Group #27216231 — subgroups

Subgroups — C₅₃×#8719 #27216231

Order 13568.

554 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 150 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×C₅₃×#382	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1749	C₂	1	2	yes	maximal
3	C₅₃×#1295	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
7	C₂×#97×C₅₃	C₂²	1	4	yes	—
8	C₅₃×#382	—	2	2	no	—
9	C₅₃×#382	C₄	1	2	yes	—
10	C₅₃×#382	C₂²	1	2	yes	—
11	C₂×#95×C₅₃	—	2	2	no	—
12	C₅₃×#336	—	2	2	no	—
13	C₂³×C₄×C₅₃	—	2	2	no	—
14	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₅₃	D₄	1	2	yes	—
16	C₂²×C₈×C₅₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
19	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₅₃	—	2	4	no	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	D₄	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²⋊C₈×C₅₃	—	2	8	no	—
25	#97×C₅₃	—	2	8	no	—
26	#95×C₅₃	—	2	4	no	—
27	#95×C₅₃	—	2	4	no	—
28	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	—	2	2	no	—
30	C₂²×C₄×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	4	no	—
33	C₂²×C₄×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	—	4	4	no	—
37	C₂²×C₄×C₅₃	—	4	4	no	—
38	C₂²×C₄×C₅₃	—	4	4	no	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₅₃	—	2	8	no	—
44	C₂×C₈×C₅₃	—	4	4	no	—
45	C₂×C₈×C₅₃	—	4	4	no	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	4	no	—
50	C₄²×C₅₃	#41	1	4	yes	—
51	C₂³×C₅₃	—	2	1	no	—
52	C₂³×C₅₃	—	2	1	no	—
53	C₂³×C₅₃	—	2	2	no	—
54	C₂³×C₅₃	#101	1	2	yes	—
55	C₂³×C₅₃	#104	1	4	yes	—
56	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	8	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	4	8	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	—	4	4	no	—
71	C₂×C₄×C₅₃	—	8	8	no	—
72	C₂×C₄×C₅₃	#100	1	2	yes	—
73	C₂×C₄×C₅₃	#117	1	2	yes	—
74	C₂×C₄×C₅₃	#123	1	4	yes	—
75	C₈×C₅₃	—	4	8	no	—
76	#8719	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	—	2	4	no	—
79	C₂²×C₅₃	—	2	4	no	—
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	#330	1	4	yes	—
82	C₂²×C₅₃	#381	1	1	yes	characteristic
83	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×#104	1	2	yes	—
86	C₂²×C₅₃	#421	1	2	yes	—
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	—	2	4	no	—
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	—	8	4	no	—
92	C₂×#382	C₂×C₅₃	1	1	yes	characteristic
93	#1749	C₂×C₅₃	1	2	yes	—
94	#1295	C₂×C₅₃	1	4	yes	complemented
95	C₂×C₅₃	—	2	4	no	—
96	C₂×C₅₃	C₂×#330	1	1	yes	characteristic
97	C₂×C₅₃	#1527	1	2	yes	—
98	C₂×C₅₃	#1534	1	1	yes	characteristic
99	C₂×C₅₃	#1821	1	1	yes	characteristic
100	C₂×C₅₃	#1826	1	2	yes	—
101	C₂²×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
102	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
103	C₂×C₂²⋊C₈	—	2	2	no	—
104	C₂×#97	C₂²×C₅₃	1	4	yes	—
105	#382	—	2	2	no	—
106	#382	C₄×C₅₃	1	2	yes	—
107	#382	C₂²×C₅₃	1	2	yes	—
108	C₂×#95	—	2	2	no	—
109	#336	—	2	2	no	—
110	C₅₃	#8719	1	1	yes	characteristic, direct product factor, Sylow
111	C₂³×C₄	—	2	2	no	—
112	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
113	C₂³×C₄	D₄×C₅₃	1	2	yes	—
114	C₂²×C₈	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	4	4	no	—
117	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
118	C₂×C₄²	—	2	4	no	—
119	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
120	C₂×C₄²	D₄×C₅₃	1	2	yes	—
121	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
122	C₂²⋊C₈	—	2	8	no	—
123	#97	—	2	8	no	—
124	#95	—	2	4	no	—
125	#95	—	2	4	no	—
126	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	4	4	no	—
135	C₂²×C₄	—	4	4	no	—
136	C₂²×C₄	—	4	4	no	—
137	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
138	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
141	C₂×C₈	—	2	8	no	—
142	C₂×C₈	—	4	4	no	—
143	C₂×C₈	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	4	no	—
148	C₄²	#41×C₅₃	1	4	yes	—
149	C₂³	—	2	1	no	—
150	C₂³	—	2	1	no	—
151	C₂³	—	2	2	no	—
152	C₂³	#101×C₅₃	1	2	yes	—
153	C₂³	#104×C₅₃	1	4	yes	—
154	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	8	no	—
170	C₂×C₄	#100×C₅₃	1	2	yes	—
171	C₂×C₄	#117×C₅₃	1	2	yes	—
172	C₂×C₄	#123×C₅₃	1	4	yes	—
173	C₈	—	4	8	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₅₃×#330	1	4	yes	—
179	C₂²	C₅₃×#381	1	1	yes	characteristic
180	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
181	C₂²	C₂×#101×C₅₃	1	1	yes	characteristic
182	C₂²	C₂×#104×C₅₃	1	2	yes	—
183	C₂²	C₅₃×#421	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	4	no	—
189	C₂	—	2	4	no	—
190	C₂	C₂×C₅₃×#330	1	1	yes	characteristic
191	C₂	C₅₃×#1527	1	2	yes	—
192	C₂	C₅₃×#1534	1	1	yes	characteristic
193	C₂	C₅₃×#1821	1	1	yes	characteristic
194	C₂	C₅₃×#1826	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#382

C₂

yes

maximal, characteristic

C₅₃×#1749

C₂

yes

maximal

C₅₃×#1295

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₅₃

C₄

yes

—

C₂²×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₅₃

—

C₂×#97×C₅₃

C₂²

yes

—

C₅₃×#382

—

C₅₃×#382

C₄

yes

—

C₅₃×#382

C₂²

yes

—

C₂×#95×C₅₃

—

C₅₃×#336

—

C₂³×C₄×C₅₃

—

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂⁴×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#101

yes

—

C₂³×C₅₃

#104

yes

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#100

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#123

yes

—

C₈×C₅₃

—

#8719

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#330

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#421

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₂×#382

C₂×C₅₃

yes

characteristic

#1749

C₂×C₅₃

yes

—

#1295

C₂×C₅₃

yes

complemented

C₂×C₅₃

—

C₂×C₅₃

C₂×#330

yes

characteristic

C₂×C₅₃

#1527

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

100

C₂×C₅₃

#1826

yes

—

101

C₂²×C₄⋊C₄

C₄×C₅₃

yes

—

102

C₂²×C₄²

C₂²×C₅₃

yes

characteristic

103

C₂×C₂²⋊C₈

—

104

C₂×#97

C₂²×C₅₃

yes

—

105

#382

—

106

#382

C₄×C₅₃

yes

—

107

#382

C₂²×C₅₃

yes

—

108

C₂×#95

—

109

#336

—

110

C₅₃

#8719

yes

characteristic, direct product factor, Sylow

111

C₂³×C₄

—

112

C₂³×C₄

C₂×C₄×C₅₃

yes

characteristic

113

C₂³×C₄

D₄×C₅₃

yes

—

114

C₂²×C₈

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

C₂×C₄×C₅₃

yes

—

118

C₂×C₄²

—

119

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

120

C₂×C₄²

D₄×C₅₃

yes

—

121

C₂×C₄²

C₂³×C₅₃

yes

characteristic

122

C₂²⋊C₈

—

123

#97

—

124

#95

—

125

#95

—

126

C₂⁴

C₂²⋊C₄×C₅₃

yes

characteristic

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

138

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

139

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

140

C₂²×C₄

C₂×D₄×C₅₃

yes

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

#41×C₅₃

yes

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

#101×C₅₃

yes

—

153

C₂³

#104×C₅₃

yes

—

154

C₂³

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#100×C₅₃

yes

—

171

C₂×C₄

#117×C₅₃

yes

—

172

C₂×C₄

#123×C₅₃

yes

—

173

C₈

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

C₅₃×#330

yes

—

179

C₂²

C₅₃×#381

yes

characteristic

180

C₂²

C₂×#100×C₅₃

yes

characteristic

181

C₂²

C₂×#101×C₅₃

yes

characteristic

182

C₂²

C₂×#104×C₅₃

yes

—

183

C₂²

C₅₃×#421

yes

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₂

—

190

C₂

C₂×C₅₃×#330

yes

characteristic

191

C₂

C₅₃×#1527

yes

—

192

C₂

C₅₃×#1534

yes

characteristic

193

C₂

C₅₃×#1821

yes

characteristic

194

C₂

C₅₃×#1826

yes

—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₅₃

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₂⁴×C₅₃

C₂²⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
3. C₅₃×#1295	C₂	semidirect
76. #8719	C₅₃	direct (N × H)
94. #1295	C₂×C₅₃	semidirect
110. C₅₃	#8719	direct (N × H)

H ≅ G/N

type

3. C₅₃×#1295

C₂

semidirect

76. #8719

C₅₃

direct (N × H)

94. #1295

C₂×C₅₃

semidirect

110. C₅₃

#8719

direct (N × H)

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

subgroup	index
C₂×C₅₃×#382	2
C₅₃×#1749	2
C₅₃×#1295	2
#8719	53

subgroup

index

C₂×C₅₃×#382

C₅₃×#1749

C₅₃×#1295

#8719

Subgroups — C₅₃×#8719 #27216231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8719 #27216231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8719 #27216231