Group #20279751 — subgroups

Subgroups — #20279751

Order 6784.

452 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133208	C₂	1	2	yes	maximal
2	#16810556	C₂	1	4	yes	maximal
3	#16810522	C₂	1	2	yes	maximal
4	#16810519	C₂	1	2	yes	maximal
5	#16810492	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
9	#13133208	—	2	8	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719919	C₂²	1	4	yes	—
16	C₂×#11719919	C₂²	1	4	yes	—
17	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
18	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
19	#13133124	C₂²	1	4	yes	—
20	#13133124	C₂²	1	2	yes	—
21	#13133124	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₃	C₂³	1	2	yes	—
31	C₂²×Dic₅₃	C₂³	1	2	yes	—
32	C₂²×Dic₅₃	C₂³	1	2	yes	—
33	C₂²×Dic₅₃	C₂³	1	2	yes	—
34	C₄⋊C₄×C₅₃	—	2	4	no	—
35	C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₄⋊C₄×C₅₃	—	4	2	no	—
37	C₄²×C₅₃	Q₈	1	4	yes	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	—	2	8	no	—
41	C₄⋊Dic₅₃	D₄	1	4	yes	—
42	C₄⋊Dic₅₃	Q₈	1	4	yes	—
43	#11719919	—	2	8	no	—
44	#11719919	—	4	4	no	—
45	C₄×Dic₅₃	—	2	4	no	—
46	C₄×Dic₅₃	—	2	4	no	—
47	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	4	no	—
51	C₂×C₄×C₅₃	—	2	4	no	—
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₅₃	#41	1	2	yes	—
59	C₂×C₄×C₅₃	#41	1	2	yes	—
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	—	2	2	no	—
62	C₂×Dic₅₃	—	2	2	no	—
63	C₂×Dic₅₃	—	2	2	no	—
64	C₂×Dic₅₃	—	2	2	no	—
65	C₂×Dic₅₃	—	2	2	no	—
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
71	C₂×Dic₅₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
76	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
77	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	4	4	no	—
81	C₄×C₅₃	#125	1	4	yes	complemented
82	C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
83	Dic₅₃	—	2	4	no	—
84	Dic₅₃	—	2	4	no	—
85	Dic₅₃	—	4	2	no	—
86	Dic₅₃	—	4	2	no	—
87	#2552	—	53	1	no	maximal, Sylow
88	C₂×C₅₃	C₂×#125	1	1	yes	characteristic
89	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₅₃	#544	1	1	yes	characteristic
91	C₂×C₅₃	#547	1	1	yes	characteristic
92	C₂×C₅₃	#550	1	1	yes	characteristic
93	C₂×C₅₃	#554	1	1	yes	characteristic
94	C₂×C₅₃	#557	1	1	yes	characteristic
95	C₂×#125	—	53	2	no	—
96	#398	—	53	4	no	—
97	#389	—	53	2	no	—
98	#389	—	53	2	no	—
99	#384	—	53	2	no	—
100	#384	D₅₃	1	1	yes	characteristic
101	C₄×C₄⋊C₄	—	53	1	no	—
102	C₄×C₄⋊C₄	—	53	1	no	—
103	C₅₃	#2552	1	1	yes	characteristic, complemented, Sylow
104	#125	—	106	8	no	—
105	C₂×C₄⋊C₄	—	53	1	no	—
106	C₂×C₄⋊C₄	—	53	1	no	—
107	C₂×C₄⋊C₄	—	53	4	no	—
108	C₂×C₄⋊C₄	—	53	4	no	—
109	C₂×C₄⋊C₄	—	53	1	no	—
110	C₂×C₄⋊C₄	—	53	1	no	—
111	C₂×C₄⋊C₄	—	53	4	no	—
112	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
115	C₂×C₄²	—	53	2	no	—
116	C₂×C₄²	—	53	2	no	—
117	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
118	#95	—	53	2	no	—
119	#95	—	53	2	no	—
120	#95	—	53	4	no	—
121	C₂²×C₄	—	53	2	no	—
122	C₂²×C₄	—	53	2	no	—
123	C₂²×C₄	—	53	2	no	—
124	C₂²×C₄	—	53	2	no	—
125	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
126	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	53	4	no	—
133	C₄⋊C₄	—	53	4	no	—
134	C₄⋊C₄	—	106	8	no	—
135	C₄⋊C₄	—	106	8	no	—
136	C₄⋊C₄	—	106	2	no	—
137	C₄⋊C₄	—	106	2	no	—
138	C₄⋊C₄	—	212	4	no	—
139	C₄²	—	106	4	no	—
140	C₄²	—	106	4	no	—
141	C₄²	Dic₁₀₆	1	4	yes	—
142	C₂³	C₂³×D₅₃	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	53	4	no	—
149	C₂×C₄	—	53	4	no	—
150	C₂×C₄	—	106	2	no	—
151	C₂×C₄	—	106	2	no	—
152	C₂×C₄	—	106	2	no	—
153	C₂×C₄	—	106	2	no	—
154	C₂×C₄	—	106	2	no	—
155	C₂×C₄	—	106	2	no	—
156	C₂×C₄	—	106	2	no	—
157	C₂×C₄	—	106	2	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
162	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
163	C₂×C₄	D₄×D₅₃	1	2	yes	—
164	C₂×C₄	#11719947	1	2	yes	—
165	C₂×C₄	Q₈×D₅₃	1	2	yes	—
166	C₂×C₄	#11719949	1	2	yes	—
167	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
169	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
170	C₂²	C₂×#11719947	1	1	yes	characteristic
171	C₂²	C₂×#11719949	1	1	yes	characteristic
172	C₂²	#13133262	1	1	yes	characteristic
173	C₂²	#13133265	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	4	no	—
176	C₄	—	106	4	no	—
177	C₄	—	106	4	no	—
178	C₄	—	212	2	no	—
179	C₄	—	212	2	no	—
180	C₄	#13133208	1	4	yes	complemented
181	C₄	#13133216	1	4	yes	—
182	C₂	C₂×#13133208	1	1	yes	characteristic
183	C₂	C₂×#13133216	1	1	yes	characteristic
184	C₂	#16811254	1	1	yes	characteristic
185	C₂	#16811276	1	1	yes	characteristic
186	C₂	#16811293	1	1	yes	characteristic
187	C₂	#16811334	1	1	yes	characteristic
188	C₂	#16811348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133208

C₂

yes

maximal

#16810556

C₂

yes

maximal

#16810522

C₂

yes

maximal

#16810519

C₂

yes

maximal

#16810492

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

C₅₃×#384

C₂

yes

maximal, characteristic

#13133208

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄⋊Dic₅₃

Q₈

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#125

yes

complemented

C₄×C₅₃

C₄⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

100

#384

D₅₃

yes

characteristic

101

C₄×C₄⋊C₄

—

102

C₄×C₄⋊C₄

—

103

C₅₃

#2552

yes

characteristic, complemented, Sylow

104

#125

—

106

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

113

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

114

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×D₅₃

yes

characteristic

118

#95

—

119

#95

—

120

#95

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂²×D₅₃

yes

—

126

C₂²×C₄

C₂²×D₅₃

yes

characteristic

127

C₂²×C₄

C₂²×D₅₃

yes

characteristic

128

C₂²×C₄

C₂²×D₅₃

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

106

—

135

C₄⋊C₄

—

106

—

136

C₄⋊C₄

—

106

—

137

C₄⋊C₄

—

106

—

138

C₄⋊C₄

—

212

—

139

C₄²

—

106

—

140

C₄²

—

106

—

141

C₄²

Dic₁₀₆

yes

—

142

C₂³

C₂³×D₅₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

106

—

152

C₂×C₄

—

106

—

153

C₂×C₄

—

106

—

154

C₂×C₄

—

106

—

155

C₂×C₄

—

106

—

156

C₂×C₄

—

106

—

157

C₂×C₄

—

106

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

106

—

160

C₂×C₄

C₂×Dic₁₀₆

yes

—

161

C₂×C₄

C₂×Dic₁₀₆

yes

—

162

C₂×C₄

C₂×Dic₁₀₆

yes

—

163

C₂×C₄

D₄×D₅₃

yes

—

164

C₂×C₄

#11719947

yes

—

165

C₂×C₄

Q₈×D₅₃

yes

—

166

C₂×C₄

#11719949

yes

—

167

C₂²

C₂×D₄×D₅₃

yes

characteristic

168

C₂²

C₂×Q₈×D₅₃

yes

characteristic

169

C₂²

C₂²×Dic₁₀₆

yes

characteristic

170

C₂²

C₂×#11719947

yes

characteristic

171

C₂²

C₂×#11719949

yes

characteristic

172

C₂²

#13133262

yes

characteristic

173

C₂²

#13133265

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

106

—

177

C₄

—

106

—

178

C₄

—

212

—

179

C₄

—

212

—

180

C₄

#13133208

yes

complemented

181

C₄

#13133216

yes

—

182

C₂

C₂×#13133208

yes

characteristic

183

C₂

C₂×#13133216

yes

characteristic

184

C₂

#16811254

yes

characteristic

185

C₂

#16811276

yes

characteristic

186

C₂

#16811293

yes

characteristic

187

C₂

#16811334

yes

characteristic

188

C₂

#16811348

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#384	C₂
hypercenter	#384	D₅₃
hypocenter	C₅₃	#2552
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#384

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
81. C₄×C₅₃	#125	semidirect
103. C₅₃	#2552	semidirect
180. C₄	#13133208	semidirect

H ≅ G/N

type

81. C₄×C₅₃

#125

semidirect

103. C₅₃

#2552

semidirect

180. C₄

#13133208

semidirect

Subgroups — #20279751

Conjugacy classes of subgroups

Complements