Group #27217878 — subgroups

Subgroups — C₅₃×#10366 #27217878

Order 13568.

402 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₅₃×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1517	C₂	1	2	yes	maximal
4	C₅₃×#1286	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#532	C₂²	1	1	yes	characteristic
7	C₅₃×#431	C₂²	1	2	yes	—
8	C₅₃×#423	C₂²	1	1	yes	characteristic
9	C₂×#104×C₅₃	—	2	2	no	—
10	C₅₃×#389	—	2	2	no	—
11	C₅₃×#387	—	2	4	no	—
12	C₅₃×#382	C₂²	1	2	yes	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#327	—	2	4	no	—
15	C₂×#41×C₅₃	—	2	1	no	—
16	C₂×M₁₆×C₅₃	—	2	1	no	—
17	C₄⋊Q₈×C₅₃	Q₈	1	1	yes	characteristic
18	#127×C₅₃	Q₈	1	1	yes	characteristic
19	#124×C₅₃	—	2	1	no	—
20	#121×C₅₃	—	2	2	no	—
21	C₄×D₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
23	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	D₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	#104×C₅₃	—	4	4	no	—
30	C₂²⋊C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	#95×C₅₃	—	2	4	no	—
33	#95×C₅₃	—	2	2	no	—
34	#95×C₅₃	D₄	1	4	yes	—
35	#41×C₅₃	—	4	2	no	—
36	#41×C₅₃	—	4	2	no	—
37	C₂×Q₈×C₅₃	—	2	1	no	—
38	C₂×D₄×C₅₃	—	2	1	no	—
39	C₂×D₄×C₅₃	—	2	2	no	—
40	C₂×D₄×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	4	1	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	4	1	no	—
49	C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₄⋊C₄×C₅₃	—	4	8	no	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₂²⋊C₄×C₅₃	—	4	2	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	—	4	4	no	—
57	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅₃	#41	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	2	yes	—
60	C₂³×C₅₃	—	4	1	no	—
61	C₂³×C₅₃	#120	1	1	yes	characteristic
62	Q₈×C₅₃	—	4	2	no	—
63	D₄×C₅₃	—	4	2	no	—
64	D₄×C₅₃	—	8	2	no	—
65	D₄×C₅₃	—	8	2	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	8	no	—
71	C₂×C₄×C₅₃	—	4	1	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	8	1	no	—
74	C₂×C₄×C₅₃	—	8	1	no	—
75	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#124	1	2	yes	—
78	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#135	1	2	yes	—
80	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
81	C₈×C₅₃	—	8	2	no	—
82	C₈×C₅₃	—	8	1	no	—
83	#10366	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	—	8	1	no	—
86	C₂²×C₅₃	—	8	2	no	—
87	C₂²×C₅₃	#393	1	1	yes	characteristic
88	C₂²×C₅₃	#449	1	2	yes	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	4	4	no	—
92	C₄×C₅₃	—	8	4	no	—
93	C₄×C₅₃	—	8	1	no	—
94	C₄×C₅₃	#478	1	2	yes	—
95	C₄×C₅₃	#488	1	1	yes	characteristic
96	C₄×C₅₃	#490	1	1	yes	characteristic
97	#2864	C₂×C₅₃	1	1	yes	characteristic, complemented
98	#1962	C₂×C₅₃	1	1	yes	characteristic
99	#1517	C₂×C₅₃	1	2	yes	—
100	#1286	C₂×C₅₃	1	2	yes	complemented
101	#1276	C₂×C₅₃	1	1	yes	characteristic, complemented
102	C₂×C₅₃	—	2	2	no	—
103	C₂×C₅₃	—	16	1	no	—
104	C₂×C₅₃	#2025	1	1	yes	characteristic
105	C₂×C₅₃	#2031	1	2	yes	—
106	#532	C₂²×C₅₃	1	1	yes	characteristic
107	#431	C₂²×C₅₃	1	2	yes	—
108	#423	C₂²×C₅₃	1	1	yes	characteristic
109	C₂×#104	—	2	2	no	—
110	#389	—	2	2	no	—
111	#387	—	2	4	no	—
112	#382	C₂²×C₅₃	1	2	yes	—
113	C₄³	C₂²×C₅₃	1	1	yes	characteristic
114	#327	—	2	4	no	—
115	C₅₃	#10366	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×#41	—	2	1	no	—
117	C₂×M₁₆	—	2	1	no	—
118	C₄⋊Q₈	Q₈×C₅₃	1	1	yes	characteristic
119	#127	Q₈×C₅₃	1	1	yes	characteristic
120	#124	—	2	1	no	—
121	#121	—	2	2	no	—
122	C₄×D₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	D₄×C₅₃	1	2	yes	—
127	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	#104	—	4	4	no	—
131	C₂²⋊C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	#95	—	2	4	no	—
134	#95	—	2	2	no	—
135	#95	D₄×C₅₃	1	4	yes	—
136	#41	—	4	2	no	—
137	#41	—	4	2	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×D₄	—	2	1	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	4	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
146	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
147	M₁₆	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	4	no	—
158	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
159	C₄²	#41×C₅₃	1	1	yes	characteristic
160	C₄²	#41×C₅₃	1	2	yes	—
161	C₂³	—	4	1	no	—
162	C₂³	#120×C₅₃	1	1	yes	characteristic
163	Q₈	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	8	2	no	—
166	D₄	—	8	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
177	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
178	C₂×C₄	#124×C₅₃	1	2	yes	—
179	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
180	C₂×C₄	#135×C₅₃	1	2	yes	—
181	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
182	C₈	—	8	2	no	—
183	C₈	—	8	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	1	no	—
186	C₂²	—	8	2	no	—
187	C₂²	C₅₃×#393	1	1	yes	characteristic
188	C₂²	C₅₃×#449	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	8	4	no	—
193	C₄	—	8	1	no	—
194	C₄	C₅₃×#478	1	2	yes	—
195	C₄	C₅₃×#488	1	1	yes	characteristic
196	C₄	C₅₃×#490	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	16	1	no	—
199	C₂	C₅₃×#2025	1	1	yes	characteristic
200	C₂	C₅₃×#2031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2864

C₂

yes

maximal, characteristic, complemented

C₅₃×#1962

C₂

yes

maximal, characteristic

C₅₃×#1517

C₂

yes

maximal

C₅₃×#1286

C₂

yes

maximal, complemented

C₅₃×#1276

C₂

yes

maximal, characteristic, complemented

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#431

C₂²

yes

—

C₅₃×#423

C₂²

yes

characteristic

C₂×#104×C₅₃

—

C₅₃×#389

—

C₅₃×#387

—

C₅₃×#382

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#327

—

C₂×#41×C₅₃

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

characteristic

#127×C₅₃

Q₈

yes

characteristic

#124×C₅₃

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#104×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

#95×C₅₃

D₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#10366

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#393

yes

characteristic

C₂²×C₅₃

#449

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#478

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic, complemented

#1962

C₂×C₅₃

yes

characteristic

#1517

C₂×C₅₃

yes

—

100

#1286

C₂×C₅₃

yes

complemented

101

#1276

C₂×C₅₃

yes

characteristic, complemented

102

C₂×C₅₃

—

103

C₂×C₅₃

—

104

C₂×C₅₃

#2025

yes

characteristic

105

C₂×C₅₃

#2031

yes

—

106

#532

C₂²×C₅₃

yes

characteristic

107

#431

C₂²×C₅₃

yes

—

108

#423

C₂²×C₅₃

yes

characteristic

109

C₂×#104

—

110

#389

—

111

#387

—

112

#382

C₂²×C₅₃

yes

—

113

C₄³

C₂²×C₅₃

yes

characteristic

114

#327

—

115

C₅₃

#10366

yes

characteristic, direct product factor, Sylow

116

C₂×#41

—

117

C₂×M₁₆

—

118

C₄⋊Q₈

Q₈×C₅₃

yes

characteristic

119

#127

Q₈×C₅₃

yes

characteristic

120

#124

—

121

#121

—

122

C₄×D₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

—

126

C₂×C₄²

D₄×C₅₃

yes

—

127

C₂×C₄²

C₂³×C₅₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

#104

—

131

C₂²⋊C₈

—

132

C₄×C₈

—

133

#95

—

134

#95

—

135

#95

D₄×C₅₃

yes

—

136

#41

—

137

#41

—

138

C₂×Q₈

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×D₄×C₅₃

yes

—

146

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₂²⋊C₄

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂×Q₈×C₅₃

yes

characteristic

159

C₄²

#41×C₅₃

yes

characteristic

160

C₄²

#41×C₅₃

yes

—

161

C₂³

—

162

C₂³

#120×C₅₃

yes

characteristic

163

Q₈

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

177

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

characteristic

178

C₂×C₄

#124×C₅₃

yes

—

179

C₂×C₄

#124×C₅₃

yes

characteristic

180

C₂×C₄

#135×C₅₃

yes

—

181

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

182

C₈

—

183

C₈

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₅₃×#393

yes

characteristic

188

C₂²

C₅₃×#449

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

C₅₃×#478

yes

—

195

C₄

C₅₃×#488

yes

characteristic

196

C₄

C₅₃×#490

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

C₅₃×#2025

yes

characteristic

200

C₂

C₅₃×#2031

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#393

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₅₃×#532

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₅₃×#2864	C₂	semidirect
4. C₅₃×#1286	C₂	semidirect
5. C₅₃×#1276	C₂	semidirect
83. #10366	C₅₃	direct (N × H)
97. #2864	C₂×C₅₃	semidirect
100. #1286	C₂×C₅₃	semidirect
101. #1276	C₂×C₅₃	semidirect
115. C₅₃	#10366	direct (N × H)

H ≅ G/N

type

1. C₅₃×#2864

C₂

semidirect

4. C₅₃×#1286

C₂

semidirect

5. C₅₃×#1276

C₂

semidirect

83. #10366

C₅₃

direct (N × H)

97. #2864

C₂×C₅₃

semidirect

100. #1286

C₂×C₅₃

semidirect

101. #1276

C₂×C₅₃

semidirect

115. C₅₃

#10366

direct (N × H)

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

subgroup	index
C₅₃×#2864	2
C₅₃×#1962	2
C₅₃×#1517	2
C₅₃×#1286	2
C₅₃×#1276	2
#10366	53

subgroup

index

C₅₃×#2864

C₅₃×#1962

C₅₃×#1517

C₅₃×#1286

C₅₃×#1276

#10366

Subgroups — C₅₃×#10366 #27217878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10366 #27217878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10366 #27217878