Group #20284198 — subgroups

Subgroups — #20284198

Order 6784.

480 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133248	C₂	1	2	yes	maximal
2	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
3	#16810830	C₂	1	4	yes	maximal, complemented
4	#16810825	C₂	1	4	yes	maximal, complemented
5	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810632	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719945	C₂²	1	1	yes	characteristic
9	#13133248	—	2	8	no	—
10	C₂×#11719925	C₂²	1	4	yes	—
11	C₂×#11719924	C₂²	1	4	yes	—
12	C₂×#11719917	C₂²	1	2	yes	—
13	C₄×Dic₁₀₆	C₂²	1	2	yes	—
14	C₄×D₂₁₂	C₂²	1	2	yes	—
15	C₄×C₂²⋊D₅₃	—	2	2	no	—
16	#13133176	C₂²	1	1	yes	characteristic
17	#13133174	—	2	1	no	—
18	C₄²×D₅₃	—	2	1	no	—
19	C₄⋊#88322	C₂²	1	4	yes	—
20	#13133141	C₂²	1	8	yes	—
21	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
22	C₄⋊Q₈×C₅₃	—	2	1	no	—
23	C₄⋊Q₈×C₅₃	—	2	1	no	—
24	C₄⋊Q₈×C₅₃	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₅₃	—	2	2	no	—
28	C₂×Q₈×C₅₃	—	2	4	no	—
29	C₂×Q₈×C₅₃	—	2	4	no	—
30	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	D₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₅₃	—	2	1	no	—
35	#11719945	—	2	2	no	—
36	#11719945	D₄	1	4	yes	—
37	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₅₃	—	2	1	no	—
39	C₂×C₄×D₅₃	—	2	2	no	—
40	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	—	2	4	no	—
42	C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂³	1	2	yes	—
46	C₂²⋊Dic₅₃	—	2	1	no	—
47	#11719925	—	2	8	no	—
48	#11719924	—	2	8	no	—
49	#11719921	—	2	2	no	—
50	C₄⋊Dic₅₃	C₂³	1	2	yes	—
51	#11719919	—	2	2	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	—	2	2	no	—
54	#11719917	—	2	4	no	—
55	C₂×#88322	C₂³	1	4	yes	—
56	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×D₅₃	—	2	1	no	—
58	Q₈×C₅₃	—	2	4	no	—
59	Q₈×C₅₃	—	2	4	no	—
60	Q₈×C₅₃	—	4	4	no	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₅₃	#41	1	2	yes	—
69	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂²⋊D₅₃	—	2	4	no	—
71	C₂×Dic₅₃	—	2	1	no	—
72	C₂×Dic₅₃	—	2	2	no	—
73	D₂₁₂	—	2	1	no	—
74	D₂₁₂	C₂×D₄	1	2	yes	—
75	C₄×D₅₃	—	2	4	no	—
76	C₄×D₅₃	—	4	4	no	—
77	Dic₁₀₆	—	2	1	no	—
78	Dic₁₀₆	C₂×D₄	1	2	yes	—
79	#88322	—	2	4	no	—
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	#121	1	2	yes	—
82	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₅₃	—	2	2	no	—
84	C₂×D₅₃	—	4	2	no	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	—	4	4	no	—
87	C₄×C₅₃	#121	1	2	yes	—
88	C₄×C₅₃	#137	1	4	yes	complemented
89	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
91	Dic₅₃	—	2	2	no	—
92	Dic₅₃	—	4	2	no	—
93	#3039	—	53	1	no	maximal, Sylow
94	C₂×C₅₃	—	2	2	no	—
95	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
96	C₂×C₅₃	C₂×#137	1	2	yes	—
97	D₅₃	—	4	2	no	—
98	C₂×#137	—	53	2	no	—
99	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
100	C₄×#41	—	53	1	no	—
101	#462	—	53	4	no	—
102	#461	—	53	4	no	—
103	#423	—	53	1	no	—
104	#419	—	53	2	no	—
105	C₅₃	#3039	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	—	53	1	no	—
107	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
108	#137	—	106	8	no	—
109	C₂×Q₁₆	—	53	4	no	—
110	C₂×SD₁₆	—	53	4	no	—
111	C₂×M₁₆	—	53	2	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂×D₅₃	1	4	yes	—
115	C₄×Q₈	—	53	2	no	—
116	C₄×D₄	—	53	2	no	—
117	C₄×D₄	—	106	2	no	—
118	#117	—	53	1	no	—
119	#117	—	106	1	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	—	106	1	no	—
122	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
123	C₄⋊C₈	—	53	4	no	—
124	#103	—	53	8	no	—
125	#41	—	53	4	no	—
126	#41	—	106	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	—	53	1	no	—
131	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
132	C₂×D₄	—	53	1	no	—
133	C₂×D₄	—	106	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	106	1	no	—
136	C₂²×C₄	—	106	2	no	—
137	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
138	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
139	Q₁₆	—	106	8	no	—
140	SD₁₆	—	106	8	no	—
141	M₁₆	—	106	4	no	—
142	C₂×C₈	—	53	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	53	2	no	—
146	C₄⋊C₄	—	106	2	no	—
147	C₂²⋊C₄	—	106	1	no	—
148	C₂²⋊C₄	—	106	2	no	—
149	C₄²	—	106	1	no	—
150	C₄²	—	106	2	no	—
151	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₄²	C₂²×D₅₃	1	2	yes	—
154	C₂³	—	106	1	no	—
155	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
156	Q₈	—	2	4	no	—
157	Q₈	—	2	4	no	—
158	Q₈	—	4	4	no	—
159	Q₈	—	53	2	no	—
160	Q₈	—	106	1	no	—
161	D₄	—	53	2	no	—
162	D₄	—	106	4	no	—
163	D₄	—	106	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	106	1	no	—
168	C₂×C₄	—	106	4	no	—
169	C₂×C₄	—	106	2	no	—
170	C₂×C₄	—	212	4	no	—
171	C₂×C₄	D₄×D₅₃	1	2	yes	—
172	C₂×C₄	#11719947	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
176	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
177	C₈	—	106	4	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	106	2	no	—
180	C₂²	—	212	2	no	—
181	C₂²	#13133188	1	2	yes	—
182	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	106	2	no	—
186	C₄	—	212	2	no	—
187	C₄	#13133188	1	2	yes	—
188	C₄	#13133248	1	4	yes	complemented
189	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
190	C₄	C₂×#11719947	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	212	2	no	—
193	C₂	C₂×#13133188	1	1	yes	characteristic
194	C₂	C₂×#13133248	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133248

C₂

yes

maximal

C₄×#11719945

C₂

yes

maximal, characteristic

#16810830

C₂

yes

maximal, complemented

#16810825

C₂

yes

maximal, complemented

#16810647

C₂

yes

maximal, characteristic, complemented

#16810632

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

characteristic

#13133248

—

C₂×#11719925

C₂²

yes

—

C₂×#11719924

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

C₄×C₂²⋊D₅₃

—

#13133176

C₂²

yes

characteristic

#13133174

—

C₄²×D₅₃

—

C₄⋊#88322

C₂²

yes

—

#13133141

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

D₄

yes

—

C₂×D₂₁₂

C₂³

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

—

#11719925

—

#11719924

—

#11719921

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

C₂×D₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

—

C₄×C₅₃

#137

yes

complemented

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#3039

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#121

yes

characteristic

C₂×C₅₃

C₂×#137

yes

—

D₅₃

—

C₂×#137

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic, complemented

100

C₄×#41

—

101

#462

—

102

#461

—

103

#423

—

104

#419

—

105

C₅₃

#3039

yes

characteristic, complemented, Sylow

106

C₂×#41

—

107

C₂²×Q₈

C₂×D₅₃

yes

—

108

#137

—

106

—

109

C₂×Q₁₆

—

110

C₂×SD₁₆

—

111

C₂×M₁₆

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×D₅₃

yes

—

115

C₄×Q₈

—

116

C₄×D₄

—

117

C₄×D₄

—

106

—

118

#117

—

119

#117

—

106

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

—

106

—

122

C₂×C₄²

C₂×D₅₃

yes

characteristic

123

C₄⋊C₈

—

124

#103

—

125

#41

—

126

#41

—

106

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

C₂²×D₅₃

yes

—

132

C₂×D₄

—

133

C₂×D₄

—

106

—

134

C₂²×C₄

—

135

C₂²×C₄

—

106

—

136

C₂²×C₄

—

106

—

137

C₂²×C₄

C₂²⋊D₅₃

yes

—

138

C₂²×C₄

C₂²×D₅₃

yes

characteristic

139

Q₁₆

—

106

—

140

SD₁₆

—

106

—

141

M₁₆

—

106

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

106

—

147

C₂²⋊C₄

—

106

—

148

C₂²⋊C₄

—

106

—

149

C₄²

—

106

—

150

C₄²

—

106

—

151

C₄²

C₂²⋊D₅₃

yes

characteristic

152

C₄²

C₂²⋊D₅₃

yes

characteristic

153

C₄²

C₂²×D₅₃

yes

—

154

C₂³

—

106

—

155

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

156

Q₈

—

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

106

—

161

D₄

—

162

D₄

—

106

—

163

D₄

—

106

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

106

—

168

C₂×C₄

—

106

—

169

C₂×C₄

—

106

—

170

C₂×C₄

—

212

—

171

C₂×C₄

D₄×D₅₃

yes

—

172

C₂×C₄

#11719947

yes

—

173

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

174

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

175

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

176

C₂×C₄

C₂³×D₅₃

yes

characteristic

177

C₈

—

106

—

178

C₂²

—

179

C₂²

—

106

—

180

C₂²

—

212

—

181

C₂²

#13133188

yes

—

182

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

106

—

186

C₄

—

212

—

187

C₄

#13133188

yes

—

188

C₄

#13133248

yes

complemented

189

C₄

C₂×D₄×D₅₃

yes

characteristic

190

C₄

C₂×#11719947

yes

characteristic

191

C₂

—

192

C₂

—

212

—

193

C₂

C₂×#13133188

yes

characteristic

194

C₂

C₂×#13133248

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#3039
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#3039

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #16810830	C₂	semidirect
4. #16810825	C₂	semidirect
5. #16810647	C₂	semidirect
6. #16810632	C₂	semidirect
7. C₂×C₄⋊Q₈×C₅₃	C₂	semidirect
88. C₄×C₅₃	#137	semidirect
99. C₂×C₄⋊Q₈	D₅₃	semidirect
105. C₅₃	#3039	semidirect
188. C₄	#13133248	semidirect

H ≅ G/N

type

3. #16810830

C₂

semidirect

4. #16810825

C₂

semidirect

5. #16810647

C₂

semidirect

6. #16810632

C₂

semidirect

7. C₂×C₄⋊Q₈×C₅₃

C₂

semidirect

88. C₄×C₅₃

#137

semidirect

99. C₂×C₄⋊Q₈

D₅₃

semidirect

105. C₅₃

#3039

semidirect

188. C₄

#13133248

semidirect

Subgroups — #20284198

Conjugacy classes of subgroups

Complements