Group #27283931 — subgroups

Subgroups — #27283931

Order 13568.

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

normal subgroups only

This group has 710 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	#20275522	C₂	1	4	yes	maximal, complemented
2	#20274872	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133149	—	2	2	no	—
6	C₂×#13133138	—	2	8	no	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	#16810493	C₂²	1	2	yes	—
9	#16810344	—	2	2	no	—
10	C₂²×D₈×C₅₃	—	2	2	no	—
11	C₂×#127×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#493	—	2	2	no	—
13	C₅₃×#493	C₄	1	4	yes	complemented
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#88322	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₅₃	—	2	4	no	—
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	#13133149	—	4	4	no	—
19	#13133138	—	4	16	no	—
20	C₄×#88322	—	2	2	no	—
21	C₄×#88322	—	2	2	no	—
22	C₂²×D₄×C₅₃	—	2	2	no	—
23	C₂²×D₄×C₅₃	—	2	4	no	—
24	C₂×D₈×C₅₃	—	2	8	no	—
25	C₂×D₈×C₅₃	—	4	4	no	—
26	C₂×D₈×C₅₃	—	4	4	no	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	#127×C₅₃	—	2	2	no	—
29	#127×C₅₃	—	2	4	no	—
30	#127×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	C₂⁴×C₅₃	—	4	2	no	—
35	C₂×D₄×C₅₃	—	2	4	no	—
36	C₂×D₄×C₅₃	—	2	8	no	—
37	C₂×D₄×C₅₃	—	4	4	no	—
38	C₂×D₄×C₅₃	—	4	2	no	—
39	C₂×D₄×C₅₃	—	4	4	no	—
40	C₂×D₄×C₅₃	—	4	4	no	—
41	C₂×D₄×C₅₃	—	4	8	no	—
42	C₂×D₄×C₅₃	—	4	8	no	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂²×Dic₅₃	—	4	2	no	—
46	D₈×C₅₃	—	4	16	no	—
47	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
48	C₂×C₈×C₅₃	D₈	1	4	yes	—
49	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅₃	#41	1	2	yes	—
53	C₄⋊Dic₅₃	—	4	4	no	—
54	C₄⋊Dic₅₃	—	4	8	no	—
55	C₂×#88322	—	2	4	no	—
56	C₂×#88322	—	2	8	no	—
57	C₂³×C₅₃	—	4	4	no	—
58	C₂³×C₅₃	—	8	2	no	—
59	C₂³×C₅₃	—	8	2	no	—
60	C₂³×C₅₃	—	8	4	no	—
61	C₂³×C₅₃	—	8	4	no	—
62	C₂³×C₅₃	#127	1	1	yes	characteristic
63	D₄×C₅₃	—	4	8	no	—
64	D₄×C₅₃	—	4	16	no	—
65	D₄×C₅₃	—	8	8	no	—
66	D₄×C₅₃	—	8	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	#124	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	8	yes	—
74	C₂×Dic₅₃	—	8	2	no	—
75	C₂×Dic₅₃	—	8	4	no	—
76	C₈×C₅₃	#102	1	8	yes	complemented
77	#88322	—	4	4	no	—
78	#9837	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	8	4	no	—
80	C₂²×C₅₃	—	8	8	no	—
81	C₂²×C₅₃	—	16	2	no	—
82	C₂²×C₅₃	—	16	2	no	—
83	C₂²×C₅₃	—	16	4	no	—
84	C₂²×C₅₃	#390	1	1	yes	characteristic
85	C₂²×C₅₃	#492	1	1	yes	characteristic
86	C₂²×C₅₃	#493	1	1	yes	characteristic
87	C₂²×C₅₃	#496	1	4	yes	—
88	C₄×C₅₃	C₂×#102	1	2	yes	—
89	C₄×C₅₃	#442	1	4	yes	—
90	C₄×C₅₃	#469	1	4	yes	—
91	C₄×C₅₃	#486	1	2	yes	—
92	Dic₅₃	—	16	2	no	—
93	C₂×#493	D₅₃	1	1	yes	characteristic
94	#1957	—	53	4	no	—
95	#1844	—	53	1	no	—
96	C₂×#322	—	53	1	no	—
97	C₂×C₅₃	—	16	4	no	—
98	C₂×C₅₃	#1715	1	2	yes	—
99	C₂×C₅₃	#1721	1	2	yes	—
100	C₂×C₅₃	#1957	1	1	yes	characteristic
101	C₂×C₅₃	#1972	1	2	yes	—
102	C₂²×D₈	—	2	2	no	—
103	C₂×#127	C₂×D₅₃	1	2	yes	—
104	#493	—	2	2	no	—
105	#493	Dic₅₃	1	4	yes	complemented
106	C₂×#106	—	106	2	no	—
107	C₂×#102	—	106	8	no	—
108	C₂×C₄×C₈	—	53	2	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	#384	—	53	2	no	—
111	#322	—	106	2	no	—
112	C₅₃	#9837	1	1	yes	characteristic, complemented, Sylow
113	C₂²×D₄	—	2	4	no	—
114	C₂²×D₄	—	2	2	no	—
115	C₂×D₈	—	2	8	no	—
116	C₂×D₈	—	4	4	no	—
117	C₂×D₈	—	4	4	no	—
118	C₂²×C₈	—	53	4	no	—
119	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
120	#127	—	2	4	no	—
121	#127	—	2	2	no	—
122	#127	C₂×Dic₅₃	1	4	yes	—
123	C₂×C₄⋊C₄	—	106	4	no	—
124	C₂×C₄⋊C₄	—	106	2	no	—
125	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
126	#106	—	212	4	no	—
127	#102	—	212	16	no	—
128	C₄×C₈	—	106	2	no	—
129	C₄×C₈	—	106	2	no	—
130	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
131	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
132	C₂⁴	—	4	2	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂×D₄	—	4	4	no	—
138	C₂×D₄	—	4	4	no	—
139	C₂×D₄	—	4	8	no	—
140	C₂×D₄	—	4	8	no	—
141	C₂²×C₄	—	212	2	no	—
142	C₂²×C₄	D₄×D₅₃	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
144	D₈	—	4	16	no	—
145	C₂×C₈	—	106	4	no	—
146	C₂×C₈	—	106	8	no	—
147	C₂×C₈	#11719922	1	4	yes	—
148	C₂×C₈	#11719923	1	4	yes	—
149	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
150	C₄⋊C₄	—	212	4	no	—
151	C₄⋊C₄	—	212	8	no	—
152	C₄²	#11719947	1	2	yes	—
153	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
154	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂³	—	4	4	no	—
156	C₂³	—	8	4	no	—
157	C₂³	—	8	4	no	—
158	C₂³	—	8	2	no	—
159	C₂³	—	8	2	no	—
160	C₂³	#13133213	1	1	yes	characteristic
161	D₄	—	4	8	no	—
162	D₄	—	4	16	no	—
163	D₄	—	8	8	no	—
164	D₄	—	8	4	no	—
165	C₂×C₄	—	424	4	no	—
166	C₂×C₄	—	424	2	no	—
167	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
168	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
169	C₂×C₄	#13133188	1	2	yes	—
170	C₂×C₄	#13133204	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719922	1	2	yes	—
172	C₂×C₄	C₂×#11719923	1	2	yes	—
173	C₂×C₄	#13133241	1	8	yes	—
174	C₈	—	212	4	no	—
175	C₈	#13133138	1	8	yes	complemented
176	C₂²	—	8	4	no	—
177	C₂²	—	8	8	no	—
178	C₂²	—	16	2	no	—
179	C₂²	—	16	2	no	—
180	C₂²	—	16	4	no	—
181	C₂²	#16810526	1	1	yes	characteristic
182	C₂²	#16810995	1	1	yes	characteristic
183	C₂²	#16810999	1	1	yes	characteristic
184	C₂²	#16811011	1	4	yes	—
185	C₄	—	848	2	no	—
186	C₄	C₂×#13133138	1	2	yes	—
187	C₄	#16810735	1	4	yes	—
188	C₄	#16810873	1	4	yes	—
189	C₄	#16810969	1	2	yes	—
190	C₂	—	16	4	no	—
191	C₂	#20274341	1	2	yes	—
192	C₂	#20274382	1	2	yes	—
193	C₂	#20275522	1	1	yes	characteristic
194	C₂	#20275592	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275522

C₂

yes

maximal, complemented

#20274872

C₂

yes

maximal, characteristic, complemented

C₂×#16810344

C₂

yes

maximal, characteristic, complemented

C₂×C₅₃×#493

C₂

yes

maximal, characteristic

C₂×#13133149

—

C₂×#13133138

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

C₂²

yes

—

#16810344

—

C₂²×D₈×C₅₃

—

C₂×#127×C₅₃

C₂²

yes

—

C₅₃×#493

—

C₅₃×#493

C₄

yes

complemented

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×#88322

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

#127×C₅₃

—

#127×C₅₃

—

#127×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

D₈×C₅₃

—

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

#41

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#127

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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×SD₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

C₂×#102

yes

—

C₄×C₅₃

#442

yes

—

C₄×C₅₃

#469

yes

—

C₄×C₅₃

#486

yes

—

Dic₅₃

—

C₂×#493

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1715

yes

—

C₂×C₅₃

#1721

yes

—

100

C₂×C₅₃

#1957

yes

characteristic

101

C₂×C₅₃

#1972

yes

—

102

C₂²×D₈

—

103

C₂×#127

C₂×D₅₃

yes

—

104

#493

—

105

#493

Dic₅₃

yes

complemented

106

C₂×#106

—

106

—

107

C₂×#102

—

106

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

110

#384

—

111

#322

—

106

—

112

C₅₃

#9837

yes

characteristic, complemented, Sylow

113

C₂²×D₄

—

114

C₂²×D₄

—

115

C₂×D₈

—

116

C₂×D₈

—

117

C₂×D₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂²⋊D₅₃

yes

—

120

#127

—

121

#127

—

122

#127

C₂×Dic₅₃

yes

—

123

C₂×C₄⋊C₄

—

106

—

124

C₂×C₄⋊C₄

—

106

—

125

C₂×C₄²

C₂²×D₅₃

yes

characteristic

126

#106

—

212

—

127

#102

—

212

—

128

C₄×C₈

—

106

—

129

C₄×C₈

—

106

—

130

C₄×C₈

C₂×Dic₅₃

yes

—

131

C₄×C₈

C₂²⋊D₅₃

yes

—

132

C₂⁴

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂²×C₄

—

212

—

142

C₂²×C₄

D₄×D₅₃

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

144

D₈

—

145

C₂×C₈

—

106

—

146

C₂×C₈

—

106

—

147

C₂×C₈

#11719922

yes

—

148

C₂×C₈

#11719923

yes

—

149

C₂×C₈

C₂²⋊Dic₅₃

yes

—

150

C₄⋊C₄

—

212

—

151

C₄⋊C₄

—

212

—

152

C₄²

#11719947

yes

—

153

C₄²

C₂²×Dic₅₃

yes

characteristic

154

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

#13133213

yes

characteristic

161

D₄

—

162

D₄

—

163

D₄

—

164

D₄

—

165

C₂×C₄

—

424

—

166

C₂×C₄

—

424

—

167

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

168

C₂×C₄

D₄×Dic₅₃

yes

—

169

C₂×C₄

#13133188

yes

—

170

C₂×C₄

#13133204

yes

characteristic

171

C₂×C₄

C₂×#11719922

yes

—

172

C₂×C₄

C₂×#11719923

yes

—

173

C₂×C₄

#13133241

yes

—

174

C₈

—

212

—

175

C₈

#13133138

yes

complemented

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

#16810526

yes

characteristic

182

C₂²

#16810995

yes

characteristic

183

C₂²

#16810999

yes

characteristic

184

C₂²

#16811011

yes

—

185

C₄

—

848

—

186

C₄

C₂×#13133138

yes

—

187

C₄

#16810735

yes

—

188

C₄

#16810873

yes

—

189

C₄

#16810969

yes

—

190

C₂

—

191

C₂

#20274341

yes

—

192

C₂

#20274382

yes

—

193

C₂

#20275522

yes

characteristic

194

C₂

#20275592

yes

—

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#493	C₂
socle	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	#9837
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂³

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting, prime radical

C₂×C₅₃×#493

C₂

socle

C₂³×C₅₃

#127

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #20275522	C₂	semidirect
2. #20274872	C₂	semidirect
3. C₂×#16810344	C₂	semidirect
13. C₅₃×#493	C₄	semidirect
76. C₈×C₅₃	#102	semidirect
105. #493	Dic₅₃	semidirect
112. C₅₃	#9837	semidirect
175. C₈	#13133138	semidirect

H ≅ G/N

type

1. #20275522

C₂

semidirect

2. #20274872

C₂

semidirect

3. C₂×#16810344

C₂

semidirect

13. C₅₃×#493

C₄

semidirect

76. C₈×C₅₃

#102

semidirect

105. #493

Dic₅₃

semidirect

112. C₅₃

#9837

semidirect

175. C₈

#13133138

semidirect

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

subgroup	index
#20275522	2
#20274872	2
C₂×#16810344	2
C₂×C₅₃×#493	2
#9837	53

subgroup

index

C₂×C₅₃×#493

Subgroups — #27283931

Conjugacy classes of subgroups

Complements

Maximal subgroups