Group #20274725 — subgroups

Subgroups — #20274725

Order 6784.

356 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 125 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⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133138	C₂	1	2	yes	maximal
3	#16810489	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810385	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₄⋊#88322	—	2	2	no	—
9	#13133138	—	2	4	no	—
10	#13133124	—	2	1	no	—
11	#13133124	—	2	1	no	—
12	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₅₃	—	2	1	no	—
14	C₄×D₄×C₅₃	—	2	2	no	—
15	C₄×D₄×C₅₃	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂⁴×C₅₃	—	2	1	no	—
20	C₂×D₄×C₅₃	—	2	1	no	—
21	C₂×D₄×C₅₃	—	2	1	no	—
22	C₂×D₄×C₅₃	—	2	1	no	—
23	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₅₃	D₄	1	2	yes	—
25	C₂×D₄×C₅₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄×C₅₃	—	2	4	no	—
28	C₂²×C₄×C₅₃	—	4	4	no	—
29	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₃	—	2	2	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²⋊C₄×C₅₃	—	2	4	no	—
36	C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₄²×C₅₃	D₄	1	2	yes	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂×#88322	—	2	2	no	—
42	C₂×#88322	—	2	2	no	—
43	C₂³×C₅₃	—	2	2	no	—
44	C₂³×C₅₃	—	2	2	no	—
45	C₂³×C₅₃	—	2	2	no	—
46	C₂³×C₅₃	—	4	2	no	—
47	C₂³×C₅₃	—	4	2	no	—
48	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
49	D₄×C₅₃	—	2	2	no	—
50	D₄×C₅₃	—	2	2	no	—
51	D₄×C₅₃	—	2	2	no	—
52	D₄×C₅₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	4	8	no	—
57	C₂×C₄×C₅₃	—	4	2	no	—
58	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂×Dic₅₃	—	4	2	no	—
67	C₂×Dic₅₃	—	4	2	no	—
68	C₂×Dic₅₃	—	4	2	no	—
69	#88322	—	4	2	no	—
70	C₂²×C₅₃	—	2	4	no	—
71	C₂²×C₅₃	—	4	2	no	—
72	C₂²×C₅₃	—	4	2	no	—
73	C₂²×C₅₃	—	4	4	no	—
74	C₂²×C₅₃	—	4	4	no	—
75	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₅₃	#121	1	1	yes	characteristic
77	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₅₃	#135	1	2	yes	—
79	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	—	4	2	no	—
83	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₅₃	#117	1	1	yes	characteristic
85	C₄×C₅₃	#121	1	1	yes	characteristic
86	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
87	Dic₅₃	—	8	2	no	—
88	#1804	—	53	1	no	maximal, Sylow
89	C₂×C₅₃	—	4	4	no	—
90	C₂×C₅₃	#380	1	1	yes	characteristic
91	C₂×C₅₃	#416	1	1	yes	characteristic
92	C₂×C₅₃	#418	1	1	yes	characteristic
93	C₂×C₅₃	#463	1	2	yes	—
94	C₂×C₅₃	#478	1	2	yes	—
95	C₂×C₄×D₄	D₅₃	1	1	yes	characteristic
96	C₂×C₄⋊C₈	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
102	C₂³×C₄	—	2	1	no	—
103	C₂²×C₈	—	53	2	no	—
104	C₄×D₄	—	2	2	no	—
105	C₄×D₄	Dic₅₃	1	4	yes	complemented
106	C₂×C₄⋊C₄	—	53	2	no	—
107	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
108	C₂×C₂²⋊C₄	—	2	1	no	—
109	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
110	C₄⋊C₈	—	106	2	no	—
111	#102	—	106	4	no	—
112	#95	—	106	1	no	—
113	#95	—	106	1	no	—
114	C₂⁴	—	2	1	no	—
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	Dic₁₀₆	1	2	yes	—
119	C₂×D₄	D₂₁₂	1	2	yes	—
120	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	4	4	no	—
124	C₂²×C₄	—	106	2	no	—
125	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₈	—	106	2	no	—
129	C₂×C₈	—	106	2	no	—
130	C₂×C₈	—	106	2	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	106	2	no	—
133	C₄⋊C₄	—	106	2	no	—
134	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₄²	C₂×Dic₅₃	1	2	yes	—
137	C₄²	C₂²⋊D₅₃	1	2	yes	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	4	2	no	—
142	C₂³	—	4	2	no	—
143	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
144	D₄	—	2	2	no	—
145	D₄	—	2	2	no	—
146	D₄	—	2	2	no	—
147	D₄	C₄⋊Dic₅₃	1	4	yes	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	212	2	no	—
154	C₂×C₄	—	212	2	no	—
155	C₂×C₄	—	212	2	no	—
156	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
157	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
158	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
159	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
160	C₂×C₄	#11719945	1	1	yes	characteristic
161	C₂×C₄	#11719945	1	1	yes	characteristic
162	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₈	—	212	2	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	4	4	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	4	4	no	—
170	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
171	C₂²	#13133191	1	1	yes	characteristic
172	C₂²	#13133196	1	1	yes	characteristic
173	C₂²	#13133237	1	2	yes	—
174	C₂²	#13133243	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	424	2	no	—
179	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
180	C₄	#13133176	1	1	yes	characteristic
181	C₄	#13133191	1	1	yes	characteristic
182	C₄	#13133196	1	1	yes	characteristic
183	C₂	—	4	4	no	—
184	C₂	#16810483	1	1	yes	characteristic
185	C₂	#16810619	1	1	yes	characteristic
186	C₂	#16810628	1	1	yes	characteristic
187	C₂	#16810838	1	2	yes	—
188	C₂	#16810923	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic, complemented

C₂×#13133138

C₂

yes

maximal

#16810489

C₂

yes

maximal, characteristic, complemented

#16810385

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

—

C₂²×D₄×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×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

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

#1804

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#478

yes

—

C₂×C₄×D₄

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#102

—

#382

—

#340

—

100

C₅₃

#1804

yes

characteristic, complemented, Sylow

101

C₂²×D₄

C₂×D₅₃

yes

characteristic

102

C₂³×C₄

—

103

C₂²×C₈

—

104

C₄×D₄

—

105

C₄×D₄

Dic₅₃

yes

complemented

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

108

C₂×C₂²⋊C₄

—

109

C₂×C₄²

C₂×D₅₃

yes

characteristic

110

C₄⋊C₈

—

106

—

111

#102

—

106

—

112

#95

—

106

—

113

#95

—

106

—

114

C₂⁴

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

Dic₁₀₆

yes

—

119

C₂×D₄

D₂₁₂

yes

—

120

C₂×D₄

C₂×Dic₅₃

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

106

—

125

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

126

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

127

C₂²×C₄

C₂²×D₅₃

yes

characteristic

128

C₂×C₈

—

106

—

129

C₂×C₈

—

106

—

130

C₂×C₈

—

106

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

C₂×Dic₅₃

yes

—

135

C₂²⋊C₄

—

136

C₄²

C₂×Dic₅₃

yes

—

137

C₄²

C₂²⋊D₅₃

yes

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

C₄⋊Dic₅₃

yes

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

—

212

—

156

C₂×C₄

C₂²⋊Dic₅₃

yes

—

157

C₂×C₄

C₂²⋊Dic₅₃

yes

—

158

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

159

C₂×C₄

C₂×D₂₁₂

yes

characteristic

160

C₂×C₄

#11719945

yes

characteristic

161

C₂×C₄

#11719945

yes

characteristic

162

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

164

C₈

—

212

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

171

C₂²

#13133191

yes

characteristic

172

C₂²

#13133196

yes

characteristic

173

C₂²

#13133237

yes

—

174

C₂²

#13133243

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

424

—

179

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

180

C₄

#13133176

yes

characteristic

181

C₄

#13133191

yes

characteristic

182

C₄

#13133196

yes

characteristic

183

C₂

—

184

C₂

#16810483

yes

characteristic

185

C₂

#16810619

yes

characteristic

186

C₂

#16810628

yes

characteristic

187

C₂

#16810838

yes

—

188

C₂

#16810923

yes

—

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

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×D₄×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄×D₄

D₅₃

hypocenter

C₅₃

#1804

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×C₄⋊#88322	C₂	semidirect
3. #16810489	C₂	semidirect
4. #16810385	C₂	semidirect
15. C₄×D₄×C₅₃	C₄	semidirect
100. C₅₃	#1804	semidirect
105. C₄×D₄	Dic₅₃	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#88322

C₂

semidirect

3. #16810489

C₂

semidirect

4. #16810385

C₂

semidirect

15. C₄×D₄×C₅₃

C₄

semidirect

100. C₅₃

#1804

semidirect

105. C₄×D₄

Dic₅₃

semidirect

Subgroups — #20274725

Conjugacy classes of subgroups

Complements