Group #27283919 — subgroups

Subgroups — #27283919

Order 13568.

526 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16811003	C₂	1	1	yes	maximal, characteristic
2	#20275675	C₂	1	2	yes	maximal
3	#20275675	C₂	1	2	yes	maximal
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	—	2	2	no	—
7	C₂×#13133217	C₂²	1	2	yes	—
8	#16811003	—	2	2	no	—
9	#16811003	C₄	1	4	yes	complemented
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	#16810385	—	2	4	no	—
13	#16810385	—	2	4	no	—
14	#16810345	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#384	C₂²	1	2	yes	—
17	C₅₃×#340	—	2	2	no	—
18	C₂²×Dic₁₀₆	—	2	2	no	—
19	C₂×Dic₂₁₂	—	2	8	no	—
20	C₂×Dic₂₁₂	—	4	8	no	—
21	C₂²×#88322	D₄	1	2	yes	—
22	C₂²×#88322	D₄	1	2	yes	—
23	#13133217	—	2	2	no	—
24	#13133217	—	2	2	no	—
25	#13133217	—	2	2	no	—
26	#13133217	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₄×#88322	—	2	1	no	—
30	C₄×#88322	—	2	1	no	—
31	C₄×#88322	—	2	1	no	—
32	C₄×#88322	—	2	1	no	—
33	C₂²×C₈×C₅₃	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	—	4	2	no	—
45	C₂×Dic₁₀₆	—	2	4	no	—
46	C₂×Dic₁₀₆	—	4	8	no	—
47	C₂×Dic₁₀₆	—	4	2	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
50	C₄⋊C₄×C₅₃	—	4	4	no	—
51	C₄⋊C₄×C₅₃	—	4	4	no	—
52	C₄⋊C₄×C₅₃	—	4	4	no	—
53	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₅₃	#41	1	1	yes	characteristic
56	C₄²×C₅₃	#41	1	1	yes	characteristic
57	C₄⋊Dic₅₃	—	2	4	no	—
58	C₄⋊Dic₅₃	—	2	4	no	—
59	C₄⋊Dic₅₃	—	4	2	no	—
60	C₄⋊Dic₅₃	—	4	2	no	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	4	no	—
65	C₂×#88322	Q₁₆	1	4	yes	complemented
66	Dic₂₁₂	—	4	16	no	—
67	C₂³×C₅₃	#127	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	8	2	no	—
69	C₂×C₄×C₅₃	—	8	2	no	—
70	C₂×C₄×C₅₃	—	8	2	no	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₅₃	#135	1	2	yes	—
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₅₃	#137	1	4	yes	—
81	C₂×Dic₅₃	—	4	4	no	—
82	C₂×Dic₅₃	—	8	4	no	—
83	Dic₁₀₆	—	4	8	no	—
84	Dic₁₀₆	—	8	4	no	—
85	C₈×C₅₃	—	2	4	no	—
86	#88322	—	2	4	no	—
87	#88322	—	4	2	no	—
88	#9835	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#390	1	1	yes	characteristic
90	C₂²×C₅₃	#494	1	1	yes	characteristic
91	C₂²×C₅₃	#495	1	1	yes	characteristic
92	C₂²×C₅₃	#496	1	2	yes	—
93	C₂²×C₅₃	#497	1	2	yes	—
94	C₄×C₅₃	—	16	2	no	—
95	C₄×C₅₃	#418	1	2	yes	—
96	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
97	C₄×C₅₃	#441	1	2	yes	—
98	C₄×C₅₃	#467	1	2	yes	—
99	C₄×C₅₃	#470	1	2	yes	—
100	C₄×C₅₃	#490	1	2	yes	—
101	Dic₅₃	—	8	4	no	—
102	C₂×#494	—	53	1	no	—
103	#1990	—	53	2	no	—
104	#1990	—	53	2	no	—
105	#1834	D₅₃	1	1	yes	characteristic
106	C₂×#322	—	53	1	no	—
107	C₂×C₅₃	#1720	1	2	yes	—
108	C₂×C₅₃	#1723	1	2	yes	—
109	C₂×C₅₃	#1961	1	1	yes	characteristic
110	C₂×C₅₃	#1969	1	1	yes	characteristic
111	C₂×C₅₃	#1971	1	1	yes	characteristic
112	C₂²×Q₁₆	—	106	2	no	—
113	C₂×C₄⋊Q₈	—	53	2	no	—
114	#494	—	53	4	no	—
115	#494	—	106	2	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
119	#384	C₂×D₅₃	1	2	yes	—
120	#340	—	2	2	no	—
121	#340	—	106	4	no	—
122	#340	—	106	4	no	—
123	#322	—	106	2	no	—
124	C₅₃	#9835	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₈	—	106	2	no	—
126	C₂×Q₁₆	—	106	8	no	—
127	C₂×Q₁₆	—	212	8	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	—	53	2	no	—
130	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
131	C₄⋊Q₈	—	53	4	no	—
132	C₄⋊Q₈	—	106	2	no	—
133	C₄⋊Q₈	—	106	2	no	—
134	C₄⋊Q₈	—	106	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	106	2	no	—
139	C₂×C₄⋊C₄	—	106	2	no	—
140	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
141	C₄×C₈	—	106	1	no	—
142	C₄×C₈	—	106	1	no	—
143	C₄×C₈	—	106	1	no	—
144	C₄×C₈	—	106	1	no	—
145	C₄×C₈	C₄×D₅₃	1	2	yes	—
146	C₄×C₈	D₂₁₂	1	2	yes	—
147	C₂×Q₈	—	106	4	no	—
148	C₂×Q₈	—	212	8	no	—
149	C₂×Q₈	—	212	2	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	212	2	no	—
152	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
153	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	Q₁₆	—	212	16	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	53	4	no	—
158	C₂×C₈	—	106	2	no	—
159	C₂×C₈	—	106	2	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	4	no	—
162	C₂×C₈	#11719921	1	4	yes	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	106	4	no	—
167	C₄⋊C₄	—	106	4	no	—
168	C₄⋊C₄	—	212	2	no	—
169	C₄⋊C₄	—	212	2	no	—
170	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
171	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
172	C₄²	#11719949	1	1	yes	characteristic
173	C₄²	#11719949	1	1	yes	characteristic
174	C₂³	#13133213	1	1	yes	characteristic
175	Q₈	—	424	4	no	—
176	Q₈	—	212	8	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	424	4	no	—
181	C₂×C₄	—	212	4	no	—
182	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
183	C₂×C₄	#13133179	1	1	yes	characteristic
184	C₂×C₄	#13133179	1	1	yes	characteristic
185	C₂×C₄	#13133192	1	1	yes	characteristic
186	C₂×C₄	#13133192	1	1	yes	characteristic
187	C₂×C₄	#13133206	1	1	yes	characteristic
188	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
189	C₂×C₄	#13133236	1	2	yes	—
190	C₂×C₄	#13133242	1	4	yes	—
191	C₂×C₄	#13133246	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	106	4	no	—
194	C₈	—	212	2	no	—
195	C₂²	#16810524	1	1	yes	characteristic
196	C₂²	#16811001	1	1	yes	characteristic
197	C₂²	#16811006	1	1	yes	characteristic
198	C₂²	#16811013	1	2	yes	—
199	C₂²	#16811014	1	2	yes	—
200	C₄	—	16	2	no	—
201	C₄	—	424	4	no	—
202	C₄	#16810627	1	2	yes	—
203	C₄	#16810717	1	2	yes	—
204	C₄	#16810729	1	2	yes	—
205	C₄	#16810865	1	2	yes	—
206	C₄	#16810882	1	2	yes	—
207	C₄	#16810989	1	2	yes	—
208	C₂	#20274374	1	2	yes	—
209	C₂	#20274394	1	2	yes	—
210	C₂	#20275538	1	1	yes	characteristic
211	C₂	#20275574	1	1	yes	characteristic
212	C₂	#20275584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811003

C₂

yes

maximal, characteristic

#20275675

C₂

yes

maximal

#20275675

C₂

yes

maximal

C₂×#16810345

C₂

yes

maximal, characteristic

C₅₃×#1834

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

—

C₂×#13133217

C₂²

yes

—

#16811003

—

#16811003

C₄

yes

complemented

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

—

—

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

—

C₅₃×#340

—

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂²×#88322

D₄

yes

—

C₂²×#88322

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

—

Q₁₆

yes

complemented

Dic₂₁₂

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#497

yes

—

C₄×C₅₃

—

C₄×C₅₃

#418

yes

—

C₄×C₅₃

C₄×Q₁₆

yes

—

C₄×C₅₃

#441

yes

—

C₄×C₅₃

#467

yes

—

C₄×C₅₃

#470

yes

—

100

C₄×C₅₃

#490

yes

—

101

Dic₅₃

—

102

C₂×#494

—

103

#1990

—

104

#1990

—

105

#1834

D₅₃

yes

characteristic

106

C₂×#322

—

107

C₂×C₅₃

#1720

yes

—

108

C₂×C₅₃

#1723

yes

—

109

C₂×C₅₃

#1961

yes

characteristic

110

C₂×C₅₃

#1969

yes

characteristic

111

C₂×C₅₃

#1971

yes

characteristic

112

C₂²×Q₁₆

—

106

—

113

C₂×C₄⋊Q₈

—

114

#494

—

115

#494

—

106

—

116

C₂×C₄×C₈

—

117

C₂×C₄×C₈

—

118

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

119

#384

C₂×D₅₃

yes

—

120

#340

—

121

#340

—

106

—

122

#340

—

106

—

123

#322

—

106

—

124

C₅₃

#9835

yes

characteristic, complemented, Sylow

125

C₂²×Q₈

—

106

—

126

C₂×Q₁₆

—

106

—

127

C₂×Q₁₆

—

212

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

C₂²⋊D₅₃

yes

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

106

—

133

C₄⋊Q₈

—

106

—

134

C₄⋊Q₈

—

106

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

106

—

139

C₂×C₄⋊C₄

—

106

—

140

C₂×C₄²

C₂²×D₅₃

yes

characteristic

141

C₄×C₈

—

106

—

142

C₄×C₈

—

106

—

143

C₄×C₈

—

106

—

144

C₄×C₈

—

106

—

145

C₄×C₈

C₄×D₅₃

yes

—

146

C₄×C₈

D₂₁₂

yes

—

147

C₂×Q₈

—

106

—

148

C₂×Q₈

—

212

—

149

C₂×Q₈

—

212

—

150

C₂²×C₄

—

151

C₂²×C₄

—

212

—

152

C₂²×C₄

D₄×D₅₃

yes

characteristic

153

C₂²×C₄

D₄×D₅₃

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

155

Q₁₆

—

212

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

106

—

159

C₂×C₈

—

106

—

160

C₂×C₈

—

106

—

161

C₂×C₈

—

106

—

162

C₂×C₈

#11719921

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

106

—

167

C₄⋊C₄

—

106

—

168

C₄⋊C₄

—

212

—

169

C₄⋊C₄

—

212

—

170

C₄²

C₂×C₄×D₅₃

yes

characteristic

171

C₄²

C₂×D₂₁₂

yes

characteristic

172

C₄²

#11719949

yes

characteristic

173

C₄²

#11719949

yes

characteristic

174

C₂³

#13133213

yes

characteristic

175

Q₈

—

424

—

176

Q₈

—

212

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

424

—

181

C₂×C₄

—

212

—

182

C₂×C₄

C₂×#11719921

yes

characteristic

183

C₂×C₄

#13133179

yes

characteristic

184

C₂×C₄

#13133179

yes

characteristic

185

C₂×C₄

#13133192

yes

characteristic

186

C₂×C₄

#13133192

yes

characteristic

187

C₂×C₄

#13133206

yes

characteristic

188

C₂×C₄

Q₁₆×D₅₃

yes

—

189

C₂×C₄

#13133236

yes

—

190

C₂×C₄

#13133242

yes

—

191

C₂×C₄

#13133246

yes

—

192

C₈

—

193

C₈

—

106

—

194

C₈

—

212

—

195

C₂²

#16810524

yes

characteristic

196

C₂²

#16811001

yes

characteristic

197

C₂²

#16811006

yes

characteristic

198

C₂²

#16811013

yes

—

199

C₂²

#16811014

yes

—

200

C₄

—

201

C₄

—

424

—

202

C₄

#16810627

yes

—

203

C₄

#16810717

yes

—

204

C₄

#16810729

yes

—

205

C₄

#16810865

yes

—

206

C₄

#16810882

yes

—

207

C₄

#16810989

yes

—

208

C₂

#20274374

yes

—

209

C₂

#20274394

yes

—

210

C₂

#20275538

yes

characteristic

211

C₂

#20275574

yes

characteristic

212

C₂

#20275584

yes

characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1834	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1834	D₅₃
hypocenter	C₅₃	#9835
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1834

C₂

socle, prime radical

C₂³×C₅₃

#127

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
9. #16811003	C₄	semidirect
65. C₂×#88322	Q₁₆	semidirect
124. C₅₃	#9835	semidirect

H ≅ G/N

type

9. #16811003

C₄

semidirect

65. C₂×#88322

Q₁₆

semidirect

124. C₅₃

#9835

semidirect

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

subgroup	index
C₂×#16811003	2
#20275675	2
C₂×#16810345	2
C₅₃×#1834	2
#9835	53

subgroup

index

C₅₃×#1834

Subgroups — #27283919

Conjugacy classes of subgroups

Complements

Maximal subgroups