Group #27300310 — subgroups

Subgroups — #27300310

Order 13568.

434 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16810578	C₂	1	1	yes	maximal, characteristic
2	#20276326	C₂	1	1	yes	maximal, characteristic
3	#20272798	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2112	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₄	1	1	yes	characteristic
6	C₂×#13133176	C₄	1	1	yes	characteristic
7	C₂×#13133157	C₂²	1	2	yes	—
8	#16810578	—	2	2	no	—
9	#16810578	—	2	2	no	—
10	C₂×#13133127	C₂²	1	2	yes	—
11	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
12	C₂²⋊#11719908	—	2	2	no	—
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
15	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
16	C₂×#11719917	—	2	4	no	—
17	C₂×#11719917	—	2	2	no	—
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	#13133176	—	2	2	no	—
20	#13133176	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
22	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
23	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
24	#13133157	—	2	4	no	—
25	C₂×#11719908	—	2	2	no	—
26	#13133127	—	2	2	no	—
27	#13133127	—	2	2	no	—
28	C₈×Dic₅₃	—	2	4	no	—
29	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	2	yes	—
33	C₂²×C₈×C₅₃	D₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	4	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	#41	1	4	yes	—
52	C₂²⋊Dic₅₃	—	2	4	no	—
53	C₄⋊Dic₅₃	—	2	4	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	—	2	2	no	—
56	#11719917	—	2	8	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	—	2	2	no	—
59	C₂×#88322	M₁₆	1	4	yes	—
60	C₁₆×C₅₃	—	4	2	no	—
61	#11719908	—	4	2	no	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	—	4	2	no	—
64	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
65	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#117	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	4	no	—
76	C₂×Dic₅₃	—	4	4	no	—
77	C₈×C₅₃	—	2	2	no	—
78	C₈×C₅₃	#123	1	2	yes	—
79	C₈×C₅₃	#123	1	2	yes	—
80	#88322	—	2	4	no	—
81	#12847	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	—	2	4	no	—
85	C₂²×C₅₃	—	4	4	no	—
86	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
87	C₂²×C₅₃	#431	1	2	yes	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	#381	1	1	yes	characteristic
90	C₄×C₅₃	#407	1	1	yes	characteristic
91	C₄×C₅₃	#432	1	2	yes	—
92	Dic₅₃	—	4	4	no	—
93	C₂×#405	—	53	1	no	—
94	#2112	—	53	1	no	—
95	#2112	D₅₃	1	1	yes	characteristic
96	#1377	—	53	4	no	—
97	C₂×C₅₃	—	2	2	no	—
98	C₂×C₅₃	—	4	2	no	—
99	C₂×C₅₃	#1817	1	1	yes	characteristic
100	C₂×C₅₃	#2115	1	2	yes	—
101	C₂²×M₁₆	—	53	1	no	—
102	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
103	C₂×#117	—	53	1	no	—
104	C₂×M₃₂	—	53	2	no	—
105	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
106	#405	—	106	2	no	—
107	#405	—	106	2	no	—
108	C₂×#97	—	53	2	no	—
109	C₂×C₄×C₈	—	53	2	no	—
110	C₂²⋊C₁₆	—	2	2	no	—
111	C₂²⋊C₁₆	—	106	2	no	—
112	C₅₃	#12847	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
114	C₂×M₁₆	—	106	4	no	—
115	C₂×M₁₆	—	106	2	no	—
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	—	53	2	no	—
118	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₂²×C₈	D₂₁₂	1	2	yes	—
120	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
121	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
122	#117	—	106	2	no	—
123	#117	—	106	2	no	—
124	C₂×C₄⋊C₄	—	53	2	no	—
125	C₂×C₂²⋊C₄	—	53	2	no	—
126	C₂×C₄²	—	53	2	no	—
127	M₃₂	—	2	4	no	—
128	M₃₂	—	106	4	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	106	2	no	—
131	#97	—	106	2	no	—
132	#97	—	106	2	no	—
133	C₄×C₈	—	106	4	no	—
134	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	53	4	no	—
138	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
139	C₂²×C₄	#11719912	1	2	yes	—
140	C₂²×C₄	#11719921	1	2	yes	—
141	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
142	M₁₆	—	106	8	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	53	4	no	—
147	C₂×C₈	—	106	2	no	—
148	C₂×C₈	—	106	2	no	—
149	C₂×C₈	#11719921	1	2	yes	—
150	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
151	C₂×C₈	#11719947	1	4	yes	—
152	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₄⋊C₄	—	106	4	no	—
154	C₂²⋊C₄	—	106	4	no	—
155	C₄²	—	106	2	no	—
156	C₄²	—	106	2	no	—
157	C₁₆	—	4	2	no	—
158	C₁₆	—	212	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	4	2	no	—
161	C₂³	#13133129	1	2	yes	—
162	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	106	4	no	—
168	C₂×C₄	—	212	4	no	—
169	C₂×C₄	#13133129	1	2	yes	—
170	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
171	C₂×C₄	#13133173	1	2	yes	—
172	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
173	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	106	4	no	—
176	C₈	#13133198	1	2	yes	—
177	C₈	#13133201	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	4	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₂×#13133129	1	1	yes	characteristic
183	C₂²	#16810673	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	212	4	no	—
186	C₄	#16810484	1	1	yes	characteristic
187	C₄	#16810586	1	1	yes	characteristic
188	C₄	#16810677	1	2	yes	—
189	C₂	—	2	2	no	—
190	C₂	—	4	2	no	—
191	C₂	#20274781	1	1	yes	characteristic
192	C₂	#20276335	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810578

C₂

yes

maximal, characteristic

#20276326

C₂

yes

maximal, characteristic

#20272798

C₂

yes

maximal, complemented

C₅₃×#2112

C₂

yes

maximal, characteristic

C₂²×#11719917

C₄

yes

characteristic

C₂×#13133176

C₄

yes

characteristic

C₂×#13133157

C₂²

yes

—

#16810578

—

#16810578

—

C₂×#13133127

C₂²

yes

—

C₂×C₈×Dic₅₃

C₄

yes

—

C₂²⋊#11719908

—

C₂³×C₈×C₅₃

C₂²

yes

characteristic

C₂×M₃₂×C₅₃

C₂²

yes

—

C₅₃×C₂²⋊C₁₆

—

C₂×#11719917

—

C₂×#11719917

—

C₂²×#88322

C₂×C₄

yes

—

#13133176

—

#13133176

—

C₂×C₄⋊Dic₅₃

C₈

yes

—

C₂×C₂²⋊Dic₅₃

C₈

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₅₃

—

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂⁴×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

M₁₆

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₈

yes

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#123

yes

—

C₈×C₅₃

#123

yes

—

#88322

—

#12847

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

#431

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

C₂×#405

—

#2112

—

#2112

D₅₃

yes

characteristic

#1377

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1817

yes

characteristic

100

C₂×C₅₃

#2115

yes

—

101

C₂²×M₁₆

—

102

C₂³×C₈

C₂×D₅₃

yes

characteristic

103

C₂×#117

—

104

C₂×M₃₂

—

105

C₂×M₃₂

C₂×D₅₃

yes

—

106

#405

—

106

—

107

#405

—

106

—

108

C₂×#97

—

109

C₂×C₄×C₈

—

110

C₂²⋊C₁₆

—

111

C₂²⋊C₁₆

—

106

—

112

C₅₃

#12847

yes

characteristic, complemented, Sylow

113

C₂³×C₄

C₄×D₅₃

yes

characteristic

114

C₂×M₁₆

—

106

—

115

C₂×M₁₆

—

106

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

C₄×D₅₃

yes

characteristic

119

C₂²×C₈

D₂₁₂

yes

—

120

C₂²×C₈

C₂²⋊D₅₃

yes

—

121

C₂²×C₈

C₂²×D₅₃

yes

characteristic

122

#117

—

106

—

123

#117

—

106

—

124

C₂×C₄⋊C₄

—

125

C₂×C₂²⋊C₄

—

126

C₂×C₄²

—

127

M₃₂

—

128

M₃₂

—

106

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

106

—

131

#97

—

106

—

132

#97

—

106

—

133

C₄×C₈

—

106

—

134

C₂⁴

C₈×D₅₃

yes

characteristic

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₈×D₅₃

yes

characteristic

139

C₂²×C₄

#11719912

yes

—

140

C₂²×C₄

#11719921

yes

—

141

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

142

M₁₆

—

106

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

106

—

148

C₂×C₈

—

106

—

149

C₂×C₈

#11719921

yes

—

150

C₂×C₈

C₂×D₂₁₂

yes

characteristic

151

C₂×C₈

#11719947

yes

—

152

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

153

C₄⋊C₄

—

106

—

154

C₂²⋊C₄

—

106

—

155

C₄²

—

106

—

156

C₄²

—

106

—

157

C₁₆

—

158

C₁₆

—

212

—

159

C₂³

—

160

C₂³

—

161

C₂³

#13133129

yes

—

162

C₂³

C₂×C₈×D₅₃

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

106

—

168

C₂×C₄

—

212

—

169

C₂×C₄

#13133129

yes

—

170

C₂×C₄

C₂×#11719921

yes

characteristic

171

C₂×C₄

#13133173

yes

—

172

C₂×C₄

M₁₆×D₅₃

yes

—

173

C₂×C₄

C₂×#11719912

yes

characteristic

174

C₈

—

175

C₈

—

106

—

176

C₈

#13133198

yes

—

177

C₈

#13133201

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

C₂×#13133129

yes

characteristic

183

C₂²

#16810673

yes

—

184

C₄

—

185

C₄

—

212

—

186

C₄

#16810484

yes

characteristic

187

C₄

#16810586

yes

characteristic

188

C₄

#16810677

yes

—

189

C₂

—

190

C₂

—

191

C₂

#20274781

yes

characteristic

192

C₂

#20276335

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2112	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2112	D₅₃
hypocenter	C₅₃	#12847
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₂×C₄

C₂×#11719921

commutator

C₂³×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#2112

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₅₃

C₂×C₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
3. #20272798	C₂	semidirect
112. C₅₃	#12847	semidirect

H ≅ G/N

type

3. #20272798

C₂

semidirect

112. C₅₃

#12847

semidirect

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

subgroup	index
C₂×#16810578	2
#20276326	2
#20272798	2
C₅₃×#2112	2
#12847	53

subgroup

index

C₅₃×#2112

Subgroups — #27300310

Conjugacy classes of subgroups

Complements

Maximal subgroups