Group #27302615 — subgroups

Subgroups — #27302615

Order 13568.

386 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 159 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×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272792	C₂	1	4	yes	maximal
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₈×Dic₁₀₆	—	2	2	no	—
8	C₈×Dic₁₀₆	—	2	2	no	—
9	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
10	C₂×#13133146	C₄	1	2	yes	—
11	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
12	C₄⋊#11719908	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#346	—	2	2	no	—
16	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₀₆	—	2	2	no	—
19	C₄×Dic₁₀₆	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
21	C₂×#11719919	C₈	1	2	yes	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	4	no	—
25	C₄⋊#88322	—	2	2	no	—
26	#13133146	—	2	4	no	—
27	C₈×Dic₅₃	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	—	2	2	no	—
33	C₂×C₁₆×C₅₃	—	2	2	no	—
34	C₂×C₁₆×C₅₃	Q₈	1	4	yes	complemented
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
41	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
42	C₂×Dic₁₀₆	—	2	2	no	—
43	C₂×Dic₁₀₆	—	2	4	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₅₃	#41	1	2	yes	—
50	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₅₃	M₁₆	1	2	yes	—
52	C₄⋊Dic₅₃	—	2	2	no	—
53	#11719919	—	2	4	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	—	2	2	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	4	no	—
58	C₁₆×C₅₃	—	2	4	no	—
59	#11719908	—	4	2	no	—
60	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#131	1	2	yes	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	—	2	4	no	—
73	C₂×Dic₅₃	M₃₂	1	4	yes	—
74	Dic₁₀₆	—	2	8	no	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₅₃	#124	1	1	yes	characteristic
78	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	#88322	—	4	2	no	—
80	#13264	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
83	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
84	C₂²×C₅₃	#504	1	2	yes	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	#391	1	1	yes	characteristic
88	C₄×C₅₃	#408	1	1	yes	characteristic
89	C₄×C₅₃	#446	1	2	yes	—
90	Dic₅₃	—	2	4	no	—
91	Dic₅₃	—	4	2	no	—
92	C₂×C₈×Q₈	—	53	1	no	—
93	C₂×C₄⋊C₁₆	—	53	1	no	—
94	C₂×#346	D₅₃	1	1	yes	characteristic
95	#1374	—	53	4	no	—
96	C₂×C₅₃	#1984	1	1	yes	characteristic
97	C₂×C₅₃	#2112	1	1	yes	characteristic
98	C₂×C₅₃	#2113	1	1	yes	characteristic
99	C₂×C₅₃	#2183	1	4	yes	—
100	C₂×C₄×Q₈	—	53	1	no	—
101	C₂²×C₁₆	—	53	2	no	—
102	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
103	C₈×Q₈	—	106	2	no	—
104	C₈×Q₈	—	106	2	no	—
105	C₂×C₄⋊C₈	—	53	2	no	—
106	C₂×C₄⋊C₈	—	53	1	no	—
107	C₂×C₄×C₈	—	53	2	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₄⋊C₁₆	—	106	2	no	—
110	#346	—	2	2	no	—
111	C₅₃	#13264	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	53	1	no	—
113	C₂²×C₈	—	53	2	no	—
114	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
116	C₄×Q₈	—	106	2	no	—
117	C₄×Q₈	—	106	2	no	—
118	C₂×C₄⋊C₄	—	53	2	no	—
119	C₂×C₄⋊C₄	—	53	1	no	—
120	C₂×C₄²	—	53	2	no	—
121	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	106	2	no	—
125	C₂×C₁₆	—	106	4	no	—
126	C₂×C₁₆	Dic₁₀₆	1	4	yes	complemented
127	C₄⋊C₈	—	106	4	no	—
128	C₄⋊C₈	—	106	2	no	—
129	C₄×C₈	—	106	2	no	—
130	C₄×C₈	—	106	2	no	—
131	C₄×C₈	D₂₁₂	1	2	yes	—
132	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
133	C₂×Q₈	—	106	4	no	—
134	C₂×Q₈	—	106	2	no	—
135	C₂²×C₄	—	53	2	no	—
136	C₂²×C₄	—	53	2	no	—
137	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	106	2	no	—
142	C₂×C₈	—	106	4	no	—
143	C₂×C₈	#11719921	1	2	yes	—
144	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
145	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
146	C₂×C₈	#11719945	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₄⋊C₄	—	106	4	no	—
149	C₄⋊C₄	—	106	2	no	—
150	C₄²	—	106	2	no	—
151	C₄²	—	106	2	no	—
152	C₄²	#11719912	1	2	yes	—
153	C₄²	#11719921	1	2	yes	—
154	C₁₆	—	2	4	no	—
155	C₁₆	—	212	2	no	—
156	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
157	Q₈	—	106	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	53	4	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	—	106	2	no	—
163	C₂×C₄	—	106	4	no	—
164	C₂×C₄	—	106	2	no	—
165	C₂×C₄	#13133129	1	2	yes	—
166	C₂×C₄	#13133129	1	2	yes	—
167	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
169	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
170	C₂×C₄	#13133223	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	212	2	no	—
173	C₈	#13133196	1	2	yes	—
174	C₈	#13133205	1	1	yes	characteristic
175	C₈	#13133217	1	1	yes	characteristic
176	C₂²	C₂×#13133129	1	1	yes	characteristic
177	C₂²	C₈×Dic₁₀₆	1	2	yes	—
178	C₂²	M₃₂×D₅₃	1	2	yes	—
179	C₂²	#16811041	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	106	4	no	—
183	C₄	—	212	2	no	—
184	C₄	#16810529	1	1	yes	characteristic
185	C₄	#16810589	1	1	yes	characteristic
186	C₄	#16810753	1	2	yes	—
187	C₂	#20275651	1	1	yes	characteristic
188	C₂	#20276325	1	1	yes	characteristic
189	C₂	#20276330	1	1	yes	characteristic
190	C₂	#20276621	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₀₆

C₂

yes

maximal, characteristic

C₂×C₄⋊#11719908

C₂

yes

maximal, characteristic

#20272792

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₀₆

C₄

yes

characteristic

C₂²×#11719908

C₂²

yes

—

C₈×Dic₁₀₆

—

C₈×Dic₁₀₆

—

C₂×C₄⋊#88322

C₄

yes

characteristic

C₂×#13133146

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

C₄⋊#11719908

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

—

C₂²×Dic₁₀₆

C₈

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₂×C₄⋊Dic₅₃

C₈

yes

characteristic

C₂×#11719919

C₈

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

Q₈

yes

complemented

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₈

yes

—

C₂²×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

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

C₂×Q₈

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

M₃₂

yes

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

#124

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

—

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

C₈×Q₈

yes

—

C₂²×C₅₃

C₂×M₃₂

yes

—

C₂²×C₅₃

#504

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

—

C₂×#346

D₅₃

yes

characteristic

#1374

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

100

C₂×C₄×Q₈

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

C₂×D₅₃

yes

—

103

C₈×Q₈

—

106

—

104

C₈×Q₈

—

106

—

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

109

C₄⋊C₁₆

—

106

—

110

#346

—

111

C₅₃

#13264

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₅₃

yes

characteristic

115

C₂²×C₈

C₂²×D₅₃

yes

characteristic

116

C₄×Q₈

—

106

—

117

C₄×Q₈

—

106

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

121

C₂×C₄²

C₄×D₅₃

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

106

—

125

C₂×C₁₆

—

106

—

126

C₂×C₁₆

Dic₁₀₆

yes

complemented

127

C₄⋊C₈

—

106

—

128

C₄⋊C₈

—

106

—

129

C₄×C₈

—

106

—

130

C₄×C₈

—

106

—

131

C₄×C₈

D₂₁₂

yes

—

132

C₄×C₈

C₂²⋊D₅₃

yes

—

133

C₂×Q₈

—

106

—

134

C₂×Q₈

—

106

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₈×D₅₃

yes

characteristic

138

C₂²×C₄

C₈×D₅₃

yes

characteristic

139

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

106

—

142

C₂×C₈

—

106

—

143

C₂×C₈

#11719921

yes

—

144

C₂×C₈

C₂×Dic₁₀₆

yes

—

145

C₂×C₈

C₂×D₂₁₂

yes

characteristic

146

C₂×C₈

#11719945

yes

—

147

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

148

C₄⋊C₄

—

106

—

149

C₄⋊C₄

—

106

—

150

C₄²

—

106

—

151

C₄²

—

106

—

152

C₄²

#11719912

yes

—

153

C₄²

#11719921

yes

—

154

C₁₆

—

155

C₁₆

—

212

—

156

C₂³

C₂×C₈×D₅₃

yes

characteristic

157

Q₈

—

106

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

106

—

162

C₂×C₄

—

106

—

163

C₂×C₄

—

106

—

164

C₂×C₄

—

106

—

165

C₂×C₄

#13133129

yes

—

166

C₂×C₄

#13133129

yes

—

167

C₂×C₄

C₂×#11719921

yes

characteristic

168

C₂×C₄

C₄×Dic₁₀₆

yes

—

169

C₂×C₄

C₂×#11719912

yes

characteristic

170

C₂×C₄

#13133223

yes

—

171

C₈

—

172

C₈

—

212

—

173

C₈

#13133196

yes

—

174

C₈

#13133205

yes

characteristic

175

C₈

#13133217

yes

characteristic

176

C₂²

C₂×#13133129

yes

characteristic

177

C₂²

C₈×Dic₁₀₆

yes

—

178

C₂²

M₃₂×D₅₃

yes

—

179

C₂²

#16811041

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

106

—

183

C₄

—

212

—

184

C₄

#16810529

yes

characteristic

185

C₄

#16810589

yes

characteristic

186

C₄

#16810753

yes

—

187

C₂

#20275651

yes

characteristic

188

C₂

#20276325

yes

characteristic

189

C₂

#20276330

yes

characteristic

190

C₂

#20276621

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₅₃×#346	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#13264
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅₃

commutator, socle, prime radical

C₂³×C₅₃

C₂²×C₈

Fitting

C₂×C₅₃×#346

C₂

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
34. C₂×C₁₆×C₅₃	Q₈	semidirect
111. C₅₃	#13264	semidirect
126. C₂×C₁₆	Dic₁₀₆	semidirect

H ≅ G/N

type

34. C₂×C₁₆×C₅₃

Q₈

semidirect

111. C₅₃

#13264

semidirect

126. C₂×C₁₆

Dic₁₀₆

semidirect

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
C₂×C₄⋊#11719908	2
#20272792	2
C₂×C₅₃×#346	2
#13264	53

subgroup

index

C₂×C₈×Dic₁₀₆

C₂×C₄⋊#11719908

#20272792

C₂×C₅₃×#346

#13264

Subgroups — #27302615

Conjugacy classes of subgroups

Complements

Maximal subgroups