Group #27302328 — subgroups

Subgroups — #27302328

Order 13568.

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

normal subgroups only

This group has 486 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	C₂×#16810428	C₂	1	4	yes	maximal
3	#20274879	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₂	—	2	1	no	—
6	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
7	C₂×#13133217	—	2	1	no	—
8	#16811003	—	2	2	no	—
9	#16811003	C₄	1	4	yes	complemented
10	C₂×#13133151	C₂²	1	2	yes	—
11	C₂×#13133149	—	2	1	no	—
12	#16810489	—	2	1	no	—
13	#16810428	—	2	8	no	—
14	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#346	—	2	2	no	—
17	C₂²×Dic₁₀₆	—	2	2	no	—
18	C₂×Dic₂₁₂	—	2	4	no	—
19	C₂×Dic₂₁₂	—	2	2	no	—
20	C₂×Dic₂₁₂	—	2	4	no	—
21	C₂×Dic₂₁₂	—	2	4	no	—
22	C₂×Dic₂₁₂	—	4	2	no	—
23	C₂×Dic₂₁₂	—	4	2	no	—
24	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
25	#13133217	—	2	4	no	—
26	#13133217	—	4	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	—	4	2	no	—
29	#13133151	—	2	4	no	—
30	#13133149	—	4	2	no	—
31	#13133124	—	4	2	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂²×C₄×C₅₃	D₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	4	2	no	—
43	C₂²×Dic₅₃	—	4	2	no	—
44	C₂×Dic₁₀₆	—	2	4	no	—
45	C₂×Dic₁₀₆	—	4	2	no	—
46	C₂×Dic₁₀₆	—	4	2	no	—
47	C₂×Dic₁₀₆	—	4	2	no	—
48	C₂×Dic₁₀₆	—	4	2	no	—
49	C₂×Dic₁₀₆	—	4	2	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	D₈	1	2	yes	—
52	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	2	yes	—
57	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₅₃	SD₁₆	1	2	yes	—
59	C₄⋊Dic₅₃	—	4	8	no	—
60	C₄⋊Dic₅₃	—	4	2	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	Dic₂₁₂	—	2	8	no	—
63	Dic₂₁₂	—	4	8	no	—
64	Dic₂₁₂	—	4	4	no	—
65	C₁₆×C₅₃	—	2	4	no	—
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	#102	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×Dic₅₃	—	4	4	no	—
76	C₂×Dic₅₃	—	8	2	no	—
77	C₂×Dic₅₃	—	8	2	no	—
78	C₂×Dic₅₃	—	8	2	no	—
79	C₂×Dic₅₃	—	8	4	no	—
80	Dic₁₀₆	—	4	4	no	—
81	Dic₁₀₆	—	4	4	no	—
82	Dic₁₀₆	—	8	2	no	—
83	Dic₁₀₆	—	8	2	no	—
84	C₈×C₅₃	#102	1	4	yes	—
85	C₈×C₅₃	C₄×D₄	1	2	yes	—
86	C₈×C₅₃	#121	1	2	yes	—
87	#13207	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
89	C₂²×C₅₃	#486	1	1	yes	characteristic
90	C₂²×C₅₃	#493	1	1	yes	characteristic
91	C₂²×C₅₃	#510	1	4	yes	—
92	C₄×C₅₃	—	2	4	no	—
93	C₄×C₅₃	#390	1	1	yes	characteristic
94	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
95	C₄×C₅₃	#486	1	1	yes	characteristic
96	C₄×C₅₃	#492	1	1	yes	characteristic
97	Dic₅₃	—	8	4	no	—
98	Dic₅₃	—	16	2	no	—
99	C₂×#494	—	53	1	no	—
100	C₂×#358	—	53	4	no	—
101	C₂×#346	D₅₃	1	1	yes	characteristic
102	#1846	—	53	1	no	—
103	C₂×C₅₃	#1957	1	1	yes	characteristic
104	C₂×C₅₃	#2141	1	2	yes	—
105	C₂×C₅₃	#2245	1	2	yes	—
106	C₂×C₅₃	#2251	1	2	yes	—
107	C₂²×Q₁₆	—	53	2	no	—
108	C₂²×Q₁₆	—	106	1	no	—
109	C₂×C₄⋊Q₈	—	106	1	no	—
110	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
111	#494	—	53	4	no	—
112	#494	—	106	2	no	—
113	C₂×#107	—	53	2	no	—
114	C₂×#106	—	106	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#382	—	106	1	no	—
117	#358	—	106	8	no	—
118	#346	—	2	2	no	—
119	C₅₃	#13207	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	106	2	no	—
121	C₂×Q₁₆	—	53	4	no	—
122	C₂×Q₁₆	—	106	4	no	—
123	C₂×Q₁₆	—	106	4	no	—
124	C₂×Q₁₆	—	106	2	no	—
125	C₂×Q₁₆	—	106	4	no	—
126	C₂×Q₁₆	—	212	2	no	—
127	C₂×Q₁₆	—	212	2	no	—
128	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
130	C₄⋊Q₈	—	106	4	no	—
131	C₄⋊Q₈	—	212	2	no	—
132	C₂×C₄⋊C₄	—	106	2	no	—
133	C₂×C₄⋊C₄	—	212	2	no	—
134	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	D₂₁₂	1	4	yes	—
138	#107	—	106	4	no	—
139	#106	—	212	2	no	—
140	C₄×C₈	C₄×D₅₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
142	#95	—	212	2	no	—
143	C₂×Q₈	—	106	4	no	—
144	C₂×Q₈	—	212	2	no	—
145	C₂×Q₈	—	212	2	no	—
146	C₂×Q₈	—	212	2	no	—
147	C₂×Q₈	—	212	2	no	—
148	C₂×Q₈	—	212	2	no	—
149	C₂²×C₄	—	212	2	no	—
150	C₂²×C₄	—	212	2	no	—
151	C₂²×C₄	D₄₂₄	1	2	yes	—
152	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
153	Q₁₆	—	106	8	no	—
154	Q₁₆	—	212	8	no	—
155	Q₁₆	—	212	4	no	—
156	C₂×C₈	#11719913	1	2	yes	—
157	C₂×C₈	D₄₂₄	1	2	yes	—
158	C₂×C₈	#11719921	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
160	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
161	C₂×C₈	#11719945	1	2	yes	—
162	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
163	C₄⋊C₄	—	212	2	no	—
164	C₄⋊C₄	—	212	2	no	—
165	C₄⋊C₄	—	212	8	no	—
166	C₄²	#11719913	1	2	yes	—
167	C₄²	#11719921	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
170	Q₈	—	424	2	no	—
171	Q₈	—	424	2	no	—
172	Q₈	—	212	4	no	—
173	Q₈	—	212	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	424	4	no	—
176	C₂×C₄	—	424	2	no	—
177	C₂×C₄	—	424	2	no	—
178	C₂×C₄	—	424	2	no	—
179	C₂×C₄	—	212	4	no	—
180	C₂×C₄	#13133139	1	4	yes	—
181	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
182	C₂×C₄	#13133205	1	1	yes	characteristic
183	C₂×C₄	#13133214	1	1	yes	characteristic
184	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
185	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
186	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
187	C₈	#13133139	1	4	yes	—
188	C₈	C₄×D₂₁₂	1	2	yes	—
189	C₈	#13133191	1	2	yes	—
190	C₂²	C₂×#13133139	1	1	yes	characteristic
191	C₂²	#16810970	1	1	yes	characteristic
192	C₂²	#16811000	1	1	yes	characteristic
193	C₂²	#16811066	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	848	2	no	—
196	C₄	—	424	4	no	—
197	C₄	#16810525	1	1	yes	characteristic
198	C₄	C₂×#13133139	1	1	yes	characteristic
199	C₄	#16810970	1	1	yes	characteristic
200	C₄	#16810996	1	1	yes	characteristic
201	C₂	#20275521	1	1	yes	characteristic
202	C₂	#20276436	1	2	yes	—
203	C₂	#20276961	1	2	yes	—
204	C₂	#20276986	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811003

C₂

yes

maximal, characteristic

C₂×#16810428

C₂

yes

maximal

#20274879

C₂

yes

maximal, characteristic

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

—

C₂²×Dic₂₁₂

C₂²

yes

—

C₂×#13133217

—

#16811003

—

#16811003

C₄

yes

complemented

C₂×#13133151

C₂²

yes

—

C₂×#13133149

—

#16810489

—

#16810428

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

—

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂×C₄

yes

—

#13133217

—

#13133217

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133151

—

#13133149

—

#13133124

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

C₄×D₄

yes

—

C₈×C₅₃

#121

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

C₂×#494

—

100

C₂×#358

—

101

C₂×#346

D₅₃

yes

characteristic

102

#1846

—

103

C₂×C₅₃

#1957

yes

characteristic

104

C₂×C₅₃

#2141

yes

—

105

C₂×C₅₃

#2245

yes

—

106

C₂×C₅₃

#2251

yes

—

107

C₂²×Q₁₆

—

108

C₂²×Q₁₆

—

106

—

109

C₂×C₄⋊Q₈

—

106

—

110

C₂²×C₁₆

C₂×D₅₃

yes

—

111

#494

—

112

#494

—

106

—

113

C₂×#107

—

114

C₂×#106

—

106

—

115

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

116

#382

—

106

—

117

#358

—

106

—

118

#346

—

119

C₅₃

#13207

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

106

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

106

—

123

C₂×Q₁₆

—

106

—

124

C₂×Q₁₆

—

106

—

125

C₂×Q₁₆

—

106

—

126

C₂×Q₁₆

—

212

—

127

C₂×Q₁₆

—

212

—

128

C₂²×C₈

D₂₁₂

yes

characteristic

129

C₂²×C₈

C₂²×D₅₃

yes

characteristic

130

C₄⋊Q₈

—

106

—

131

C₄⋊Q₈

—

212

—

132

C₂×C₄⋊C₄

—

106

—

133

C₂×C₄⋊C₄

—

212

—

134

C₂×C₄²

D₂₁₂

yes

characteristic

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

D₂₁₂

yes

—

138

#107

—

106

—

139

#106

—

212

—

140

C₄×C₈

C₄×D₅₃

yes

—

141

C₄×C₈

C₂²⋊D₅₃

yes

—

142

#95

—

212

—

143

C₂×Q₈

—

106

—

144

C₂×Q₈

—

212

—

145

C₂×Q₈

—

212

—

146

C₂×Q₈

—

212

—

147

C₂×Q₈

—

212

—

148

C₂×Q₈

—

212

—

149

C₂²×C₄

—

212

—

150

C₂²×C₄

—

212

—

151

C₂²×C₄

D₄₂₄

yes

—

152

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

153

Q₁₆

—

106

—

154

Q₁₆

—

212

—

155

Q₁₆

—

212

—

156

C₂×C₈

#11719913

yes

—

157

C₂×C₈

D₄₂₄

yes

—

158

C₂×C₈

#11719921

yes

—

159

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

160

C₂×C₈

C₂×D₂₁₂

yes

—

161

C₂×C₈

#11719945

yes

—

162

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

163

C₄⋊C₄

—

212

—

164

C₄⋊C₄

—

212

—

165

C₄⋊C₄

—

212

—

166

C₄²

#11719913

yes

—

167

C₄²

#11719921

yes

—

168

C₁₆

—

169

C₂³

C₂×D₄₂₄

yes

characteristic

170

Q₈

—

424

—

171

Q₈

—

424

—

172

Q₈

—

212

—

173

Q₈

—

212

—

174

C₂×C₄

—

175

C₂×C₄

—

424

—

176

C₂×C₄

—

424

—

177

C₂×C₄

—

424

—

178

C₂×C₄

—

424

—

179

C₂×C₄

—

212

—

180

C₂×C₄

#13133139

yes

—

181

C₂×C₄

C₂×#11719921

yes

characteristic

182

C₂×C₄

#13133205

yes

characteristic

183

C₂×C₄

#13133214

yes

characteristic

184

C₂×C₄

C₂×D₄₂₄

yes

characteristic

185

C₂×C₄

C₂×#11719913

yes

characteristic

186

C₂×C₄

C₂×#11719913

yes

characteristic

187

C₈

#13133139

yes

—

188

C₈

C₄×D₂₁₂

yes

—

189

C₈

#13133191

yes

—

190

C₂²

C₂×#13133139

yes

characteristic

191

C₂²

#16810970

yes

characteristic

192

C₂²

#16811000

yes

characteristic

193

C₂²

#16811066

yes

—

194

C₄

—

195

C₄

—

848

—

196

C₄

—

424

—

197

C₄

#16810525

yes

characteristic

198

C₄

C₂×#13133139

yes

characteristic

199

C₄

#16810970

yes

characteristic

200

C₄

#16810996

yes

characteristic

201

C₂

#20275521

yes

characteristic

202

C₂

#20276436

yes

—

203

C₂

#20276961

yes

—

204

C₂

#20276986

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#13207
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#346

C₂

socle, prime radical

C₂³×C₅₃

C₂×D₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
9. #16811003	C₄	semidirect
119. C₅₃	#13207	semidirect

H ≅ G/N

type

9. #16811003

C₄

semidirect

119. C₅₃

#13207

semidirect

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

subgroup	index
C₂×#16811003	2
C₂×#16810428	2
#20274879	2
C₂×C₅₃×#346	2
#13207	53

subgroup

index

C₂×C₅₃×#346

Subgroups — #27302328

Conjugacy classes of subgroups

Complements

Maximal subgroups