Group #27277050 — subgroups

Subgroups — #27277050

Order 13568.

362 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20273478	C₂	1	2	yes	maximal
2	#20273059	C₂	1	2	yes	maximal
3	#20272585	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810680	C₂²	1	4	yes	—
6	#16810376	—	2	2	no	—
7	#16810374	—	2	2	no	—
8	#16810349	—	2	4	no	—
9	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₅₃	—	2	2	no	—
11	C₈×Q₈×C₅₃	—	2	2	no	—
12	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊#88322	—	2	4	no	—
17	C₂²⋊#88322	—	2	4	no	—
18	#13133128	—	2	4	no	—
19	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×Q₈×C₅₃	—	2	2	no	—
24	C₄×Q₈×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	4	no	—
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₂×Q₈×C₅₃	—	2	4	no	—
36	C₂×Q₈×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	4	1	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	C₂×D₄	1	2	yes	—
54	C₄²×C₅₃	#41	1	2	yes	—
55	C₂×#88322	—	4	4	no	—
56	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	Q₈×C₅₃	—	2	8	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	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	4	1	no	—
65	C₂×C₄×C₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	#108	1	4	yes	—
67	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#121	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	2	2	no	—
73	C₈×C₅₃	—	4	2	no	—
74	C₈×C₅₃	#101	1	2	yes	—
75	#88322	—	8	4	no	—
76	#8623	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	#380	1	1	yes	characteristic
80	C₂²×C₅₃	C₂×#108	1	2	yes	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	4	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	4	2	no	—
86	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
87	C₄×C₅₃	#413	1	1	yes	characteristic
88	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
89	#1568	—	53	2	no	—
90	#1467	—	53	2	no	—
91	#1290	—	53	2	no	—
92	C₂×C₅₃	—	2	2	no	—
93	C₂×C₅₃	#1801	1	1	yes	characteristic
94	C₂×C₅₃	#1810	1	1	yes	characteristic
95	C₂×C₅₃	#1812	1	1	yes	characteristic
96	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
97	C₈×Q₈	—	2	2	no	—
98	C₈×Q₈	—	2	2	no	—
99	#432	—	53	4	no	—
100	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
101	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
103	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
104	#335	—	106	2	no	—
105	#334	—	106	2	no	—
106	#324	—	106	4	no	—
107	C₅₃	#8623	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
111	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
112	C₄×Q₈	—	2	2	no	—
113	C₄×Q₈	—	2	2	no	—
114	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
115	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
116	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
117	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	106	4	no	—
121	C₂²⋊C₈	—	106	4	no	—
122	#97	—	106	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
126	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	—	2	2	no	—
129	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
130	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
131	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
133	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	212	4	no	—
139	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
141	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#11719945	1	2	yes	—
147	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
148	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
149	Q₈	—	2	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	#13133153	1	4	yes	—
159	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
160	C₂×C₄	#13133176	1	1	yes	characteristic
161	C₂×C₄	#13133191	1	2	yes	—
162	C₂×C₄	#13133196	1	2	yes	—
163	C₈	—	2	1	no	—
164	C₈	—	2	2	no	—
165	C₈	—	4	2	no	—
166	C₈	—	424	4	no	—
167	C₈	#13133136	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16810483	1	1	yes	characteristic
171	C₂²	C₂×#13133153	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×#13133136	1	1	yes	characteristic
178	C₄	#16810607	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	#20274711	1	1	yes	characteristic
181	C₂	#20274753	1	1	yes	characteristic
182	C₂	#20274761	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273478

C₂

yes

maximal

#20273059

C₂

yes

maximal

#20272585

C₂

yes

maximal

C₂×C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic

#16810680

C₂²

yes

—

#16810376

—

#16810374

—

#16810349

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₈×Q₈×C₅₃

—

C₈×Q₈×C₅₃

—

C₂×C₄⋊C₈×C₅₃

C₄

yes

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

C₂²×Q₈×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

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

—

C₂×C₄×C₅₃

#108

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#101

yes

—

#88322

—

#8623

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#380

yes

characteristic

C₂²×C₅₃

C₂×#108

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₈×Q₈

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#432

—

100

C₂×C₄⋊C₈

Dic₅₃

yes

—

101

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

102

C₂×C₄×C₈

Dic₅₃

yes

—

103

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

104

#335

—

106

—

105

#334

—

106

—

106

#324

—

106

—

107

C₅₃

#8623

yes

characteristic, complemented, Sylow

108

C₂²×Q₈

Dic₁₀₆

yes

characteristic

109

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

110

C₂²×C₈

C₂×Dic₅₃

yes

—

111

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

112

C₄×Q₈

—

113

C₄×Q₈

—

114

C₂×C₄⋊C₄

Dic₁₀₆

yes

characteristic

115

C₂×C₄⋊C₄

D₂₁₂

yes

—

116

C₂×C₄²

C₂×Dic₅₃

yes

—

117

C₂×C₄²

C₂²×D₅₃

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

106

—

121

C₂²⋊C₈

—

106

—

122

#97

—

106

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

C₂²⋊D₅₃

yes

—

126

C₄×C₈

C₂²⋊D₅₃

yes

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

C₄⋊Dic₅₃

yes

—

130

C₂²×C₄

C₄⋊Dic₅₃

yes

—

131

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

132

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

133

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

212

—

139

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

140

C₂×C₈

C₂²⋊Dic₅₃

yes

—

141

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄²

—

145

C₄²

—

146

C₄²

#11719945

yes

—

147

C₄²

C₂×C₂²⋊D₅₃

yes

—

148

C₂³

C₂×C₄⋊Dic₅₃

yes

characteristic

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#13133153

yes

—

159

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

160

C₂×C₄

#13133176

yes

characteristic

161

C₂×C₄

#13133191

yes

—

162

C₂×C₄

#13133196

yes

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

424

—

167

C₈

#13133136

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

#16810483

yes

characteristic

171

C₂²

C₂×#13133153

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

C₂×#13133136

yes

characteristic

178

C₄

#16810607

yes

characteristic

179

C₂

—

180

C₂

#20274711

yes

characteristic

181

C₂

#20274753

yes

characteristic

182

C₂

#20274761

yes

characteristic

	subgroup	G/H
center	C₂²	#16810483
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#8623
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

subgroup

G/H

center

C₂²

#16810483

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

#380

prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

hypercenter

C₂×C₈×Q₈

D₅₃

hypocenter

C₅₃

#8623

LCS

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

N	H ≅ G/N	type
107. C₅₃	#8623	semidirect

H ≅ G/N

type

107. C₅₃

#8623

semidirect

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

subgroup	index
#20273478	2
#20273059	2
#20272585	2
C₂×C₈×Q₈×C₅₃	2
#8623	53

subgroup

index

C₂×C₈×Q₈×C₅₃

Subgroups — #27277050

Conjugacy classes of subgroups

Complements

Maximal subgroups