Group #20275848 — subgroups

Subgroups — #20275848

Order 6784.

326 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13133137	C₂	1	2	yes	maximal
2	#16810525	C₂	1	1	yes	maximal, characteristic
3	#16810385	C₂	1	2	yes	maximal, complemented
4	#16810378	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719921	—	2	1	no	—
10	C₂×#11719921	—	2	1	no	—
11	#13133137	—	2	4	no	—
12	C₂²×Q₈×C₅₃	—	2	1	no	—
13	C₄⋊Q₈×C₅₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×D₅₃	—	2	1	no	—
18	C₂×Q₈×C₅₃	—	2	4	no	—
19	C₂×Q₈×C₅₃	—	4	2	no	—
20	C₂×Q₈×C₅₃	—	4	2	no	—
21	C₂²×C₄×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	2	1	no	—
26	C₂×D₂₁₂	—	2	2	no	—
27	C₂×D₂₁₂	—	2	2	no	—
28	C₂×D₂₁₂	—	2	1	no	—
29	C₂×D₂₁₂	Q₈	1	2	yes	complemented
30	C₄⋊C₄×C₅₃	—	2	4	no	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄⋊C₄×C₅₃	D₄	1	4	yes	—
33	C₄²×C₅₃	—	2	2	no	—
34	#11719921	—	4	2	no	—
35	#11719921	—	4	2	no	—
36	C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×#88322	—	2	2	no	—
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₅₃	—	2	2	no	—
42	C₂²×D₅₃	—	4	1	no	—
43	C₂²×D₅₃	—	4	1	no	—
44	C₂²×D₅₃	—	4	1	no	—
45	C₂²×D₅₃	—	4	1	no	—
46	C₂²×D₅₃	—	4	1	no	—
47	C₂²×D₅₃	—	4	1	no	—
48	Q₈×C₅₃	—	4	8	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	—	4	2	no	—
55	C₂×C₄×C₅₃	—	4	2	no	—
56	C₂×C₄×C₅₃	D₈	1	2	yes	—
57	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#41	1	2	yes	—
61	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂×Dic₅₃	—	4	1	no	—
63	C₂×Dic₅₃	—	4	1	no	—
64	C₂×Dic₅₃	—	4	1	no	—
65	D₂₁₂	—	2	4	no	—
66	D₂₁₂	—	4	2	no	—
67	#88322	—	4	2	no	—
68	C₂²×C₅₃	#120	1	1	yes	characteristic
69	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₅₃	#124	1	1	yes	characteristic
71	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂²×C₅₃	#137	1	1	yes	characteristic
75	C₂×D₅₃	—	4	2	no	—
76	C₂×D₅₃	—	4	2	no	—
77	C₂×D₅₃	—	4	2	no	—
78	C₂×D₅₃	—	8	1	no	—
79	C₂×D₅₃	—	8	1	no	—
80	C₂×D₅₃	—	8	1	no	—
81	C₂×D₅₃	—	8	1	no	—
82	C₄×C₅₃	—	2	2	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	4	4	no	—
85	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
86	C₄×C₅₃	#124	1	2	yes	—
87	Dic₅₃	—	8	1	no	—
88	#2023	—	53	1	no	maximal, Sylow
89	C₂×C₅₃	#393	1	1	yes	characteristic
90	C₂×C₅₃	#447	1	1	yes	characteristic
91	C₂×C₅₃	#448	1	1	yes	characteristic
92	C₂×C₅₃	#474	1	1	yes	characteristic
93	C₂×C₅₃	#476	1	1	yes	characteristic
94	C₂×C₅₃	#486	1	1	yes	characteristic
95	C₂×C₅₃	#489	1	1	yes	characteristic
96	D₅₃	—	8	2	no	—
97	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
98	C₂×#102	—	53	2	no	—
99	#390	—	53	1	no	—
100	#340	—	53	2	no	—
101	#336	—	53	1	no	—
102	C₅₃	#2023	1	1	yes	characteristic, complemented, Sylow
103	C₂²×Q₈	—	2	1	no	—
104	C₂²×D₄	—	53	1	no	—
105	C₂²×C₈	—	53	2	no	—
106	C₄⋊Q₈	—	2	4	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	—	53	1	no	—
109	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
110	C₂×C₂²⋊C₄	—	106	1	no	—
111	C₂×C₂²⋊C₄	—	106	1	no	—
112	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
113	#102	—	106	4	no	—
114	C₂⁴	—	106	1	no	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	2	no	—
117	C₂×Q₈	—	4	2	no	—
118	C₂×D₄	—	53	2	no	—
119	C₂×D₄	—	106	2	no	—
120	C₂×D₄	—	106	1	no	—
121	C₂×D₄	—	106	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	106	1	no	—
124	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
127	C₂×C₈	—	106	2	no	—
128	C₂×C₈	—	106	2	no	—
129	C₂×C₈	—	106	2	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	106	2	no	—
133	C₄⋊C₄	C₂²⋊D₅₃	1	4	yes	—
134	C₂²⋊C₄	—	212	2	no	—
135	C₂²⋊C₄	—	212	2	no	—
136	C₄²	—	2	2	no	—
137	C₂³	—	106	2	no	—
138	C₂³	—	212	1	no	—
139	C₂³	—	212	1	no	—
140	C₂³	—	212	1	no	—
141	C₂³	—	212	1	no	—
142	C₂³	—	212	1	no	—
143	C₂³	—	212	1	no	—
144	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
145	Q₈	—	4	8	no	—
146	D₄	—	106	4	no	—
147	D₄	—	212	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	212	1	no	—
156	C₂×C₄	—	212	1	no	—
157	C₂×C₄	—	212	1	no	—
158	C₂×C₄	#11719922	1	2	yes	—
159	C₂×C₄	#11719924	1	2	yes	—
160	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
161	C₂×C₄	#11719949	1	2	yes	—
162	C₂×C₄	#11719949	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
164	C₈	—	212	2	no	—
165	C₂²	—	424	1	no	—
166	C₂²	—	424	1	no	—
167	C₂²	—	424	1	no	—
168	C₂²	—	424	1	no	—
169	C₂²	—	212	2	no	—
170	C₂²	—	212	2	no	—
171	C₂²	—	212	2	no	—
172	C₂²	#13133186	1	1	yes	characteristic
173	C₂²	#13133193	1	1	yes	characteristic
174	C₂²	#13133206	1	1	yes	characteristic
175	C₂²	C₂×#11719922	1	1	yes	characteristic
176	C₂²	C₂×#11719924	1	1	yes	characteristic
177	C₂²	#13133243	1	1	yes	characteristic
178	C₂²	#13133248	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	4	no	—
182	C₄	—	424	1	no	—
183	C₄	#13133193	1	2	yes	—
184	C₄	#13133206	1	2	yes	—
185	C₂	—	424	2	no	—
186	C₂	#16810534	1	1	yes	characteristic
187	C₂	#16810757	1	1	yes	characteristic
188	C₂	#16810762	1	1	yes	characteristic
189	C₂	#16810900	1	1	yes	characteristic
190	C₂	#16810910	1	1	yes	characteristic
191	C₂	#16810971	1	1	yes	characteristic
192	C₂	#16810985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133137

C₂

yes

maximal

#16810525

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal, complemented

#16810378

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

Q₈

yes

complemented

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#124

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic, complemented

C₂×#102

—

#390

—

100

#340

—

101

#336

—

102

C₅₃

#2023

yes

characteristic, complemented, Sylow

103

C₂²×Q₈

—

104

C₂²×D₄

—

105

C₂²×C₈

—

106

C₄⋊Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

110

C₂×C₂²⋊C₄

—

106

—

111

C₂×C₂²⋊C₄

—

106

—

112

C₂×C₄²

C₂×D₅₃

yes

characteristic

113

#102

—

106

—

114

C₂⁴

—

106

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×D₄

—

119

C₂×D₄

—

106

—

120

C₂×D₄

—

106

—

121

C₂×D₄

—

106

—

122

C₂²×C₄

—

123

C₂²×C₄

—

106

—

124

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

125

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

126

C₂²×C₄

C₂²×D₅₃

yes

characteristic

127

C₂×C₈

—

106

—

128

C₂×C₈

—

106

—

129

C₂×C₈

—

106

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

C₂²⋊D₅₃

yes

—

134

C₂²⋊C₄

—

212

—

135

C₂²⋊C₄

—

212

—

136

C₄²

—

137

C₂³

—

106

—

138

C₂³

—

212

—

139

C₂³

—

212

—

140

C₂³

—

212

—

141

C₂³

—

212

—

142

C₂³

—

212

—

143

C₂³

—

212

—

144

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

145

Q₈

—

146

D₄

—

106

—

147

D₄

—

212

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

212

—

156

C₂×C₄

—

212

—

157

C₂×C₄

—

212

—

158

C₂×C₄

#11719922

yes

—

159

C₂×C₄

#11719924

yes

—

160

C₂×C₄

Q₈×D₅₃

yes

characteristic

161

C₂×C₄

#11719949

yes

—

162

C₂×C₄

#11719949

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

164

C₈

—

212

—

165

C₂²

—

424

—

166

C₂²

—

424

—

167

C₂²

—

424

—

168

C₂²

—

424

—

169

C₂²

—

212

—

170

C₂²

—

212

—

171

C₂²

—

212

—

172

C₂²

#13133186

yes

characteristic

173

C₂²

#13133193

yes

characteristic

174

C₂²

#13133206

yes

characteristic

175

C₂²

C₂×#11719922

yes

characteristic

176

C₂²

C₂×#11719924

yes

characteristic

177

C₂²

#13133243

yes

characteristic

178

C₂²

#13133248

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

424

—

183

C₄

#13133193

yes

—

184

C₄

#13133206

yes

—

185

C₂

—

424

—

186

C₂

#16810534

yes

characteristic

187

C₂

#16810757

yes

characteristic

188

C₂

#16810762

yes

characteristic

189

C₂

#16810900

yes

characteristic

190

C₂

#16810910

yes

characteristic

191

C₂

#16810971

yes

characteristic

192

C₂

#16810985

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#2023
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#2023

LCS

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #16810385	C₂	semidirect
4. #16810378	C₂	semidirect
5. C₂×C₄⋊Q₈×C₅₃	C₂	semidirect
29. C₂×D₂₁₂	Q₈	semidirect
97. C₂×C₄⋊Q₈	D₅₃	semidirect
102. C₅₃	#2023	semidirect

H ≅ G/N

type

3. #16810385

C₂

semidirect

4. #16810378

C₂

semidirect

5. C₂×C₄⋊Q₈×C₅₃

C₂

semidirect

29. C₂×D₂₁₂

Q₈

semidirect

97. C₂×C₄⋊Q₈

D₅₃

semidirect

102. C₅₃

#2023

semidirect

Subgroups — #20275848

Conjugacy classes of subgroups

Complements