Group #27284214 — subgroups

Subgroups — #27284214

Order 13568.

430 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 167 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	#20274832	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272598	C₂	1	4	yes	maximal
4	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	2	yes	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	—	2	2	no	—
10	C₂×#125×C₅₃	—	2	1	no	—
11	C₅₃×#389	—	2	2	no	—
12	C₅₃×#389	C₄	1	4	yes	—
13	C₅₃×#384	C₂²	1	2	yes	—
14	C₅₃×#382	—	2	1	no	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	—	2	2	no	—
18	C₄⋊#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₄⋊#88322	Q₈	1	4	yes	—
21	C₄×#88322	—	4	2	no	—
22	#125×C₅₃	—	4	4	no	—
23	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#95×C₅₃	—	2	4	no	—
33	C₂²×C₄×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₅₃	—	4	8	no	—
40	C₄⋊C₄×C₅₃	—	4	4	no	—
41	C₄⋊C₄×C₅₃	—	4	4	no	—
42	C₄⋊C₄×C₅₃	—	4	4	no	—
43	C₄²×C₅₃	—	2	4	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	4	2	no	—
46	C₄²×C₅₃	SD₁₆	1	2	yes	—
47	C₄²×C₅₃	Q₁₆	1	2	yes	—
48	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
49	C₄²×C₅₃	#41	1	2	yes	—
50	C₂×#88322	—	2	4	no	—
51	C₂×#88322	—	4	2	no	—
52	C₂×#88322	—	4	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	—	4	4	no	—
55	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	4	4	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	4	4	no	—
65	C₂×C₄×C₅₃	#103	1	4	yes	—
66	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#135	1	4	yes	—
73	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
74	#88322	—	4	4	no	—
75	#88322	—	8	2	no	—
76	#9897	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	#391	1	1	yes	characteristic
78	C₂²×C₅₃	#413	1	2	yes	—
79	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
80	C₂²×C₅₃	#416	1	2	yes	—
81	C₂²×C₅₃	#418	1	1	yes	characteristic
82	C₄×C₅₃	—	2	4	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	4	2	no	—
85	C₄×C₅₃	—	8	4	no	—
86	C₄×C₅₃	#476	1	2	yes	—
87	C₄×C₅₃	#477	1	2	yes	—
88	C₄×C₅₃	#488	1	2	yes	—
89	C₄×C₅₃	#499	1	2	yes	—
90	#2859	D₅₃	1	1	yes	characteristic
91	#1831	—	53	1	no	—
92	C₄×C₄⋊C₈	—	53	1	no	—
93	#1296	—	53	4	no	—
94	C₂×C₅₃	#1542	1	2	yes	—
95	C₂×C₅₃	#1553	1	2	yes	—
96	C₂×C₅₃	#1987	1	1	yes	characteristic
97	C₂×C₅₃	#1991	1	1	yes	characteristic
98	C₂×C₅₃	#1998	1	1	yes	characteristic
99	C₂×#125	—	2	1	no	—
100	C₂×C₄⋊C₈	—	53	2	no	—
101	C₂×C₄⋊C₈	—	53	2	no	—
102	C₂×C₄×C₈	—	106	1	no	—
103	#389	—	2	2	no	—
104	#389	Dic₅₃	1	4	yes	—
105	#384	C₂×D₅₃	1	2	yes	—
106	#382	—	2	1	no	—
107	C₄³	C₂×D₅₃	1	1	yes	characteristic
108	#336	—	106	2	no	—
109	#336	—	106	1	no	—
110	C₅₃	#9897	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	106	2	no	—
112	C₂²×C₈	—	106	2	no	—
113	#125	—	4	4	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
118	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
119	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
121	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
123	C₄⋊C₈	—	53	4	no	—
124	C₄⋊C₈	—	106	2	no	—
125	C₄⋊C₈	—	106	4	no	—
126	C₄×C₈	—	212	2	no	—
127	#95	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
130	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
131	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂×C₈	—	106	4	no	—
135	C₂×C₈	—	212	2	no	—
136	C₂×C₈	—	212	4	no	—
137	C₂×C₈	—	212	2	no	—
138	C₂×C₈	—	212	2	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#11719924	1	2	yes	—
147	C₄²	#11719925	1	2	yes	—
148	C₄²	Q₈×D₅₃	1	2	yes	—
149	C₄²	#11719949	1	2	yes	—
150	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	#13133141	1	4	yes	—
161	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
162	C₂×C₄	#13133193	1	2	yes	—
163	C₂×C₄	#13133206	1	1	yes	characteristic
164	C₂×C₄	#13133215	1	1	yes	characteristic
165	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
166	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
167	C₂×C₄	#13133237	1	4	yes	—
168	C₂×C₄	#13133243	1	2	yes	—
169	C₈	—	424	2	no	—
170	C₈	—	212	4	no	—
171	C₂²	#16810530	1	1	yes	characteristic
172	C₂²	#16810607	1	2	yes	—
173	C₂²	C₂×#13133141	1	1	yes	characteristic
174	C₂²	#16810619	1	2	yes	—
175	C₂²	#16810628	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	4	no	—
180	C₄	#16810910	1	2	yes	—
181	C₄	#16810915	1	2	yes	—
182	C₄	#16810980	1	2	yes	—
183	C₄	#16811022	1	2	yes	—
184	C₂	#20273372	1	2	yes	—
185	C₂	#20273420	1	2	yes	—
186	C₂	#20275664	1	1	yes	characteristic
187	C₂	#20275681	1	1	yes	characteristic
188	C₂	#20275709	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal, characteristic

C₄×C₄⋊#88322

C₂

yes

maximal, characteristic

#20272598

C₂

yes

maximal

C₅₃×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄×#88322

—

#16810378

—

#16810378

—

C₂×#125×C₅₃

—

C₅₃×#389

—

C₅₃×#389

C₄

yes

—

C₅₃×#384

C₂²

yes

—

C₅₃×#382

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄⋊#88322

Q₈

yes

—

C₄×#88322

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

C₄×Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#103

yes

characteristic

C₂²×C₅₃

#416

yes

—

C₂²×C₅₃

#418

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#476

yes

—

C₄×C₅₃

#477

yes

—

C₄×C₅₃

#488

yes

—

C₄×C₅₃

#499

yes

—

#2859

D₅₃

yes

characteristic

#1831

—

C₄×C₄⋊C₈

—

#1296

—

C₂×C₅₃

#1542

yes

—

C₂×C₅₃

#1553

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄×C₈

—

106

—

103

#389

—

104

#389

Dic₅₃

yes

—

105

#384

C₂×D₅₃

yes

—

106

#382

—

107

C₄³

C₂×D₅₃

yes

characteristic

108

#336

—

106

—

109

#336

—

106

—

110

C₅₃

#9897

yes

characteristic, complemented, Sylow

111

C₂²×C₈

—

106

—

112

C₂²×C₈

—

106

—

113

#125

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

118

C₂×C₄²

C₂×Dic₅₃

yes

—

119

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

120

C₂×C₄²

C₂²⋊D₅₃

yes

—

121

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

122

C₂×C₄²

C₂²×D₅₃

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

106

—

125

C₄⋊C₈

—

106

—

126

C₄×C₈

—

212

—

127

#95

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

130

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

131

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

132

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

133

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

134

C₂×C₈

—

106

—

135

C₂×C₈

—

212

—

136

C₂×C₈

—

212

—

137

C₂×C₈

—

212

—

138

C₂×C₈

—

212

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#11719924

yes

—

147

C₄²

#11719925

yes

—

148

C₄²

Q₈×D₅₃

yes

—

149

C₄²

#11719949

yes

—

150

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

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

#13133141

yes

—

161

C₂×C₄

Q₈×Dic₅₃

yes

—

162

C₂×C₄

#13133193

yes

—

163

C₂×C₄

#13133206

yes

characteristic

164

C₂×C₄

#13133215

yes

characteristic

165

C₂×C₄

C₂×#11719924

yes

characteristic

166

C₂×C₄

C₂×#11719925

yes

characteristic

167

C₂×C₄

#13133237

yes

—

168

C₂×C₄

#13133243

yes

—

169

C₈

—

424

—

170

C₈

—

212

—

171

C₂²

#16810530

yes

characteristic

172

C₂²

#16810607

yes

—

173

C₂²

C₂×#13133141

yes

characteristic

174

C₂²

#16810619

yes

—

175

C₂²

#16810628

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#16810910

yes

—

181

C₄

#16810915

yes

—

182

C₄

#16810980

yes

—

183

C₄

#16811022

yes

—

184

C₂

#20273372

yes

—

185

C₂

#20273420

yes

—

186

C₂

#20275664

yes

characteristic

187

C₂

#20275681

yes

characteristic

188

C₂

#20275709

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2859	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2859	D₅₃
hypocenter	C₅₃	#9897
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2859

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
110. C₅₃	#9897	semidirect

H ≅ G/N

type

110. C₅₃

#9897

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274832	2
C₄×C₄⋊#88322	2
#20272598	2
C₅₃×#2859	2
#9897	53

subgroup

index

#20274832

C₄×C₄⋊#88322

#20272598

C₅₃×#2859

#9897

Subgroups — #27284214

Conjugacy classes of subgroups

Complements

Maximal subgroups