Group #27216517 — subgroups

Subgroups — C₅₃×#9005 #27216517

Order 13568.

416 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 118 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×#3135	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2980	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1543	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1497	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1302	C₂	1	2	yes	maximal, complemented
6	C₅₃×#533	C₂²	1	1	yes	characteristic
7	C₅₃×#532	C₂²	1	1	yes	characteristic
8	C₅₃×#491	—	2	1	no	—
9	C₅₃×#490	—	2	1	no	—
10	C₅₃×#489	C₄	1	4	yes	—
11	C₅₃×#435	—	2	4	no	—
12	C₅₃×#432	C₂²	1	2	yes	—
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₅₃×#416	—	2	2	no	—
15	C₅₃×#408	—	2	2	no	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#333	—	2	1	no	—
18	C₅₃×#331	—	2	1	no	—
19	C₅₃×Q₈⋊C₈	—	2	4	no	—
20	C₂×#41×C₅₃	D₄	1	2	yes	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	1	no	—
24	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
26	#127×C₅₃	D₄	1	1	yes	characteristic
27	#125×C₅₃	—	2	1	no	—
28	#124×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²⋊Q₈×C₅₃	—	2	2	no	—
30	#121×C₅₃	—	2	2	no	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₄×D₄×C₅₃	—	2	2	no	—
33	#117×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
37	#103×C₅₃	—	2	4	no	—
38	#102×C₅₃	—	2	4	no	—
39	C₂²⋊C₈×C₅₃	—	2	4	no	—
40	C₂²⋊C₈×C₅₃	—	2	2	no	—
41	C₂²⋊C₈×C₅₃	—	2	2	no	—
42	#97×C₅₃	—	2	1	no	—
43	#97×C₅₃	—	2	2	no	—
44	#97×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	—	2	2	no	—
46	#41×C₅₃	—	4	2	no	—
47	#41×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	#41	1	2	yes	—
56	M₁₆×C₅₃	—	8	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	—	8	2	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₂²⋊C₄×C₅₃	—	2	4	no	—
67	C₂²⋊C₄×C₅₃	—	4	1	no	—
68	C₄²×C₅₃	—	4	1	no	—
69	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	2	yes	—
72	C₂³×C₅₃	—	2	2	no	—
73	C₂³×C₅₃	#123	1	1	yes	characteristic
74	Q₈×C₅₃	—	4	2	no	—
75	Q₈×C₅₃	—	4	4	no	—
76	D₄×C₅₃	—	4	2	no	—
77	D₄×C₅₃	—	8	2	no	—
78	D₄×C₅₃	—	8	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#121	1	2	yes	—
89	C₈×C₅₃	—	4	2	no	—
90	C₈×C₅₃	—	4	4	no	—
91	C₈×C₅₃	—	8	2	no	—
92	#9005	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	—	4	4	no	—
96	C₂²×C₅₃	—	4	2	no	—
97	C₂²×C₅₃	#386	1	1	yes	characteristic
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	2	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	4	2	no	—
102	C₄×C₅₃	—	8	2	no	—
103	C₄×C₅₃	#455	1	2	yes	—
104	#3135	C₂×C₅₃	1	1	yes	characteristic
105	#2980	C₂×C₅₃	1	1	yes	characteristic
106	#1543	C₂×C₅₃	1	1	yes	characteristic
107	#1497	C₂×C₅₃	1	2	yes	complemented
108	#1302	C₂×C₅₃	1	2	yes	complemented
109	C₂×C₅₃	—	4	1	no	—
110	C₂×C₅₃	—	8	2	no	—
111	C₂×C₅₃	#1881	1	2	yes	—
112	C₂×C₅₃	#1885	1	1	yes	characteristic
113	#533	C₂²×C₅₃	1	1	yes	characteristic
114	#532	C₂²×C₅₃	1	1	yes	characteristic
115	#491	—	2	1	no	—
116	#490	—	2	1	no	—
117	#489	C₄×C₅₃	1	4	yes	—
118	#435	—	2	4	no	—
119	#432	C₂²×C₅₃	1	2	yes	—
120	#431	C₂²×C₅₃	1	2	yes	—
121	#416	—	2	2	no	—
122	#408	—	2	2	no	—
123	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
124	#333	—	2	1	no	—
125	#331	—	2	1	no	—
126	Q₈⋊C₈	—	2	4	no	—
127	C₅₃	#9005	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×#41	D₄×C₅₃	1	2	yes	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂²×C₈	—	2	1	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
133	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
134	#127	D₄×C₅₃	1	1	yes	characteristic
135	#125	—	2	1	no	—
136	#124	C₂×C₄×C₅₃	1	2	yes	—
137	C₂²⋊Q₈	—	2	2	no	—
138	#121	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	C₄×D₄	—	2	2	no	—
141	#117	—	2	1	no	—
142	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
145	#103	—	2	4	no	—
146	#102	—	2	4	no	—
147	C₂²⋊C₈	—	2	4	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	C₂²⋊C₈	—	2	2	no	—
150	#97	—	2	1	no	—
151	#97	—	2	2	no	—
152	#97	C₂×C₄×C₅₃	1	2	yes	—
153	C₄×C₈	—	2	2	no	—
154	#41	—	4	2	no	—
155	#41	—	4	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	C₂²⋊C₄×C₅₃	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
163	C₂²×C₄	#41×C₅₃	1	2	yes	—
164	M₁₆	—	8	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	8	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	4	1	no	—
176	C₄²	—	4	1	no	—
177	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
178	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
179	C₄²	C₂×D₄×C₅₃	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	#123×C₅₃	1	1	yes	characteristic
182	Q₈	—	4	2	no	—
183	Q₈	—	4	4	no	—
184	D₄	—	4	2	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
194	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
195	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
196	C₂×C₄	#121×C₅₃	1	2	yes	—
197	C₈	—	4	2	no	—
198	C₈	—	4	4	no	—
199	C₈	—	8	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	4	4	no	—
203	C₂²	—	4	2	no	—
204	C₂²	C₅₃×#386	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	2	no	—
209	C₄	—	8	2	no	—
210	C₄	C₅₃×#455	1	2	yes	—
211	C₂	—	4	1	no	—
212	C₂	—	8	2	no	—
213	C₂	C₅₃×#1881	1	2	yes	—
214	C₂	C₅₃×#1885	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3135

C₂

yes

maximal, characteristic

C₅₃×#2980

C₂

yes

maximal, characteristic

C₅₃×#1543

C₂

yes

maximal, characteristic

C₅₃×#1497

C₂

yes

maximal, complemented

C₅₃×#1302

C₂

yes

maximal, complemented

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#491

—

C₅₃×#490

—

C₅₃×#489

C₄

yes

—

C₅₃×#435

—

C₅₃×#432

C₂²

yes

—

C₅₃×#431

C₂²

yes

—

C₅₃×#416

—

C₅₃×#408

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#333

—

C₅₃×#331

—

C₅₃×Q₈⋊C₈

—

C₂×#41×C₅₃

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

D₄

yes

characteristic

#127×C₅₃

D₄

yes

characteristic

#125×C₅₃

—

#124×C₅₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₅₃

—

#121×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×D₄×C₅₃

—

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

#103×C₅₃

—

#102×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

#97×C₅₃

—

#97×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂²⋊C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×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

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

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×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

—

#9005

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#386

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

100

C₄×C₅₃

—

101

C₄×C₅₃

—

102

C₄×C₅₃

—

103

C₄×C₅₃

#455

yes

—

104

#3135

C₂×C₅₃

yes

characteristic

105

#2980

C₂×C₅₃

yes

characteristic

106

#1543

C₂×C₅₃

yes

characteristic

107

#1497

C₂×C₅₃

yes

complemented

108

#1302

C₂×C₅₃

yes

complemented

109

C₂×C₅₃

—

110

C₂×C₅₃

—

111

C₂×C₅₃

#1881

yes

—

112

C₂×C₅₃

#1885

yes

characteristic

113

#533

C₂²×C₅₃

yes

characteristic

114

#532

C₂²×C₅₃

yes

characteristic

115

#491

—

116

#490

—

117

#489

C₄×C₅₃

yes

—

118

#435

—

119

#432

C₂²×C₅₃

yes

—

120

#431

C₂²×C₅₃

yes

—

121

#416

—

122

#408

—

123

C₂×#97

C₂²×C₅₃

yes

characteristic

124

#333

—

125

#331

—

126

Q₈⋊C₈

—

127

C₅₃

#9005

yes

characteristic, direct product factor, Sylow

128

C₂×#41

D₄×C₅₃

yes

—

129

C₂×M₁₆

—

130

C₂²×C₈

—

131

C₂²×C₈

—

132

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

133

C₄⋊Q₈

D₄×C₅₃

yes

characteristic

134

#127

D₄×C₅₃

yes

characteristic

135

#125

—

136

#124

C₂×C₄×C₅₃

yes

—

137

C₂²⋊Q₈

—

138

#121

—

139

C₄×Q₈

—

140

C₄×D₄

—

141

#117

—

142

C₂×C₄²

C₂³×C₅₃

yes

characteristic

143

C₄⋊C₈

—

144

C₄⋊C₈

D₄×C₅₃

yes

—

145

#103

—

146

#102

—

147

C₂²⋊C₈

—

148

C₂²⋊C₈

—

149

C₂²⋊C₈

—

150

#97

—

151

#97

—

152

#97

C₂×C₄×C₅₃

yes

—

153

C₄×C₈

—

154

#41

—

155

#41

—

156

C₂×Q₈

—

157

C₂×Q₈

C₂²⋊C₄×C₅₃

yes

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

C₂²⋊C₄×C₅₃

yes

—

161

C₂²×C₄

—

162

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

163

C₂²×C₄

#41×C₅₃

yes

—

164

M₁₆

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

C₂²×C₄×C₅₃

yes

characteristic

178

C₄²

C₂×D₄×C₅₃

yes

characteristic

179

C₄²

C₂×D₄×C₅₃

yes

—

180

C₂³

—

181

C₂³

#123×C₅₃

yes

characteristic

182

Q₈

—

183

Q₈

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

194

C₂×C₄

C₄×D₄×C₅₃

yes

—

195

C₂×C₄

#120×C₅₃

yes

characteristic

196

C₂×C₄

#121×C₅₃

yes

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

C₅₃×#386

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

C₅₃×#455

yes

—

211

C₂

—

212

C₂

—

213

C₂

C₅₃×#1881

yes

—

214

C₂

C₅₃×#1885

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	C₅₃×#532	C₂²
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

socle

C₂²×C₅₃

#386

prime radical

C₅₃×#532

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
4. C₅₃×#1497	C₂	semidirect
5. C₅₃×#1302	C₂	semidirect
92. #9005	C₅₃	direct (N × H)
107. #1497	C₂×C₅₃	semidirect
108. #1302	C₂×C₅₃	semidirect
127. C₅₃	#9005	direct (N × H)

H ≅ G/N

type

4. C₅₃×#1497

C₂

semidirect

5. C₅₃×#1302

C₂

semidirect

92. #9005

C₅₃

direct (N × H)

107. #1497

C₂×C₅₃

semidirect

108. #1302

C₂×C₅₃

semidirect

127. C₅₃

#9005

direct (N × H)

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

subgroup	index
C₅₃×#3135	2
C₅₃×#2980	2
C₅₃×#1543	2
C₅₃×#1497	2
C₅₃×#1302	2
#9005	53

subgroup

index

C₅₃×#3135

C₅₃×#2980

C₅₃×#1543

C₅₃×#1497

C₅₃×#1302

#9005

Subgroups — C₅₃×#9005 #27216517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9005 #27216517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9005 #27216517