Group #20276198 — subgroups

Subgroups — #20276198

Order 6784.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133149	C₂	1	1	yes	maximal, characteristic
3	#16810521	C₂	1	1	yes	maximal, characteristic
4	#16810489	C₂	1	1	yes	maximal, characteristic
5	#16810386	C₂	1	1	yes	maximal, characteristic
6	#16810386	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#11719919	—	2	1	no	—
13	#13133151	—	2	2	no	—
14	#13133149	—	2	2	no	—
15	#13133124	—	2	1	no	—
16	#13133124	—	2	1	no	—
17	#13133124	—	2	1	no	—
18	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₃	—	2	1	no	—
25	C₂²×Dic₅₃	—	2	1	no	—
26	C₂²×Dic₅₃	—	2	1	no	—
27	C₂²×Dic₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	Q₈	1	2	yes	—
33	C₂×C₈×C₅₃	Q₈	1	2	yes	—
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₄⋊Dic₅₃	—	2	1	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄⋊Dic₅₃	—	2	1	no	—
43	C₄⋊Dic₅₃	D₄	1	2	yes	—
44	#11719919	—	4	2	no	—
45	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	—	4	1	no	—
60	C₂×Dic₅₃	—	4	1	no	—
61	C₂×Dic₅₃	—	4	1	no	—
62	C₂×Dic₅₃	—	4	1	no	—
63	C₂×Dic₅₃	—	4	1	no	—
64	C₂×Dic₅₃	—	4	1	no	—
65	C₂×Dic₅₃	—	4	1	no	—
66	C₂×Dic₅₃	—	4	1	no	—
67	C₂×Dic₅₃	—	4	1	no	—
68	C₂×Dic₅₃	—	4	1	no	—
69	C₂×Dic₅₃	—	4	1	no	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	#121	1	1	yes	characteristic
73	C₂²×C₅₃	#123	1	1	yes	characteristic
74	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂²×C₅₃	#135	1	1	yes	characteristic
76	C₂²×C₅₃	#135	1	1	yes	characteristic
77	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₅₃	#137	1	1	yes	characteristic
79	C₄×C₅₃	—	4	1	no	—
80	C₄×C₅₃	—	4	1	no	—
81	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₄×C₅₃	#125	1	1	yes	characteristic
84	C₄×C₅₃	#125	1	1	yes	characteristic
85	Dic₅₃	—	4	2	no	—
86	Dic₅₃	—	8	1	no	—
87	Dic₅₃	—	8	1	no	—
88	Dic₅₃	—	8	1	no	—
89	#2085	—	53	1	no	maximal, Sylow
90	C₂×C₅₃	#398	1	1	yes	characteristic
91	C₂×C₅₃	#463	1	1	yes	characteristic
92	C₂×C₅₃	#464	1	1	yes	characteristic
93	C₂×C₅₃	#482	1	1	yes	characteristic
94	C₂×C₅₃	#485	1	1	yes	characteristic
95	C₂×C₅₃	#499	1	1	yes	characteristic
96	C₂×C₅₃	#501	1	1	yes	characteristic
97	C₂×#107	—	53	1	no	—
98	C₂×#106	—	53	1	no	—
99	#389	—	53	1	no	—
100	#382	—	53	1	no	—
101	#340	—	53	1	no	—
102	#340	—	53	1	no	—
103	#336	D₅₃	1	1	yes	characteristic
104	C₅₃	#2085	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	53	1	no	—
108	C₂×C₄⋊C₄	—	53	1	no	—
109	C₂×C₄⋊C₄	—	53	1	no	—
110	C₂×C₄⋊C₄	—	53	1	no	—
111	C₂×C₄⋊C₄	—	106	1	no	—
112	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
113	#107	—	106	2	no	—
114	#106	—	106	2	no	—
115	#95	—	106	1	no	—
116	#95	—	106	1	no	—
117	#95	—	106	1	no	—
118	C₂²×C₄	—	106	1	no	—
119	C₂²×C₄	—	106	1	no	—
120	C₂²×C₄	—	106	1	no	—
121	C₂²×C₄	—	106	1	no	—
122	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
123	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	Dic₁₀₆	1	2	yes	—
130	C₂×C₈	Dic₁₀₆	1	2	yes	—
131	C₄⋊C₄	—	53	2	no	—
132	C₄⋊C₄	—	106	1	no	—
133	C₄⋊C₄	—	106	1	no	—
134	C₄⋊C₄	—	106	1	no	—
135	C₄⋊C₄	—	106	1	no	—
136	C₄⋊C₄	—	106	1	no	—
137	C₄⋊C₄	—	106	1	no	—
138	C₄⋊C₄	—	106	1	no	—
139	C₄⋊C₄	—	212	2	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	106	2	no	—
150	C₂×C₄	—	212	1	no	—
151	C₂×C₄	—	212	1	no	—
152	C₂×C₄	—	212	1	no	—
153	C₂×C₄	—	212	1	no	—
154	C₂×C₄	—	212	1	no	—
155	C₂×C₄	—	212	1	no	—
156	C₂×C₄	—	212	1	no	—
157	C₂×C₄	—	212	1	no	—
158	C₂×C₄	—	212	1	no	—
159	C₂×C₄	—	212	1	no	—
160	C₂×C₄	—	212	1	no	—
161	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
163	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
164	C₂×C₄	#11719947	1	1	yes	characteristic
165	C₂×C₄	#11719947	1	1	yes	characteristic
166	C₂×C₄	#11719949	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	2	2	no	—
169	C₂²	#13133192	1	1	yes	characteristic
170	C₂²	#13133201	1	1	yes	characteristic
171	C₂²	#13133217	1	1	yes	characteristic
172	C₂²	#13133239	1	1	yes	characteristic
173	C₂²	#13133239	1	1	yes	characteristic
174	C₂²	#13133245	1	1	yes	characteristic
175	C₂²	#13133250	1	1	yes	characteristic
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	424	1	no	—
179	C₄	—	424	1	no	—
180	C₄	—	424	1	no	—
181	C₄	—	212	2	no	—
182	C₄	#13133194	1	1	yes	characteristic
183	C₄	#13133194	1	1	yes	characteristic
184	C₄	#13133208	1	1	yes	characteristic
185	C₄	#13133208	1	1	yes	characteristic
186	C₂	#16810555	1	1	yes	characteristic
187	C₂	#16810839	1	1	yes	characteristic
188	C₂	#16810846	1	1	yes	characteristic
189	C₂	#16810948	1	1	yes	characteristic
190	C₂	#16810965	1	1	yes	characteristic
191	C₂	#16811025	1	1	yes	characteristic
192	C₂	#16811032	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133149

C₂

yes

maximal, characteristic

#16810521

C₂

yes

maximal, characteristic

#16810489

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal, characteristic

C₅₃×#336

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₄⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

100

#382

—

101

#340

—

102

#340

—

103

#336

D₅₃

yes

characteristic

104

C₅₃

#2085

yes

characteristic, complemented, Sylow

105

C₂²×C₈

C₂×D₅₃

yes

characteristic

106

C₂²×C₈

C₂×D₅₃

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

106

—

112

C₂×C₄²

C₂×D₅₃

yes

characteristic

113

#107

—

106

—

114

#106

—

106

—

115

#95

—

106

—

116

#95

—

106

—

117

#95

—

106

—

118

C₂²×C₄

—

106

—

119

C₂²×C₄

—

106

—

120

C₂²×C₄

—

106

—

121

C₂²×C₄

—

106

—

122

C₂²×C₄

D₂₁₂

yes

characteristic

123

C₂²×C₄

D₂₁₂

yes

characteristic

124

C₂²×C₄

C₂²×D₅₃

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

Dic₁₀₆

yes

—

130

C₂×C₈

Dic₁₀₆

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

106

—

135

C₄⋊C₄

—

106

—

136

C₄⋊C₄

—

106

—

137

C₄⋊C₄

—

106

—

138

C₄⋊C₄

—

106

—

139

C₄⋊C₄

—

212

—

140

C₄²

—

141

C₄²

—

142

C₂³

C₂×D₂₁₂

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

212

—

151

C₂×C₄

—

212

—

152

C₂×C₄

—

212

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

—

212

—

156

C₂×C₄

—

212

—

157

C₂×C₄

—

212

—

158

C₂×C₄

—

212

—

159

C₂×C₄

—

212

—

160

C₂×C₄

—

212

—

161

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

162

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

163

C₂×C₄

D₄×D₅₃

yes

characteristic

164

C₂×C₄

#11719947

yes

characteristic

165

C₂×C₄

#11719947

yes

characteristic

166

C₂×C₄

#11719949

yes

characteristic

167

C₈

—

168

C₈

—

169

C₂²

#13133192

yes

characteristic

170

C₂²

#13133201

yes

characteristic

171

C₂²

#13133217

yes

characteristic

172

C₂²

#13133239

yes

characteristic

173

C₂²

#13133239

yes

characteristic

174

C₂²

#13133245

yes

characteristic

175

C₂²

#13133250

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

424

—

179

C₄

—

424

—

180

C₄

—

424

—

181

C₄

—

212

—

182

C₄

#13133194

yes

characteristic

183

C₄

#13133194

yes

characteristic

184

C₄

#13133208

yes

characteristic

185

C₄

#13133208

yes

characteristic

186

C₂

#16810555

yes

characteristic

187

C₂

#16810839

yes

characteristic

188

C₂

#16810846

yes

characteristic

189

C₂

#16810948

yes

characteristic

190

C₂

#16810965

yes

characteristic

191

C₂

#16811025

yes

characteristic

192

C₂

#16811032

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#2085
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
104. C₅₃	#2085	semidirect

H ≅ G/N

type

104. C₅₃

#2085

semidirect

Subgroups — #20276198

Conjugacy classes of subgroups

Complements