Group #27287200 — subgroups

Subgroups — #27287200

Order 13568.

402 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20283102	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273275	C₂	1	2	yes	maximal
3	#20272575	C₂	1	2	yes	maximal, complemented
4	#20272548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	#16810521	—	2	2	no	—
8	#16810507	—	2	4	no	—
9	#16810489	C₂²	1	2	yes	—
10	#16810357	—	2	4	no	—
11	C₅₃×#532	C₂²	1	1	yes	characteristic
12	C₅₃×#423	C₂²	1	1	yes	characteristic
13	C₂×#104×C₅₃	—	2	2	no	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅₃	—	2	2	no	—
16	C₂×#11719919	—	2	4	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₂²⋊#88322	—	2	2	no	—
19	C₄×#88322	—	2	2	no	—
20	#13133124	—	2	4	no	—
21	#13133124	—	2	2	no	—
22	#13133124	D₄	1	4	yes	—
23	C₂×#41×C₅₃	—	2	1	no	—
24	C₂×M₁₆×C₅₃	—	2	1	no	—
25	C₄⋊Q₈×C₅₃	Q₈	1	1	yes	characteristic
26	#127×C₅₃	Q₈	1	1	yes	characteristic
27	#124×C₅₃	—	2	1	no	—
28	#121×C₅₃	—	2	2	no	—
29	C₄×D₄×C₅₃	—	2	2	no	—
30	C₂×C₄²×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	#104×C₅₃	—	4	4	no	—
35	#41×C₅₃	—	4	2	no	—
36	#41×C₅₃	—	4	2	no	—
37	C₂×Q₈×C₅₃	—	2	1	no	—
38	C₂×D₄×C₅₃	—	2	1	no	—
39	C₂×D₄×C₅₃	—	2	2	no	—
40	C₂×D₄×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	4	1	no	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	2	4	no	—
46	M₁₆×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	4	1	no	—
48	C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂²⋊C₄×C₅₃	—	4	2	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	—	4	4	no	—
53	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	1	yes	characteristic
55	C₄²×C₅₃	#41	1	2	yes	—
56	C₄⋊Dic₅₃	—	4	2	no	—
57	C₄⋊Dic₅₃	—	4	2	no	—
58	#11719919	—	4	8	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂³×C₅₃	—	4	1	no	—
61	C₂³×C₅₃	#120	1	1	yes	characteristic
62	Q₈×C₅₃	—	4	2	no	—
63	D₄×C₅₃	—	4	2	no	—
64	D₄×C₅₃	—	8	2	no	—
65	D₄×C₅₃	—	8	2	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	1	no	—
71	C₂×C₄×C₅₃	—	8	1	no	—
72	C₂×C₄×C₅₃	—	8	1	no	—
73	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#124	1	2	yes	—
76	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#135	1	2	yes	—
78	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×Dic₅₃	—	4	8	no	—
80	C₂×Dic₅₃	—	4	4	no	—
81	C₈×C₅₃	—	8	1	no	—
82	#88322	—	8	2	no	—
83	#10366	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	—	8	1	no	—
86	C₂²×C₅₃	—	8	2	no	—
87	C₂²×C₅₃	#393	1	1	yes	characteristic
88	C₂²×C₅₃	#449	1	2	yes	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	4	4	no	—
92	C₄×C₅₃	—	8	1	no	—
93	C₄×C₅₃	#478	1	2	yes	—
94	C₄×C₅₃	#488	1	1	yes	characteristic
95	C₄×C₅₃	#490	1	1	yes	characteristic
96	Dic₅₃	—	8	4	no	—
97	#2864	—	53	1	no	—
98	#1962	D₅₃	1	1	yes	characteristic
99	#1517	—	53	2	no	—
100	#1286	—	53	2	no	—
101	#1276	—	53	1	no	—
102	C₂×C₅₃	—	2	2	no	—
103	C₂×C₅₃	—	16	1	no	—
104	C₂×C₅₃	#2025	1	1	yes	characteristic
105	C₂×C₅₃	#2031	1	2	yes	—
106	#532	C₂×D₅₃	1	1	yes	characteristic
107	#431	—	53	2	no	—
108	#423	C₂×D₅₃	1	1	yes	characteristic
109	C₂×#104	—	2	2	no	—
110	#389	—	106	2	no	—
111	#387	—	106	4	no	—
112	#382	—	53	2	no	—
113	C₄³	C₂×D₅₃	1	1	yes	characteristic
114	#327	—	106	4	no	—
115	C₅₃	#10366	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	—	2	1	no	—
117	C₂×M₁₆	—	2	1	no	—
118	C₄⋊Q₈	Dic₁₀₆	1	1	yes	characteristic
119	#127	Dic₁₀₆	1	1	yes	characteristic
120	#124	—	2	1	no	—
121	#121	—	2	2	no	—
122	C₄×D₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	106	4	no	—
124	C₂×C₄⋊C₄	—	106	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	106	2	no	—
130	#104	—	4	4	no	—
131	C₂²⋊C₈	—	106	2	no	—
132	C₄×C₈	—	106	2	no	—
133	#95	—	53	4	no	—
134	#95	—	106	4	no	—
135	#95	—	106	2	no	—
136	#41	—	4	2	no	—
137	#41	—	4	2	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	1	no	—
141	C₂×D₄	—	4	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	4	1	no	—
144	C₂²×C₄	—	106	4	no	—
145	C₂²×C₄	D₄×D₅₃	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	M₁₆	—	4	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	212	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	212	8	no	—
152	C₄⋊C₄	—	212	2	no	—
153	C₄⋊C₄	—	212	2	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	4	no	—
158	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
159	C₄²	#11719945	1	1	yes	characteristic
160	C₄²	#11719947	1	2	yes	—
161	C₂³	—	4	1	no	—
162	C₂³	#13133185	1	1	yes	characteristic
163	Q₈	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	8	2	no	—
166	D₄	—	8	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	212	8	no	—
175	C₂×C₄	—	212	4	no	—
176	C₂×C₄	#13133194	1	2	yes	—
177	C₂×C₄	#13133196	1	1	yes	characteristic
178	C₂×C₄	#13133203	1	2	yes	—
179	C₂×C₄	#13133204	1	1	yes	characteristic
180	C₂×C₄	#13133237	1	2	yes	—
181	C₂×C₄	#13133243	1	2	yes	—
182	C₈	—	8	1	no	—
183	C₈	—	424	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	1	no	—
186	C₂²	—	8	2	no	—
187	C₂²	#16810535	1	1	yes	characteristic
188	C₂²	#16810766	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	8	1	no	—
193	C₄	—	424	4	no	—
194	C₄	#16810923	1	2	yes	—
195	C₄	#16810977	1	1	yes	characteristic
196	C₄	#16810987	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	16	1	no	—
199	C₂	#20275861	1	1	yes	characteristic
200	C₂	#20275887	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283102

C₂

yes

maximal, characteristic, complemented

#20273275

C₂

yes

maximal

#20272575

C₂

yes

maximal, complemented

#20272548

C₂

yes

maximal, characteristic, complemented

C₅₃×#1962

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

#16810521

—

#16810507

—

#16810489

C₂²

yes

—

#16810357

—

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#423

C₂²

yes

characteristic

C₂×#104×C₅₃

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

—

—

—

—

—

—

—

D₄

yes

—

C₂×#41×C₅₃

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

characteristic

#127×C₅₃

Q₈

yes

characteristic

#124×C₅₃

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

#104×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

#120

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×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#10366

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#393

yes

characteristic

C₂²×C₅₃

#449

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#478

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2864

—

#1962

D₅₃

yes

characteristic

#1517

—

100

#1286

—

101

#1276

—

102

C₂×C₅₃

—

103

C₂×C₅₃

—

104

C₂×C₅₃

#2025

yes

characteristic

105

C₂×C₅₃

#2031

yes

—

106

#532

C₂×D₅₃

yes

characteristic

107

#431

—

108

#423

C₂×D₅₃

yes

characteristic

109

C₂×#104

—

110

#389

—

106

—

111

#387

—

106

—

112

#382

—

113

C₄³

C₂×D₅₃

yes

characteristic

114

#327

—

106

—

115

C₅₃

#10366

yes

characteristic, complemented, Sylow

116

C₂×#41

—

117

C₂×M₁₆

—

118

C₄⋊Q₈

Dic₁₀₆

yes

characteristic

119

#127

Dic₁₀₆

yes

characteristic

120

#124

—

121

#121

—

122

C₄×D₄

—

123

C₂×C₄⋊C₄

—

106

—

124

C₂×C₄⋊C₄

—

106

—

125

C₂×C₄²

—

126

C₂×C₄²

C₂²⋊D₅₃

yes

—

127

C₂×C₄²

C₂²×D₅₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

106

—

130

#104

—

131

C₂²⋊C₈

—

106

—

132

C₄×C₈

—

106

—

133

#95

—

134

#95

—

106

—

135

#95

—

106

—

136

#41

—

137

#41

—

138

C₂×Q₈

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

106

—

145

C₂²×C₄

D₄×D₅₃

yes

—

146

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

212

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

212

—

152

C₄⋊C₄

—

212

—

153

C₄⋊C₄

—

212

—

154

C₂²⋊C₄

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂×Dic₁₀₆

yes

characteristic

159

C₄²

#11719945

yes

characteristic

160

C₄²

#11719947

yes

—

161

C₂³

—

162

C₂³

#13133185

yes

characteristic

163

Q₈

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

212

—

175

C₂×C₄

—

212

—

176

C₂×C₄

#13133194

yes

—

177

C₂×C₄

#13133196

yes

characteristic

178

C₂×C₄

#13133203

yes

—

179

C₂×C₄

#13133204

yes

characteristic

180

C₂×C₄

#13133237

yes

—

181

C₂×C₄

#13133243

yes

—

182

C₈

—

183

C₈

—

424

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#16810535

yes

characteristic

188

C₂²

#16810766

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

424

—

194

C₄

#16810923

yes

—

195

C₄

#16810977

yes

characteristic

196

C₄

#16810987

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#20275861

yes

characteristic

200

C₂

#20275887

yes

—

	subgroup	G/H
center	C₂²	#16810535
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1962	C₂
socle	C₂²×C₅₃	#393
prime radical	C₅₃×#532	C₂²
hypercenter	#1962	D₅₃
hypocenter	C₅₃	#10366
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

subgroup

G/H

center

C₂²

#16810535

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1962

C₂

socle

C₂²×C₅₃

#393

prime radical

C₅₃×#532

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

N	H ≅ G/N	type
1. #20283102	C₂	semidirect
3. #20272575	C₂	semidirect
4. #20272548	C₂	semidirect
115. C₅₃	#10366	semidirect

H ≅ G/N

type

1. #20283102

C₂

semidirect

3. #20272575

C₂

semidirect

4. #20272548

C₂

semidirect

115. C₅₃

#10366

semidirect

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

subgroup	index
#20283102	2
#20273275	2
#20272575	2
#20272548	2
C₅₃×#1962	2
#10366	53

subgroup

index

C₅₃×#1962

Subgroups — #27287200

Conjugacy classes of subgroups

Complements

Maximal subgroups