Group #20283347 — subgroups

Subgroups — #20283347

Order 6784.

484 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 197 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×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16810679	C₂	1	4	yes	maximal, complemented
3	#16810675	C₂	1	4	yes	maximal, complemented
4	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810586	C₂	1	2	yes	maximal
6	M₁₆×Dic₅₃	C₂	1	2	yes	maximal, complemented
7	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₅₃	C₄	1	2	yes	—
9	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	2	yes	—
11	C₂×C₄×Dic₅₃	C₄	1	2	yes	—
12	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₄²×D₅₃	C₄	1	2	yes	—
14	C₄²×D₅₃	C₄	1	2	yes	—
15	C₄²×D₅₃	C₂²	1	2	yes	—
16	C₄²×D₅₃	C₂²	1	2	yes	—
17	C₄⋊#88322	C₂²	1	4	yes	complemented
18	#13133129	C₂²	1	8	yes	—
19	#13133127	C₂²	1	4	yes	complemented
20	C₈×Dic₅₃	C₂²	1	4	yes	complemented
21	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	4	yes	complemented
24	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₅₃	—	2	2	no	—
30	C₂×C₄×D₅₃	—	2	2	no	—
31	C₂×C₄×D₅₃	—	2	8	no	—
32	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₅₃	C₂³	1	2	yes	—
38	M₁₆×C₅₃	—	2	4	no	—
39	C₂×C₈×C₅₃	C₂³	1	4	yes	—
40	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₂³	1	2	yes	—
43	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
44	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
45	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
46	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
47	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
48	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
49	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
50	#11719917	—	2	4	no	—
51	C₂×#88322	C₂³	1	4	yes	—
52	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×D₅₃	—	2	2	no	—
54	C₂²×D₅₃	—	2	4	no	—
55	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₅₃	—	2	4	no	—
57	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₅₃	#41	1	2	yes	—
61	C₂×C₄×C₅₃	#41	1	2	yes	—
62	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₅₃	—	2	2	no	—
64	C₂×Dic₅₃	—	2	4	no	—
65	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
66	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
67	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
68	C₄×D₅₃	—	2	4	no	—
69	C₄×D₅₃	—	2	4	no	—
70	C₄×D₅₃	—	2	4	no	—
71	C₄×D₅₃	—	2	2	no	—
72	C₄×D₅₃	—	2	2	no	—
73	C₄×D₅₃	—	2	4	no	—
74	C₄×D₅₃	M₁₆	1	8	yes	complemented
75	C₈×C₅₃	—	2	4	no	—
76	#88322	—	2	4	no	—
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	#117	1	2	yes	—
79	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
80	C₂×D₅₃	—	2	8	no	—
81	C₂×D₅₃	—	2	2	no	—
82	C₂×D₅₃	—	2	2	no	—
83	C₂×D₅₃	C₂×M₁₆	1	4	yes	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	#117	1	2	yes	—
86	C₄×C₅₃	C₂×M₁₆	1	4	yes	—
87	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
88	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
89	Dic₅₃	—	2	2	no	—
90	Dic₅₃	—	2	2	no	—
91	Dic₅₃	C₂×M₁₆	1	4	yes	—
92	#2920	—	53	1	no	maximal, Sylow
93	C₂×C₅₃	—	2	2	no	—
94	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
95	C₂×C₅₃	C₂²×M₁₆	1	2	yes	—
96	D₅₃	—	2	4	no	—
97	C₂²×C₄²	—	53	1	no	—
98	#432	—	53	4	no	—
99	#431	—	53	4	no	—
100	#423	—	53	1	no	—
101	#423	D₅₃	1	1	yes	characteristic, complemented
102	#407	—	53	2	no	—
103	C₄×M₁₆	—	53	2	no	—
104	C₅₃	#2920	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₄	—	53	2	no	—
106	C₂³×C₄	—	53	1	no	—
107	C₂×M₁₆	—	53	2	no	—
108	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
109	C₂×C₄²	—	53	2	no	—
110	C₂×C₄²	—	53	1	no	—
111	C₂×C₄²	—	53	2	no	—
112	C₂×C₄²	—	53	2	no	—
113	C₂×C₄²	—	53	2	no	—
114	C₂×C₄²	—	53	2	no	—
115	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
116	C₄⋊C₈	—	53	4	no	—
117	C₄⋊C₈	C₂×D₅₃	1	4	yes	complemented
118	C₂²⋊C₈	—	53	8	no	—
119	#97	—	53	4	no	—
120	C₄×C₈	—	53	4	no	—
121	C₂⁴	—	53	1	no	—
122	C₂²×C₄	—	53	2	no	—
123	C₂²×C₄	—	53	2	no	—
124	C₂²×C₄	—	53	1	no	—
125	C₂²×C₄	—	53	2	no	—
126	C₂²×C₄	—	53	2	no	—
127	C₂²×C₄	—	53	2	no	—
128	C₂²×C₄	—	53	2	no	—
129	C₂²×C₄	—	53	2	no	—
130	C₂²×C₄	—	106	8	no	—
131	C₂²×C₄	—	106	2	no	—
132	C₂²×C₄	—	106	2	no	—
133	C₂²×C₄	C₄×D₅₃	1	2	yes	—
134	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
135	M₁₆	—	2	4	no	—
136	M₁₆	—	106	4	no	—
137	C₂×C₈	—	53	4	no	—
138	C₂×C₈	C₂²×D₅₃	1	4	yes	—
139	C₄²	—	53	2	no	—
140	C₄²	—	53	2	no	—
141	C₄²	—	53	2	no	—
142	C₄²	—	53	2	no	—
143	C₄²	—	53	2	no	—
144	C₄²	—	53	1	no	—
145	C₄²	—	53	1	no	—
146	C₄²	C₄×D₅₃	1	1	yes	characteristic
147	C₄²	C₄×D₅₃	1	1	yes	characteristic
148	C₄²	C₂²×D₅₃	1	2	yes	—
149	C₂³	—	53	2	no	—
150	C₂³	—	106	4	no	—
151	C₂³	—	106	2	no	—
152	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	53	2	no	—
155	C₂×C₄	—	53	8	no	—
156	C₂×C₄	—	53	2	no	—
157	C₂×C₄	—	53	2	no	—
158	C₂×C₄	—	106	4	no	—
159	C₂×C₄	—	106	4	no	—
160	C₂×C₄	—	106	4	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	—	106	4	no	—
163	C₂×C₄	—	106	4	no	—
164	C₂×C₄	—	106	2	no	—
165	C₂×C₄	—	106	2	no	—
166	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
167	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
169	C₂×C₄	#11719947	1	2	yes	—
170	C₂×C₄	#11719949	1	2	yes	—
171	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
172	C₈	—	2	4	no	—
173	C₈	—	106	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	53	4	no	—
176	C₂²	—	106	8	no	—
177	C₂²	—	106	2	no	—
178	C₂²	—	106	2	no	—
179	C₂²	#13133175	1	2	yes	—
180	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	53	4	no	—
183	C₄	—	106	2	no	—
184	C₄	—	106	2	no	—
185	C₄	#13133175	1	2	yes	—
186	C₄	M₁₆×D₅₃	1	4	yes	—
187	C₄	C₂×#11719947	1	1	yes	characteristic
188	C₄	C₂×#11719949	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	106	4	no	—
191	C₂	C₂×#13133175	1	1	yes	characteristic
192	C₂	C₂×M₁₆×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₅₃

C₂

yes

maximal, characteristic

#16810679

C₂

yes

maximal, complemented

#16810675

C₂

yes

maximal, complemented

#16810647

C₂

yes

maximal, characteristic, complemented

#16810586

C₂

yes

maximal

M₁₆×Dic₅₃

C₂

yes

maximal, complemented

C₅₃×#423

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₄

yes

—

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₂×C₄×Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₄²×D₅₃

C₄

yes

—

C₄²×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

—

C₄⋊#88322

C₂²

yes

complemented

#13133129

C₂²

yes

—

#13133127

C₂²

yes

complemented

C₈×Dic₅₃

C₂²

yes

complemented

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

complemented

C₂³×D₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

#11719917

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

M₁₆

yes

complemented

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×M₁₆

yes

—

#2920

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#117

yes

characteristic

C₂×C₅₃

C₂²×M₁₆

yes

—

D₅₃

—

C₂²×C₄²

—

#432

—

#431

—

100

#423

—

101

#423

D₅₃

yes

characteristic, complemented

102

#407

—

103

C₄×M₁₆

—

104

C₅₃

#2920

yes

characteristic, complemented, Sylow

105

C₂³×C₄

—

106

C₂³×C₄

—

107

C₂×M₁₆

—

108

C₂×M₁₆

C₂×D₅₃

yes

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×D₅₃

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

C₂×D₅₃

yes

complemented

118

C₂²⋊C₈

—

119

#97

—

120

C₄×C₈

—

121

C₂⁴

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

106

—

131

C₂²×C₄

—

106

—

132

C₂²×C₄

—

106

—

133

C₂²×C₄

C₄×D₅₃

yes

—

134

C₂²×C₄

C₂²×D₅₃

yes

characteristic

135

M₁₆

—

136

M₁₆

—

106

—

137

C₂×C₈

—

138

C₂×C₈

C₂²×D₅₃

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₄×D₅₃

yes

characteristic

147

C₄²

C₄×D₅₃

yes

characteristic

148

C₄²

C₂²×D₅₃

yes

—

149

C₂³

—

150

C₂³

—

106

—

151

C₂³

—

106

—

152

C₂³

C₂×C₄×D₅₃

yes

characteristic

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

106

—

162

C₂×C₄

—

106

—

163

C₂×C₄

—

106

—

164

C₂×C₄

—

106

—

165

C₂×C₄

—

106

—

166

C₂×C₄

C₂×C₄×D₅₃

yes

—

167

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

168

C₂×C₄

C₂×C₄×D₅₃

yes

—

169

C₂×C₄

#11719947

yes

—

170

C₂×C₄

#11719949

yes

—

171

C₂×C₄

C₂³×D₅₃

yes

characteristic

172

C₈

—

173

C₈

—

106

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

106

—

177

C₂²

—

106

—

178

C₂²

—

106

—

179

C₂²

#13133175

yes

—

180

C₂²

C₂²×C₄×D₅₃

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

106

—

184

C₄

—

106

—

185

C₄

#13133175

yes

—

186

C₄

M₁₆×D₅₃

yes

—

187

C₄

C₂×#11719947

yes

characteristic

188

C₄

C₂×#11719949

yes

characteristic

189

C₂

—

190

C₂

—

106

—

191

C₂

C₂×#13133175

yes

characteristic

192

C₂

C₂×M₁₆×D₅₃

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#423	C₂
hypercenter	#423	D₅₃
hypocenter	C₅₃	#2920
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#423

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #16810679	C₂	semidirect
3. #16810675	C₂	semidirect
4. #16810647	C₂	semidirect
6. M₁₆×Dic₅₃	C₂	semidirect
7. C₅₃×#423	C₂	semidirect
17. C₄⋊#88322	C₂²	semidirect
19. #13133127	C₂²	semidirect
20. C₈×Dic₅₃	C₂²	semidirect
23. C₄⋊C₈×C₅₃	C₂²	semidirect
74. C₄×D₅₃	M₁₆	semidirect
101. #423	D₅₃	semidirect
104. C₅₃	#2920	semidirect
117. C₄⋊C₈	C₂×D₅₃	semidirect

H ≅ G/N

type

2. #16810679

C₂

semidirect

3. #16810675

C₂

semidirect

4. #16810647

C₂

semidirect

6. M₁₆×Dic₅₃

C₂

semidirect

7. C₅₃×#423

C₂

semidirect

17. C₄⋊#88322

C₂²

semidirect

19. #13133127

C₂²

semidirect

20. C₈×Dic₅₃

C₂²

semidirect

23. C₄⋊C₈×C₅₃

C₂²

semidirect

74. C₄×D₅₃

M₁₆

semidirect

101. #423

D₅₃

semidirect

104. C₅₃

#2920

semidirect

117. C₄⋊C₈

C₂×D₅₃

semidirect

Subgroups — #20283347

Conjugacy classes of subgroups

Complements