₅₃	—	2	2	no	—
37	C₄×D₄×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₂²⋊C₈×C₅₃	—	2	2	no	—
41	#97×C₅₃	—	2	2	no	—
42	#41×C₅₃	—	4	2	no	—
43	#41×C₅₃	—	4	2	no	—
44	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂×D₄×C₅₃	—	2	2	no	—
46	C₂×D₄×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	#41	1	2	yes	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₂²⋊C₄×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	C₂×D₄	1	2	yes	—
55	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	#11719917	—	4	4	no	—
58	#11719917	—	4	2	no	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	4	1	no	—
62	C₂×#88322	—	4	4	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	C₂²⋊C₄	1	2	yes	—
65	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	#117	1	1	yes	characteristic
69	Q₈×C₅₃	—	4	2	no	—
70	D₄×C₅₃	—	4	2	no	—
71	D₄×C₅₃	—	4	4	no	—
72	D₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#121	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
83	C₈×C₅₃	—	8	2	no	—
84	#88322	—	2	1	no	—
85	#88322	—	2	2	no	—
86	#88322	—	4	2	no	—
87	#88322	—	8	2	no	—
88	#88322	#100	1	2	yes	—
89	#8632	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	—	4	2	no	—
93	C₂²×C₅₃	—	4	4	no	—
94	C₂²×C₅₃	#380	1	1	yes	characteristic
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	2	2	no	—
98	C₄×C₅₃	—	4	2	no	—
99	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
100	C₄×C₅₃	#413	1	1	yes	characteristic
101	#2931	—	53	1	no	—
102	#1568	—	53	2	no	—
103	#1536	—	53	1	no	—
104	#1536	D₅₃	1	1	yes	characteristic
105	#1307	—	53	2	no	—
106	C₂×C₅₃	—	4	1	no	—
107	C₂×C₅₃	—	8	2	no	—
108	C₂×C₅₃	#1800	1	1	yes	characteristic
109	C₂×C₅₃	#1814	1	2	yes	—
110	C₂×#131	—	53	2	no	—
111	#532	C₂×D₅₃	1	1	yes	characteristic
112	#436	—	106	2	no	—
113	C₈×D₄	—	106	2	no	—
114	#432	—	53	2	no	—
115	#432	C₂×D₅₃	1	2	yes	—
116	#423	—	53	1	no	—
117	#408	—	53	2	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	#335	—	106	2	no	—
120	#334	—	106	2	no	—
121	#329	—	2	2	no	—
122	#329	—	106	2	no	—
123	C₅₃	#8632	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	C₄×D₅₃	1	2	yes	—
125	#131	—	212	4	no	—
126	C₂×M₁₆	—	53	2	no	—
127	C₂×M₁₆	—	106	2	no	—
128	C₂²×C₈	—	53	1	no	—
129	C₂²×C₈	—	53	1	no	—
130	C₂²×C₈	—	106	2	no	—
131	C₄⋊Q₈	Dic₁₀₆	1	1	yes	characteristic
132	#127	Dic₁₀₆	1	1	yes	characteristic
133	#124	C₂²⋊D₅₃	1	2	yes	—
134	#121	—	2	2	no	—
135	C₄×D₄	—	2	2	no	—
136	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	106	2	no	—
139	C₄⋊C₈	—	106	2	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	106	4	no	—
142	C₂²⋊C₈	—	106	2	no	—
143	#97	—	2	2	no	—
144	#97	—	106	2	no	—
145	C₄×C₈	—	53	2	no	—
146	C₄×C₈	—	53	2	no	—
147	#41	—	4	2	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	#11719919	1	2	yes	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	#11719919	1	2	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
155	C₂²×C₄	#11719947	1	2	yes	—
156	M₁₆	—	212	4	no	—
157	M₁₆	—	212	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	53	1	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	—	212	4	no	—
165	C₂×C₈	—	212	1	no	—
166	C₂×C₈	—	212	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
170	C₄²	D₄×D₅₃	1	2	yes	—
171	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂³	—	2	2	no	—
173	C₂³	#13133173	1	1	yes	characteristic
174	Q₈	—	4	2	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
186	C₂×C₄	#13133189	1	2	yes	—
187	C₂×C₄	#13133194	1	2	yes	—
188	C₈	—	8	2	no	—
189	C₈	—	53	2	no	—
190	C₈	—	106	1	no	—
191	C₈	—	106	2	no	—
192	C₈	—	424	2	no	—
193	C₈	—	212	2	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	#16810481	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	#100×D₅₃	1	1	yes	characteristic
204	C₄	#16810604	1	1	yes	characteristic
205	C₂	—	4	1	no	—
206	C₂	—	8	2	no	—
207	C₂	#20274705	1	1	yes	characteristic
208	C₂	#20274767	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283427

C₂

yes

maximal, characteristic

#20273476

C₂

yes

maximal, complemented

#20273350

C₂

yes

maximal, characteristic

#20272625

C₂

yes

maximal, complemented

C₅₃×#1536

C₂

yes

maximal, characteristic

C₂×#13133224

C₄

yes

—

#16810701

—

D₄×#88322

—

#16810680

C₂²

yes

—

#16810647

C₂²

yes

characteristic

#16810590

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

—

—

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#432

C₂²

yes

—

C₅₃×#329

—

#13133224

—

C₂×#11719917

—

C₂×#11719917

C₂×C₄

yes

—

C₂²×#88322

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#88322

D₄

yes

—

C₂×#41×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

Q₈

yes

characteristic

#127×C₅₃

Q₈

yes

characteristic

#124×C₅₃

D₄

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₄⋊C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

—

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

#117

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

—

—

—

—

—

yes

—

#8632

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#380

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#100

yes

characteristic

100

C₄×C₅₃

#413

yes

characteristic

101

#2931

—

102

#1568

—

103

#1536

—

104

#1536

D₅₃

yes

characteristic

105

#1307

—

106

C₂×C₅₃

—

107

C₂×C₅₃

—

108

C₂×C₅₃

#1800

yes

characteristic

109

C₂×C₅₃

#1814

yes

—

110

C₂×#131

—

111

#532

C₂×D₅₃

yes

characteristic

112

#436

—

106

—

113

C₈×D₄

—

106

—

114

#432

—

115

#432

C₂×D₅₃

yes

—

116

#423

—

117

#408

—

118

C₂×C₄×C₈

—

119

#335

—

106

—

120

#334

—

106

—

121

#329

—

122

#329

—

106

—

123

C₅₃

#8632

yes

characteristic, complemented, Sylow

124

C₂×#41

C₄×D₅₃

yes

—

125

#131

—

212

—

126

C₂×M₁₆

—

127

C₂×M₁₆

—

106

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

—

106

—

131

C₄⋊Q₈

Dic₁₀₆

yes

characteristic

132

#127

Dic₁₀₆

yes

characteristic

133

#124

C₂²⋊D₅₃

yes

—

134

#121

—

135

C₄×D₄

—

136

C₂×C₄²

C₂²×D₅₃

yes

characteristic

137

C₄⋊C₈

—

138

C₄⋊C₈

—

106

—

139

C₄⋊C₈

—

106

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

106

—

142

C₂²⋊C₈

—

106

—

143

#97

—

144

#97

—

106

—

145

C₄×C₈

—

146

C₄×C₈

—

147

#41

—

148

#41

—

149

C₂×Q₈

#11719919

yes

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

#11719919

yes

—

153

C₂²×C₄

—

154

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

155

C₂²×C₄

#11719947

yes

—

156

M₁₆

—

212

—

157

M₁₆

—

212

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

106

—

163

C₂×C₈

—

106

—

164

C₂×C₈

—

212

—

165

C₂×C₈

—

212

—

166

C₂×C₈

—

212

—

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

C₂×Dic₁₀₆

yes

characteristic

170

C₄²

D₄×D₅₃

yes

—

171

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

172

C₂³

—

173

C₂³

#13133173

yes

characteristic

174

Q₈

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂²⋊C₄×D₅₃

yes

characteristic

185

C₂×C₄

C₂×#11719919

yes

characteristic

186

C₂×C₄

#13133189

yes

—

187

C₂×C₄

#13133194

yes

—

188

C₈

—

189

C₈

—

190

C₈

—

106

—

191

C₈

—

106

—

192

C₈

—

424

—

193

C₈

—

212

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#16810481

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

#100×D₅₃

yes

characteristic

204

C₄

#16810604

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#20274705

yes

characteristic

208

C₂

#20274767

yes

—

	subgroup	G/H
center	C₂²	#16810481
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1536	C₂
socle	C₂²×C₅₃	#380
prime radical	C₅₃×#532	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1536	D₅₃
hypocenter	C₅₃	#8632
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

subgroup

G/H

center

C₂²

#16810481

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1536

C₂

socle

C₂²×C₅₃

#380

prime radical

C₅₃×#532

C₂²

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

N	H ≅ G/N	type
2. #20273476	C₂	semidirect
4. #20272625	C₂	semidirect
123. C₅₃	#8632	semidirect

H ≅ G/N

type

2. #20273476

C₂

semidirect

4. #20272625

C₂

semidirect

123. C₅₃

#8632

semidirect

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

subgroup	index
#20283427	2
#20273476	2
#20273350	2
#20272625	2
C₅₃×#1536	2
#8632	53

subgroup

index

C₅₃×#1536

Subgroups — #27277090

Conjugacy classes of subgroups

Complements

Maximal subgroups