Group #27216580 — subgroups

Subgroups — C₅₃×#9068 #27216580

Order 13568.

404 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 126 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×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2986	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1526	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1485	C₂	1	2	yes	maximal
5	C₅₃×#1278	C₂	1	2	yes	maximal
6	C₅₃×#533	C₂²	1	1	yes	characteristic
7	C₅₃×#533	C₂²	1	1	yes	characteristic
8	C₅₃×#500	C₄	1	2	yes	—
9	C₅₃×#499	—	2	2	no	—
10	C₅₃×#498	C₄	1	2	yes	—
11	C₅₃×#446	—	2	2	no	—
12	C₈×Q₈×C₅₃	—	2	2	no	—
13	C₅₃×#433	—	2	2	no	—
14	C₅₃×#431	C₂²	1	2	yes	—
15	C₅₃×#427	—	2	2	no	—
16	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#330	—	2	2	no	—
19	C₅₃×Q₈⋊C₈	—	2	2	no	—
20	C₅₃×Q₈⋊C₈	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	2	1	no	—
23	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	#125×C₅₃	—	2	1	no	—
26	#125×C₅₃	D₄	1	2	yes	—
27	C₂²⋊Q₈×C₅₃	—	2	2	no	—
28	C₂²⋊Q₈×C₅₃	—	2	2	no	—
29	C₄×Q₈×C₅₃	—	2	2	no	—
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	#117×C₅₃	—	2	1	no	—
32	#117×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#107×C₅₃	—	2	4	no	—
35	#106×C₅₃	—	2	4	no	—
36	C₄⋊C₈×C₅₃	—	2	4	no	—
37	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
38	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
39	C₂²⋊C₈×C₅₃	—	2	2	no	—
40	#97×C₅₃	—	2	2	no	—
41	#97×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	—	2	1	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	2	yes	—
52	M₁₆×C₅₃	—	2	4	no	—
53	M₁₆×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	—	2	4	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
66	C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂²⋊C₄×C₅₃	—	4	1	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	4	1	no	—
70	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	2	yes	—
73	C₂³×C₅₃	#123	1	1	yes	characteristic
74	Q₈×C₅₃	—	4	2	no	—
75	Q₈×C₅₃	—	4	2	no	—
76	Q₈×C₅₃	—	4	2	no	—
77	Q₈×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#121	1	2	yes	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	—	4	4	no	—
90	C₈×C₅₃	—	4	2	no	—
91	C₈×C₅₃	—	4	2	no	—
92	#9068	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	#386	1	1	yes	characteristic
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	4	4	no	—
99	C₄×C₅₃	—	8	2	no	—
100	C₄×C₅₃	#443	1	2	yes	—
101	C₄×C₅₃	#444	1	2	yes	—
102	C₄×C₅₃	#456	1	2	yes	—
103	#3160	C₂×C₅₃	1	1	yes	characteristic
104	#2986	C₂×C₅₃	1	1	yes	characteristic
105	#1526	C₂×C₅₃	1	1	yes	characteristic
106	#1485	C₂×C₅₃	1	2	yes	—
107	#1278	C₂×C₅₃	1	2	yes	—
108	C₂×C₅₃	—	4	1	no	—
109	C₂×C₅₃	#1876	1	2	yes	—
110	C₂×C₅₃	#1886	1	1	yes	characteristic
111	#533	C₂²×C₅₃	1	1	yes	characteristic
112	#533	C₂²×C₅₃	1	1	yes	characteristic
113	#500	C₄×C₅₃	1	2	yes	—
114	#499	—	2	2	no	—
115	#498	C₄×C₅₃	1	2	yes	—
116	#446	—	2	2	no	—
117	C₈×Q₈	—	2	2	no	—
118	#433	—	2	2	no	—
119	#431	C₂²×C₅₃	1	2	yes	—
120	#427	—	2	2	no	—
121	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
123	#330	—	2	2	no	—
124	Q₈⋊C₈	—	2	2	no	—
125	Q₈⋊C₈	—	2	2	no	—
126	C₅₃	#9068	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×M₁₆	—	2	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
130	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
131	#125	—	2	1	no	—
132	#125	D₄×C₅₃	1	2	yes	—
133	C₂²⋊Q₈	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₄×Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	#117	—	2	1	no	—
138	#117	D₄×C₅₃	1	2	yes	—
139	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
140	#107	—	2	4	no	—
141	#106	—	2	4	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
144	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
145	C₂²⋊C₈	—	2	2	no	—
146	#97	—	2	2	no	—
147	#97	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
153	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
157	C₂²×C₄	#41×C₅₃	1	2	yes	—
158	M₁₆	—	2	4	no	—
159	M₁₆	—	4	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
177	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
178	C₄²	C₂×D₄×C₅₃	1	2	yes	—
179	C₂³	#123×C₅₃	1	1	yes	characteristic
180	Q₈	—	4	2	no	—
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	4	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
190	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
191	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
192	C₂×C₄	#121×C₅₃	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	4	4	no	—
196	C₈	—	4	2	no	—
197	C₈	—	4	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₅₃×#386	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	4	no	—
204	C₄	—	8	2	no	—
205	C₄	C₅₃×#443	1	2	yes	—
206	C₄	C₅₃×#444	1	2	yes	—
207	C₄	C₅₃×#456	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	C₅₃×#1876	1	2	yes	—
210	C₂	C₅₃×#1886	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3160

C₂

yes

maximal, characteristic

C₅₃×#2986

C₂

yes

maximal, characteristic

C₅₃×#1526

C₂

yes

maximal, characteristic

C₅₃×#1485

C₂

yes

maximal

C₅₃×#1278

C₂

yes

maximal

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#500

C₄

yes

—

C₅₃×#499

—

C₅₃×#498

C₄

yes

—

C₅₃×#446

—

C₈×Q₈×C₅₃

—

C₅₃×#433

—

C₅₃×#431

C₂²

yes

—

C₅₃×#427

—

C₄×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#330

—

C₅₃×Q₈⋊C₈

—

C₅₃×Q₈⋊C₈

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

#125×C₅₃

—

#125×C₅₃

D₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

#117×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

#106×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

C₄⋊C₈×C₅₃

D₄

yes

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#9068

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#386

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

100

C₄×C₅₃

#443

yes

—

101

C₄×C₅₃

#444

yes

—

102

C₄×C₅₃

#456

yes

—

103

#3160

C₂×C₅₃

yes

characteristic

104

#2986

C₂×C₅₃

yes

characteristic

105

#1526

C₂×C₅₃

yes

characteristic

106

#1485

C₂×C₅₃

yes

—

107

#1278

C₂×C₅₃

yes

—

108

C₂×C₅₃

—

109

C₂×C₅₃

#1876

yes

—

110

C₂×C₅₃

#1886

yes

characteristic

111

#533

C₂²×C₅₃

yes

characteristic

112

#533

C₂²×C₅₃

yes

characteristic

113

#500

C₄×C₅₃

yes

—

114

#499

—

115

#498

C₄×C₅₃

yes

—

116

#446

—

117

C₈×Q₈

—

118

#433

—

119

#431

C₂²×C₅₃

yes

—

120

#427

—

121

C₄×M₁₆

C₂²×C₅₃

yes

—

122

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

123

#330

—

124

Q₈⋊C₈

—

125

Q₈⋊C₈

—

126

C₅₃

#9068

yes

characteristic, direct product factor, Sylow

127

C₂×M₁₆

—

128

C₂²×C₈

—

129

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

130

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

131

#125

—

132

#125

D₄×C₅₃

yes

—

133

C₂²⋊Q₈

—

134

C₂²⋊Q₈

—

135

C₄×Q₈

—

136

C₄×Q₈

—

137

#117

—

138

#117

D₄×C₅₃

yes

—

139

C₂×C₄²

C₂³×C₅₃

yes

characteristic

140

#107

—

141

#106

—

142

C₄⋊C₈

—

143

C₄⋊C₈

D₄×C₅₃

yes

—

144

C₄⋊C₈

D₄×C₅₃

yes

—

145

C₂²⋊C₈

—

146

#97

—

147

#97

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

153

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

157

C₂²×C₄

#41×C₅₃

yes

—

158

M₁₆

—

159

M₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

C₂²⋊C₄×C₅₃

yes

—

172

C₂²⋊C₄

—

173

C₂²⋊C₄

—

174

C₄²

—

175

C₄²

—

176

C₄²

C₂²×C₄×C₅₃

yes

characteristic

177

C₄²

C₂×D₄×C₅₃

yes

characteristic

178

C₄²

C₂×D₄×C₅₃

yes

—

179

C₂³

#123×C₅₃

yes

characteristic

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

190

C₂×C₄

C₄×D₄×C₅₃

yes

—

191

C₂×C₄

#120×C₅₃

yes

characteristic

192

C₂×C₄

#121×C₅₃

yes

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

198

C₂²

—

199

C₂²

—

200

C₂²

C₅₃×#386

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

C₅₃×#443

yes

—

206

C₄

C₅₃×#444

yes

—

207

C₄

C₅₃×#456

yes

—

208

C₂

—

209

C₂

C₅₃×#1876

yes

—

210

C₂

C₅₃×#1886

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#120
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#120

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂²×C₅₃

C₂³×C₅₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
92. #9068	C₅₃	direct (N × H)
126. C₅₃	#9068	direct (N × H)

H ≅ G/N

type

92. #9068

C₅₃

direct (N × H)

126. C₅₃

#9068

direct (N × H)

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

subgroup	index
C₅₃×#3160	2
C₅₃×#2986	2
C₅₃×#1526	2
C₅₃×#1485	2
C₅₃×#1278	2
#9068	53

subgroup

index

C₅₃×#3160

C₅₃×#2986

C₅₃×#1526

C₅₃×#1485

C₅₃×#1278

#9068

Subgroups — C₅₃×#9068 #27216580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9068 #27216580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9068 #27216580