Group #20274726 — subgroups

Subgroups — #20274726

Order 6784.

356 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133139	C₂	1	2	yes	maximal
3	#16810489	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810386	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
7	C₂²×C₄×D₅₃	—	2	1	no	—
8	C₄×D₂₁₂	—	2	2	no	—
9	C₄×D₂₁₂	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#11719921	—	2	1	no	—
13	#13133139	—	2	4	no	—
14	#13133124	—	2	1	no	—
15	#13133124	—	2	1	no	—
16	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₅₃	—	2	2	no	—
19	C₂³×D₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	1	no	—
24	C₂²×Dic₅₃	—	2	2	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
29	C₂×D₂₁₂	D₄	1	2	yes	—
30	C₂×D₂₁₂	Q₈	1	2	yes	—
31	C₂×C₄×D₅₃	—	2	4	no	—
32	C₂×C₄×D₅₃	—	4	4	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₄²×C₅₃	D₄	1	2	yes	—
38	#11719921	—	2	4	no	—
39	C₄⋊Dic₅₃	—	2	1	no	—
40	C₄⋊Dic₅₃	—	2	2	no	—
41	C₄⋊Dic₅₃	—	2	2	no	—
42	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
43	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×D₅₃	—	2	2	no	—
45	C₂²×D₅₃	—	2	2	no	—
46	C₂²×D₅₃	—	2	2	no	—
47	C₂²×D₅₃	—	4	2	no	—
48	C₂²×D₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	—	4	2	no	—
61	C₂×Dic₅₃	—	4	2	no	—
62	C₂×Dic₅₃	—	4	2	no	—
63	C₂×Dic₅₃	—	4	2	no	—
64	D₂₁₂	—	2	2	no	—
65	D₂₁₂	—	2	2	no	—
66	D₂₁₂	—	2	2	no	—
67	D₂₁₂	C₄⋊C₄	1	4	yes	—
68	C₄×D₅₃	—	4	8	no	—
69	C₈×C₅₃	—	4	2	no	—
70	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₅₃	#121	1	1	yes	characteristic
72	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂²×C₅₃	#135	1	2	yes	—
74	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×D₅₃	—	2	4	no	—
76	C₂×D₅₃	—	4	2	no	—
77	C₂×D₅₃	—	4	2	no	—
78	C₂×D₅₃	—	4	4	no	—
79	C₂×D₅₃	—	4	4	no	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₅₃	#117	1	1	yes	characteristic
84	C₄×C₅₃	#121	1	1	yes	characteristic
85	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	Dic₅₃	—	4	2	no	—
87	Dic₅₃	—	8	2	no	—
88	#1804	—	53	1	no	maximal, Sylow
89	C₂×C₅₃	#380	1	1	yes	characteristic
90	C₂×C₅₃	#416	1	1	yes	characteristic
91	C₂×C₅₃	#418	1	1	yes	characteristic
92	C₂×C₅₃	#463	1	2	yes	—
93	C₂×C₅₃	#478	1	2	yes	—
94	D₅₃	—	4	4	no	—
95	C₂×C₄×D₄	—	53	1	no	—
96	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic, complemented
97	C₂×#102	—	53	2	no	—
98	#382	—	53	1	no	—
99	#340	—	53	2	no	—
100	C₅₃	#1804	1	1	yes	characteristic, complemented, Sylow
101	C₂²×D₄	—	53	1	no	—
102	C₂³×C₄	—	106	1	no	—
103	C₂²×C₈	C₂×D₅₃	1	2	yes	—
104	C₄×D₄	—	53	4	no	—
105	C₄×D₄	—	106	2	no	—
106	C₂×C₄⋊C₄	—	53	1	no	—
107	C₂×C₄⋊C₄	—	53	2	no	—
108	C₂×C₂²⋊C₄	—	106	1	no	—
109	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
110	C₄⋊C₈	—	2	2	no	—
111	#102	—	106	4	no	—
112	#95	—	106	1	no	—
113	#95	—	106	1	no	—
114	C₂⁴	—	106	1	no	—
115	C₂×D₄	—	53	2	no	—
116	C₂×D₄	—	53	2	no	—
117	C₂×D₄	—	53	2	no	—
118	C₂×D₄	—	106	1	no	—
119	C₂×D₄	—	106	1	no	—
120	C₂×D₄	—	106	1	no	—
121	C₂²×C₄	—	106	1	no	—
122	C₂²×C₄	—	106	4	no	—
123	C₂²×C₄	—	106	2	no	—
124	C₂²×C₄	—	212	4	no	—
125	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
126	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₄⋊C₄	—	53	2	no	—
132	C₄⋊C₄	—	106	1	no	—
133	C₄⋊C₄	—	106	2	no	—
134	C₄⋊C₄	—	106	2	no	—
135	C₂²⋊C₄	—	106	4	no	—
136	C₄²	C₄×D₅₃	1	2	yes	—
137	C₄²	C₂²⋊D₅₃	1	2	yes	—
138	C₂³	—	106	2	no	—
139	C₂³	—	106	2	no	—
140	C₂³	—	106	2	no	—
141	C₂³	—	212	2	no	—
142	C₂³	—	212	2	no	—
143	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
144	D₄	—	53	4	no	—
145	D₄	—	106	2	no	—
146	D₄	—	106	2	no	—
147	D₄	—	106	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	106	2	no	—
151	C₂×C₄	—	212	8	no	—
152	C₂×C₄	—	212	2	no	—
153	C₂×C₄	—	212	2	no	—
154	C₂×C₄	—	212	2	no	—
155	C₂×C₄	—	212	2	no	—
156	C₂×C₄	#11719921	1	2	yes	—
157	C₂×C₄	#11719921	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
159	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
160	C₂×C₄	#11719947	1	1	yes	characteristic
161	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
162	C₂×C₄	#11719949	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₂²	—	106	4	no	—
166	C₂²	—	212	4	no	—
167	C₂²	—	212	2	no	—
168	C₂²	—	212	2	no	—
169	C₂²	—	212	4	no	—
170	C₂²	C₂×#11719921	1	1	yes	characteristic
171	C₂²	#13133192	1	1	yes	characteristic
172	C₂²	#13133197	1	1	yes	characteristic
173	C₂²	#13133239	1	2	yes	—
174	C₂²	#13133245	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	424	2	no	—
178	C₄	—	212	2	no	—
179	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
180	C₄	#13133175	1	1	yes	characteristic
181	C₄	#13133188	1	1	yes	characteristic
182	C₄	#13133193	1	1	yes	characteristic
183	C₂	—	212	4	no	—
184	C₂	#16810480	1	1	yes	characteristic
185	C₂	#16810620	1	1	yes	characteristic
186	C₂	#16810629	1	1	yes	characteristic
187	C₂	#16810839	1	2	yes	—
188	C₂	#16810924	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₂₁₂

C₂

yes

maximal, characteristic

C₂×#13133139

C₂

yes

maximal

#16810489

C₂

yes

maximal, characteristic, complemented

#16810386

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂²×C₄×D₅₃

—

C₄×D₂₁₂

—

C₄×D₂₁₂

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×D₂₁₂

D₄

yes

—

C₂×D₂₁₂

Q₈

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

C₄⋊C₄

yes

—

C₄×D₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#478

yes

—

D₅₃

—

C₂×C₄×D₄

—

C₂×C₄⋊C₈

D₅₃

yes

characteristic, complemented

C₂×#102

—

#382

—

#340

—

100

C₅₃

#1804

yes

characteristic, complemented, Sylow

101

C₂²×D₄

—

102

C₂³×C₄

—

106

—

103

C₂²×C₈

C₂×D₅₃

yes

—

104

C₄×D₄

—

105

C₄×D₄

—

106

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₂²⋊C₄

—

106

—

109

C₂×C₄²

C₂×D₅₃

yes

characteristic

110

C₄⋊C₈

—

111

#102

—

106

—

112

#95

—

106

—

113

#95

—

106

—

114

C₂⁴

—

106

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

106

—

119

C₂×D₄

—

106

—

120

C₂×D₄

—

106

—

121

C₂²×C₄

—

106

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

—

106

—

124

C₂²×C₄

—

212

—

125

C₂²×C₄

D₂₁₂

yes

characteristic

126

C₂²×C₄

D₂₁₂

yes

characteristic

127

C₂²×C₄

C₂²×D₅₃

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

106

—

135

C₂²⋊C₄

—

106

—

136

C₄²

C₄×D₅₃

yes

—

137

C₄²

C₂²⋊D₅₃

yes

—

138

C₂³

—

106

—

139

C₂³

—

106

—

140

C₂³

—

106

—

141

C₂³

—

212

—

142

C₂³

—

212

—

143

C₂³

C₂×D₂₁₂

yes

characteristic

144

D₄

—

145

D₄

—

106

—

146

D₄

—

106

—

147

D₄

—

106

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

212

—

152

C₂×C₄

—

212

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

—

212

—

156

C₂×C₄

#11719921

yes

—

157

C₂×C₄

#11719921

yes

—

158

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

159

C₂×C₄

D₄×D₅₃

yes

characteristic

160

C₂×C₄

#11719947

yes

characteristic

161

C₂×C₄

Q₈×D₅₃

yes

characteristic

162

C₂×C₄

#11719949

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

164

C₈

—

165

C₂²

—

106

—

166

C₂²

—

212

—

167

C₂²

—

212

—

168

C₂²

—

212

—

169

C₂²

—

212

—

170

C₂²

C₂×#11719921

yes

characteristic

171

C₂²

#13133192

yes

characteristic

172

C₂²

#13133197

yes

characteristic

173

C₂²

#13133239

yes

—

174

C₂²

#13133245

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

424

—

178

C₄

—

212

—

179

C₄

C₄⋊C₄×D₅₃

yes

characteristic

180

C₄

#13133175

yes

characteristic

181

C₄

#13133188

yes

characteristic

182

C₄

#13133193

yes

characteristic

183

C₂

—

212

—

184

C₂

#16810480

yes

characteristic

185

C₂

#16810620

yes

characteristic

186

C₂

#16810629

yes

characteristic

187

C₂

#16810839

yes

—

188

C₂

#16810924

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1804
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₅₃

hypocenter

C₅₃

#1804

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #16810489	C₂	semidirect
4. #16810386	C₂	semidirect
5. C₂×C₄⋊C₈×C₅₃	C₂	semidirect
9. C₄×D₂₁₂	C₄	semidirect
96. C₂×C₄⋊C₈	D₅₃	semidirect
100. C₅₃	#1804	semidirect

H ≅ G/N

type

3. #16810489

C₂

semidirect

4. #16810386

C₂

semidirect

5. C₂×C₄⋊C₈×C₅₃

C₂

semidirect

9. C₄×D₂₁₂

C₄

semidirect

96. C₂×C₄⋊C₈

D₅₃

semidirect

100. C₅₃

#1804

semidirect

Subgroups — #20274726

Conjugacy classes of subgroups

Complements