Group #20291375 — subgroups

Subgroups — D₅₃⋊#430 #20291375

Order 6784.

460 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 200 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×D₅₃⋊M₁₆	C₂	1	2	yes	maximal
2	#16811638	C₂	1	8	yes	maximal, complemented
3	D₅₃⋊#108	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719936	C₂²	1	4	yes	—
6	D₅₃⋊M₁₆	—	2	2	no	—
7	D₅₃⋊M₁₆	—	2	2	no	—
8	D₅₃⋊M₁₆	C₂²	1	4	yes	—
9	D₅₃⋊M₁₆	C₂²	1	4	yes	—
10	C₂×#11719940	—	2	2	no	—
11	C₂×D₅₃⋊C₈	—	2	2	no	—
12	#13133288	C₂²	1	8	yes	complemented
13	#13133287	C₂²	1	8	yes	complemented
14	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719917	C₄	1	1	yes	characteristic
16	C₂×#11719912	C₄	1	2	yes	—
17	M₁₆×D₅₃	C₄	1	4	yes	—
18	M₁₆×D₅₃	C₂²	1	4	yes	—
19	C₂×C₈×D₅₃	C₂²	1	2	yes	—
20	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
21	C₂³×D₅₃	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	D₄	1	2	yes	—
27	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
29	C₂×C₄×D₅₃	Q₈	1	2	yes	—
30	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₅₃	C₂³	1	2	yes	—
32	#11719940	—	2	4	no	—
33	#11719940	—	2	4	no	—
34	C₂×#88324	—	2	4	no	—
35	#11719936	—	2	4	no	—
36	#11719936	C₂³	1	8	yes	—
37	D₅₃⋊C₈	—	2	4	no	—
38	D₅₃⋊C₈	—	2	4	no	—
39	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	#11719917	C₂×C₄	1	4	yes	—
42	C₂×#88322	C₂×C₄	1	2	yes	—
43	#11719912	C₂×C₄	1	8	yes	—
44	C₈×D₅₃	C₂×C₄	1	4	yes	—
45	C₈×D₅₃	C₂³	1	4	yes	—
46	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×D₅₃	—	4	2	no	—
48	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×D₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂²×D₅₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
59	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
61	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
62	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
63	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₄×D₅₃	C₂²×C₄	1	2	yes	—
65	C₄×D₅₃	C₂×D₄	1	2	yes	—
66	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄×D₅₃	C₂×Q₈	1	2	yes	—
68	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
70	#88324	—	2	8	no	—
71	C₈×C₅₃	C₂²×C₄	1	4	yes	—
72	#88322	C₂²×C₄	1	4	yes	—
73	C₂²×C₅₃	—	4	1	no	—
74	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×D₅₃	—	2	4	no	—
77	C₂×D₅₃	—	2	2	no	—
78	C₂×D₅₃	—	4	1	no	—
79	C₂×D₅₃	—	4	2	no	—
80	C₂×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×D₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
86	Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
88	Dic₅₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×#430	—	53	1	no	maximal, Sylow
90	C₂×C₅₃	—	2	2	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	D₅₃	—	2	2	no	—
94	D₅₃	—	2	1	no	—
95	D₅₃	#430	1	2	yes	complemented
96	C₂²×M₁₆	—	53	1	no	—
97	C₂²×M₁₆	—	53	2	no	—
98	#430	—	53	8	no	—
99	C₂×#108	—	53	4	no	—
100	C₅₃	C₂×#430	1	1	yes	characteristic, complemented, Sylow
101	C₂³×C₄	—	53	1	no	—
102	C₂×M₁₆	—	53	1	no	—
103	C₂×M₁₆	—	53	4	no	—
104	C₂×M₁₆	—	53	4	no	—
105	C₂×M₁₆	—	53	2	no	—
106	C₂×M₁₆	—	53	4	no	—
107	C₂×M₁₆	—	53	4	no	—
108	C₂×M₁₆	—	53	4	no	—
109	C₂×M₁₆	—	106	2	no	—
110	C₂×M₁₆	—	106	2	no	—
111	C₂×M₁₆	—	106	2	no	—
112	C₂×M₁₆	#6157	1	1	yes	characteristic
113	C₂²×C₈	—	53	2	no	—
114	C₂²×C₈	—	106	2	no	—
115	#108	—	53	8	no	—
116	#108	—	53	8	no	—
117	C₂⁴	—	53	1	no	—
118	C₂²×C₄	—	53	1	no	—
119	C₂²×C₄	—	53	2	no	—
120	C₂²×C₄	—	53	1	no	—
121	C₂²×C₄	—	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	1	no	—
126	C₂²×C₄	—	53	2	no	—
127	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
128	M₁₆	—	53	4	no	—
129	M₁₆	—	53	8	no	—
130	M₁₆	—	53	8	no	—
131	M₁₆	—	106	4	no	—
132	M₁₆	—	106	4	no	—
133	M₁₆	—	106	4	no	—
134	M₁₆	C₂×#6157	1	4	yes	—
135	C₂×C₈	—	53	2	no	—
136	C₂×C₈	—	53	4	no	—
137	C₂×C₈	—	53	4	no	—
138	C₂×C₈	—	106	4	no	—
139	C₂×C₈	—	106	4	no	—
140	C₂×C₈	—	106	4	no	—
141	C₂×C₈	C₂×#6157	1	2	yes	—
142	C₂³	—	53	2	no	—
143	C₂³	—	53	1	no	—
144	C₂³	—	53	1	no	—
145	C₂³	—	53	2	no	—
146	C₂³	—	212	2	no	—
147	C₂³	C₄⋊#6157	1	1	yes	characteristic
148	C₂×C₄	—	53	1	no	—
149	C₂×C₄	—	53	2	no	—
150	C₂×C₄	—	53	2	no	—
151	C₂×C₄	—	53	1	no	—
152	C₂×C₄	—	53	2	no	—
153	C₂×C₄	—	53	2	no	—
154	C₂×C₄	—	53	2	no	—
155	C₂×C₄	—	53	2	no	—
156	C₂×C₄	—	53	2	no	—
157	C₂×C₄	—	53	1	no	—
158	C₂×C₄	—	53	1	no	—
159	C₂×C₄	—	53	1	no	—
160	C₂×C₄	—	53	2	no	—
161	C₂×C₄	—	53	1	no	—
162	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
163	C₂×C₄	C₄⋊#6157	1	2	yes	—
164	C₂×C₄	C₂²×#6157	1	2	yes	—
165	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
166	C₈	—	53	4	no	—
167	C₈	—	106	8	no	—
168	C₈	C₂²×#6157	1	4	yes	—
169	C₂²	—	4	1	no	—
170	C₂²	—	53	1	no	—
171	C₂²	—	53	2	no	—
172	C₂²	—	53	1	no	—
173	C₂²	—	106	4	no	—
174	C₂²	—	106	2	no	—
175	C₂²	—	212	1	no	—
176	C₂²	—	212	2	no	—
177	C₂²	C₂×C₄⋊#6157	1	2	yes	—
178	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
179	C₄	—	53	1	no	—
180	C₄	—	53	2	no	—
181	C₄	—	53	1	no	—
182	C₄	C₂×C₄⋊#6157	1	2	yes	—
183	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
184	C₄	C₂³×#6157	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	—	2	1	no	—
187	C₂	—	53	2	no	—
188	C₂	—	106	2	no	—
189	C₂	—	106	1	no	—
190	C₂	C₂²×C₄⋊#6157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₅₃⋊M₁₆

C₂

yes

maximal

#16811638

C₂

yes

maximal, complemented

D₅₃⋊#108

C₂

yes

maximal, complemented

C₂×M₁₆×D₅₃

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

D₅₃⋊M₁₆

—

D₅₃⋊M₁₆

—

D₅₃⋊M₁₆

C₂²

yes

—

D₅₃⋊M₁₆

C₂²

yes

—

C₂×#11719940

—

C₂×D₅₃⋊C₈

—

#13133288

C₂²

yes

complemented

#13133287

C₂²

yes

complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂×#11719917

C₄

yes

characteristic

C₂×#11719912

C₄

yes

—

M₁₆×D₅₃

C₄

yes

—

M₁₆×D₅₃

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₄

yes

characteristic

C₂³×D₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

—

—

—

—

—

C₂³

yes

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

#11719917

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

#11719912

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂³

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂²×D₅₃

C₄⋊C₄

yes

characteristic

C₂²×D₅₃

C₂×Q₈

yes

characteristic

C₂²×D₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

C₂²×C₄

yes

characteristic

C₄×D₅₃

C₂²×C₄

yes

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

C₂⁴

yes

characteristic

#88324

—

C₈×C₅₃

C₂²×C₄

yes

—

#88322

C₂²×C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₄⋊C₄

yes

—

C₂×D₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂³×C₄

yes

characteristic

Dic₅₃

C₂×C₄⋊C₄

yes

characteristic

Dic₅₃

C₂×C₄⋊C₄

yes

—

Dic₅₃

C₂²×D₄

yes

characteristic

C₂×#430

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

D₅₃

—

D₅₃

—

D₅₃

#430

yes

complemented

C₂²×M₁₆

—

C₂²×M₁₆

—

#430

—

C₂×#108

—

100

C₅₃

C₂×#430

yes

characteristic, complemented, Sylow

101

C₂³×C₄

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

C₂×M₁₆

—

107

C₂×M₁₆

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

106

—

110

C₂×M₁₆

—

106

—

111

C₂×M₁₆

—

106

—

112

C₂×M₁₆

#6157

yes

characteristic

113

C₂²×C₈

—

114

C₂²×C₈

—

106

—

115

#108

—

116

#108

—

117

C₂⁴

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂×#6157

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

131

M₁₆

—

106

—

132

M₁₆

—

106

—

133

M₁₆

—

106

—

134

M₁₆

C₂×#6157

yes

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

106

—

139

C₂×C₈

—

106

—

140

C₂×C₈

—

106

—

141

C₂×C₈

C₂×#6157

yes

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

212

—

147

C₂³

C₄⋊#6157

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₄⋊#6157

yes

characteristic

163

C₂×C₄

C₄⋊#6157

yes

—

164

C₂×C₄

C₂²×#6157

yes

—

165

C₂×C₄

C₂²×#6157

yes

characteristic

166

C₈

—

167

C₈

—

106

—

168

C₈

C₂²×#6157

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

106

—

174

C₂²

—

106

—

175

C₂²

—

212

—

176

C₂²

—

212

—

177

C₂²

C₂×C₄⋊#6157

yes

—

178

C₂²

C₂×C₄⋊#6157

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

C₂×C₄⋊#6157

yes

—

183

C₄

C₂×C₄⋊#6157

yes

characteristic

184

C₄

C₂³×#6157

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

—

188

C₂

—

106

—

189

C₂

—

106

—

190

C₂

C₂²×C₄⋊#6157

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₄⋊#6157
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₄	C₂³×#6157
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₂²×C₄⋊C₄
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	C₂×#430
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₂×C₄⋊#6157

commutator

C₄×C₅₃

C₂³×C₄

Frattini

C₄

C₂³×#6157

Fitting

C₂×M₁₆×C₅₃

C₄

socle

C₂×C₅₃

C₂²×C₄⋊C₄

C₂×M₁₆

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. #16811638	C₂	semidirect
3. D₅₃⋊#108	C₂	semidirect
12. #13133288	C₂²	semidirect
13. #13133287	C₂²	semidirect
95. D₅₃	#430	semidirect
100. C₅₃	C₂×#430	semidirect

H ≅ G/N

type

2. #16811638

C₂

semidirect

3. D₅₃⋊#108

C₂

semidirect

12. #13133288

C₂²

semidirect

13. #13133287

C₂²

semidirect

95. D₅₃

#430

semidirect

100. C₅₃

C₂×#430

semidirect

Subgroups — D₅₃⋊#430 #20291375

Conjugacy classes of subgroups

Complements

Subgroups — D53⋊#430 #20291375

Conjugacy classes of subgroups

Complements

Subgroups — D₅₃⋊#430 #20291375