Group #27218188 — subgroups

Subgroups — C₅₃×#10676 #27218188

Order 13568.

410 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 162 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×#1990	C₂	1	2	yes	maximal
2	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1834	C₂	1	2	yes	maximal
5	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#107×C₅₃	—	2	2	no	—
8	C₂×#103×C₅₃	—	2	2	no	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#384	C₂²	1	1	yes	characteristic
12	C₅₃×#384	C₂²	1	2	yes	—
13	C₅₃×#340	—	2	4	no	—
14	C₅₃×#340	—	2	4	no	—
15	C₂²×Q₈×C₅₃	—	2	1	no	—
16	C₂²×C₈×C₅₃	D₄	1	4	yes	—
17	C₂²×C₈×C₅₃	D₄	1	2	yes	—
18	C₄⋊Q₈×C₅₃	—	2	1	no	—
19	C₄⋊Q₈×C₅₃	—	2	2	no	—
20	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	#107×C₅₃	—	4	4	no	—
29	#103×C₅₃	—	4	4	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₂×Q₈×C₅₃	—	2	2	no	—
34	C₂×Q₈×C₅₃	—	4	1	no	—
35	C₂×Q₈×C₅₃	—	4	2	no	—
36	C₂×Q₈×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	—	4	1	no	—
38	C₂²×C₄×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	—	4	1	no	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	D₈	1	4	yes	—
48	C₄⋊C₄×C₅₃	—	2	4	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₄⋊C₄×C₅₃	—	4	2	no	—
51	C₄⋊C₄×C₅₃	—	4	4	no	—
52	C₄⋊C₄×C₅₃	—	4	4	no	—
53	C₄⋊C₄×C₅₃	—	4	4	no	—
54	C₄⋊C₄×C₅₃	—	4	4	no	—
55	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₅₃	#41	1	1	yes	characteristic
57	C₄²×C₅₃	#41	1	2	yes	—
58	C₂³×C₅₃	#127	1	1	yes	characteristic
59	Q₈×C₅₃	—	4	4	no	—
60	Q₈×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	—	4	2	no	—
62	C₂×C₄×C₅₃	—	8	1	no	—
63	C₂×C₄×C₅₃	—	8	1	no	—
64	C₂×C₄×C₅₃	—	8	2	no	—
65	C₂×C₄×C₅₃	—	8	1	no	—
66	C₂×C₄×C₅₃	—	8	2	no	—
67	C₂×C₄×C₅₃	—	8	2	no	—
68	C₂×C₄×C₅₃	—	8	2	no	—
69	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#123	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₅₃	#135	1	4	yes	—
75	C₂×C₄×C₅₃	#137	1	2	yes	—
76	C₂×C₄×C₅₃	#137	1	4	yes	—
77	C₈×C₅₃	—	2	4	no	—
78	C₈×C₅₃	—	4	4	no	—
79	#10676	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₅₃	#399	1	1	yes	characteristic
81	C₂²×C₅₃	#493	1	1	yes	characteristic
82	C₂²×C₅₃	#495	1	2	yes	—
83	C₂²×C₅₃	#497	1	1	yes	characteristic
84	C₂²×C₅₃	#497	1	2	yes	—
85	C₄×C₅₃	—	8	2	no	—
86	C₄×C₅₃	—	16	2	no	—
87	C₄×C₅₃	—	16	1	no	—
88	C₄×C₅₃	#474	1	2	yes	—
89	C₄×C₅₃	#479	1	4	yes	—
90	C₄×C₅₃	#480	1	2	yes	—
91	C₄×C₅₃	#485	1	4	yes	—
92	#1990	C₂×C₅₃	1	2	yes	—
93	#1956	C₂×C₅₃	1	1	yes	characteristic
94	#1845	C₂×C₅₃	1	1	yes	characteristic
95	#1834	C₂×C₅₃	1	2	yes	—
96	#1273	C₂×C₅₃	1	1	yes	characteristic
97	C₂×C₅₃	#1709	1	1	yes	characteristic
98	C₂×C₅₃	#1718	1	2	yes	—
99	C₂×C₅₃	#1722	1	1	yes	characteristic
100	C₂×C₅₃	#2093	1	1	yes	characteristic
101	C₂×C₅₃	#2097	1	2	yes	—
102	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
103	C₂×#107	—	2	2	no	—
104	C₂×#103	—	2	2	no	—
105	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
107	#384	C₂²×C₅₃	1	1	yes	characteristic
108	#384	C₂²×C₅₃	1	2	yes	—
109	#340	—	2	4	no	—
110	#340	—	2	4	no	—
111	C₅₃	#10676	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×C₈	D₄×C₅₃	1	4	yes	—
114	C₂²×C₈	D₄×C₅₃	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	Q₈×C₅₃	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	1	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
125	#107	—	4	4	no	—
126	#103	—	4	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	D₈×C₅₃	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
153	C₄²	#41×C₅₃	1	1	yes	characteristic
154	C₄²	#41×C₅₃	1	2	yes	—
155	C₂³	#127×C₅₃	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
168	C₂×C₄	#123×C₅₃	1	2	yes	—
169	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
170	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
171	C₂×C₄	#135×C₅₃	1	4	yes	—
172	C₂×C₄	#137×C₅₃	1	4	yes	—
173	C₂×C₄	#137×C₅₃	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	4	4	no	—
176	C₂²	C₅₃×#399	1	1	yes	characteristic
177	C₂²	C₅₃×#493	1	1	yes	characteristic
178	C₂²	C₅₃×#495	1	2	yes	—
179	C₂²	C₅₃×#497	1	1	yes	characteristic
180	C₂²	C₅₃×#497	1	2	yes	—
181	C₄	—	8	2	no	—
182	C₄	—	16	2	no	—
183	C₄	—	16	1	no	—
184	C₄	C₅₃×#474	1	2	yes	—
185	C₄	C₅₃×#479	1	4	yes	—
186	C₄	C₅₃×#480	1	2	yes	—
187	C₄	C₅₃×#485	1	4	yes	—
188	C₂	C₅₃×#1709	1	1	yes	characteristic
189	C₂	C₅₃×#1718	1	2	yes	—
190	C₂	C₅₃×#1722	1	1	yes	characteristic
191	C₂	C₅₃×#2093	1	1	yes	characteristic
192	C₂	C₅₃×#2097	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1990

C₂

yes

maximal

C₅₃×#1956

C₂

yes

maximal, characteristic

C₅₃×#1845

C₂

yes

maximal, characteristic

C₅₃×#1834

C₂

yes

maximal

C₅₃×#1273

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×#107×C₅₃

—

C₂×#103×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

—

C₅₃×#340

—

C₅₃×#340

—

C₂²×Q₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

#103×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₂³×C₅₃

#127

yes

characteristic

Q₈×C₅₃

—

Q₈×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

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#10676

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#497

yes

characteristic

C₂²×C₅₃

#497

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#474

yes

—

C₄×C₅₃

#479

yes

—

C₄×C₅₃

#480

yes

—

C₄×C₅₃

#485

yes

—

#1990

C₂×C₅₃

yes

—

#1956

C₂×C₅₃

yes

characteristic

#1845

C₂×C₅₃

yes

characteristic

#1834

C₂×C₅₃

yes

—

#1273

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1722

yes

characteristic

100

C₂×C₅₃

#2093

yes

characteristic

101

C₂×C₅₃

#2097

yes

—

102

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

characteristic

103

C₂×#107

—

104

C₂×#103

—

105

C₂×C₄×C₈

C₂²×C₅₃

yes

—

106

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

107

#384

C₂²×C₅₃

yes

characteristic

108

#384

C₂²×C₅₃

yes

—

109

#340

—

110

#340

—

111

C₅₃

#10676

yes

characteristic, direct product factor, Sylow

112

C₂²×Q₈

—

113

C₂²×C₈

D₄×C₅₃

yes

—

114

C₂²×C₈

D₄×C₅₃

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

Q₈×C₅₃

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂³×C₅₃

yes

characteristic

125

#107

—

126

#103

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

138

C₂²×C₄

C₂×D₄×C₅₃

yes

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

D₈×C₅₃

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×Q₈×C₅₃

yes

characteristic

153

C₄²

#41×C₅₃

yes

characteristic

154

C₄²

#41×C₅₃

yes

—

155

C₂³

#127×C₅₃

yes

characteristic

156

Q₈

—

157

Q₈

—

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

C₂²⋊Q₈×C₅₃

yes

characteristic

167

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

168

C₂×C₄

#123×C₅₃

yes

—

169

C₂×C₄

#123×C₅₃

yes

characteristic

170

C₂×C₄

C₂×D₈×C₅₃

yes

—

171

C₂×C₄

#135×C₅₃

yes

—

172

C₂×C₄

#137×C₅₃

yes

—

173

C₂×C₄

#137×C₅₃

yes

—

174

C₈

—

175

C₈

—

176

C₂²

C₅₃×#399

yes

characteristic

177

C₂²

C₅₃×#493

yes

characteristic

178

C₂²

C₅₃×#495

yes

—

179

C₂²

C₅₃×#497

yes

characteristic

180

C₂²

C₅₃×#497

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

C₅₃×#474

yes

—

185

C₄

C₅₃×#479

yes

—

186

C₄

C₅₃×#480

yes

—

187

C₄

C₅₃×#485

yes

—

188

C₂

C₅₃×#1709

yes

characteristic

189

C₂

C₅₃×#1718

yes

—

190

C₂

C₅₃×#1722

yes

characteristic

191

C₂

C₅₃×#2093

yes

characteristic

192

C₂

C₅₃×#2097

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅₃

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
79. #10676	C₅₃	direct (N × H)
111. C₅₃	#10676	direct (N × H)

H ≅ G/N

type

79. #10676

C₅₃

direct (N × H)

111. C₅₃

#10676

direct (N × H)

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

subgroup	index
C₅₃×#1990	2
C₅₃×#1956	2
C₅₃×#1845	2
C₅₃×#1834	2
C₅₃×#1273	2
#10676	53

subgroup

index

C₅₃×#1990

C₅₃×#1956

C₅₃×#1845

C₅₃×#1834

C₅₃×#1273

#10676

Subgroups — C₅₃×#10676 #27218188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10676 #27218188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10676 #27218188