Group #27283985 — subgroups

Subgroups — #27283985

Order 13568.

498 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20285294	C₂	1	1	yes	maximal, characteristic
2	#20275542	C₂	1	2	yes	maximal, complemented
3	#20274460	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273165	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811173	C₂²	1	2	yes	—
7	#16811028	C₄	1	1	yes	characteristic, complemented
8	#16811000	C₄	1	1	yes	characteristic, complemented
9	#16810970	C₄	1	2	yes	complemented
10	#16810859	—	2	2	no	—
11	C₄×D₄₂₄	—	2	2	no	—
12	C₂×#13133143	—	2	4	no	—
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810377	—	2	2	no	—
16	#16810350	—	2	4	no	—
17	C₅₃×#423	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#335	—	2	2	no	—
20	C₂×#11719945	—	2	2	no	—
21	C₂×D₄₂₄	—	2	2	no	—
22	C₂²×#88322	—	2	1	no	—
23	#13133217	C₂×C₄	1	2	yes	—
24	#13133214	C₂×C₄	1	2	yes	—
25	#13133205	—	2	2	no	—
26	#13133191	—	2	2	no	—
27	#13133191	—	2	2	no	—
28	C₄×D₂₁₂	—	2	2	no	—
29	C₄×D₂₁₂	—	2	2	no	—
30	C₂×C₄×Dic₅₃	—	2	1	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
33	#13133151	—	2	2	no	—
34	#13133143	—	4	8	no	—
35	#13133139	—	2	4	no	—
36	C₄×#88322	—	2	1	no	—
37	C₄×#88322	D₄	1	2	yes	—
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂²×C₈×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₂²⋊D₅₃	—	4	2	no	—
50	C₂²×Dic₅₃	—	2	2	no	—
51	#11719945	—	4	4	no	—
52	#11719945	—	4	4	no	—
53	C₂×D₂₁₂	—	2	2	no	—
54	C₂×D₂₁₂	—	2	2	no	—
55	C₂×D₂₁₂	—	2	2	no	—
56	C₂×C₄×D₅₃	—	4	2	no	—
57	C₂×Dic₁₀₆	—	2	2	no	—
58	M₁₆×C₅₃	—	4	4	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	#41	1	1	yes	characteristic
67	#11719921	—	4	2	no	—
68	#11719921	—	4	2	no	—
69	C₄⋊Dic₅₃	—	2	2	no	—
70	C₄⋊Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₄×Dic₅₃	—	4	2	no	—
73	C₄×Dic₅₃	—	4	2	no	—
74	C₄×Dic₅₃	D₈	1	4	yes	complemented
75	C₂×#88322	—	2	2	no	—
76	C₂×#88322	—	4	2	no	—
77	D₄₂₄	—	4	4	no	—
78	C₂³×C₅₃	#121	1	1	yes	characteristic
79	C₂²×D₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₅₃	#135	1	2	yes	—
90	C₂²⋊D₅₃	—	8	4	no	—
91	C₂×Dic₅₃	—	2	4	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	—	4	2	no	—
94	C₂×Dic₅₃	—	4	2	no	—
95	D₂₁₂	—	4	4	no	—
96	D₂₁₂	—	4	4	no	—
97	D₂₁₂	—	4	4	no	—
98	C₄×D₅₃	—	8	2	no	—
99	C₄×D₅₃	—	8	2	no	—
100	Dic₁₀₆	—	4	4	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	—	2	2	no	—
103	C₈×C₅₃	—	8	2	no	—
104	#88322	—	4	2	no	—
105	#9847	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	#390	1	1	yes	characteristic
109	C₂²×C₅₃	#466	1	2	yes	—
110	C₂×D₅₃	—	8	2	no	—
111	C₂×D₅₃	—	8	2	no	—
112	C₂×D₅₃	—	8	2	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	#421	1	2	yes	—
116	C₄×C₅₃	C₄×D₈	1	2	yes	—
117	C₄×C₅₃	#444	1	2	yes	—
118	C₄×C₅₃	#493	1	1	yes	characteristic
119	C₄×C₅₃	#495	1	1	yes	characteristic
120	Dic₅₃	—	4	4	no	—
121	Dic₅₃	—	8	2	no	—
122	#3169	—	53	1	no	—
123	#1962	—	53	2	no	—
124	#1744	—	53	1	no	—
125	#1567	D₅₃	1	1	yes	characteristic, complemented
126	#1492	—	53	2	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	#1963	1	1	yes	characteristic
129	C₂×C₅₃	#1970	1	1	yes	characteristic
130	C₂×C₅₃	#1975	1	1	yes	characteristic
131	D₅₃	—	16	2	no	—
132	#532	—	53	2	no	—
133	#500	—	53	1	no	—
134	#493	—	53	1	no	—
135	#486	—	53	2	no	—
136	#466	—	106	2	no	—
137	C₄×D₈	—	106	2	no	—
138	#423	C₂×D₅₃	1	2	yes	—
139	C₂×#104	—	106	4	no	—
140	C₂×C₄×C₈	—	53	1	no	—
141	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
142	C₄³	—	53	1	no	—
143	#336	—	106	2	no	—
144	#335	—	2	2	no	—
145	D₄⋊C₈	—	106	4	no	—
146	C₅₃	#9847	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	—	106	2	no	—
148	C₂×D₈	—	106	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂²×C₈	—	106	1	no	—
151	C₂²×C₈	D₂₁₂	1	2	yes	—
152	C₄⋊Q₈	—	53	2	no	—
153	#127	—	53	2	no	—
154	#124	—	106	2	no	—
155	#121	—	106	2	no	—
156	#121	—	106	2	no	—
157	C₄×D₄	—	106	2	no	—
158	C₄×D₄	—	106	2	no	—
159	C₂×C₄²	—	53	2	no	—
160	C₂×C₄²	—	106	1	no	—
161	C₂×C₄²	—	106	1	no	—
162	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
163	#107	—	106	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	#104	—	212	8	no	—
167	#102	—	106	4	no	—
168	C₄×C₈	—	53	2	no	—
169	C₄×C₈	—	106	1	no	—
170	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
171	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
173	#41	—	212	4	no	—
174	#41	—	212	4	no	—
175	C₂×Q₈	—	106	2	no	—
176	C₂×D₄	—	106	2	no	—
177	C₂×D₄	—	106	2	no	—
178	C₂×D₄	—	106	2	no	—
179	C₂×D₄	—	212	2	no	—
180	C₂²×C₄	—	106	2	no	—
181	C₂²×C₄	—	212	2	no	—
182	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₄	#11719949	1	1	yes	characteristic
185	D₈	—	212	4	no	—
186	M₁₆	—	4	4	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	106	2	no	—
190	C₂×C₈	—	212	2	no	—
191	C₂×C₈	#11719921	1	2	yes	—
192	C₂×C₈	#11719921	1	2	yes	—
193	C₄⋊C₄	—	106	2	no	—
194	C₄⋊C₄	—	106	2	no	—
195	C₂²⋊C₄	—	212	2	no	—
196	C₂²⋊C₄	—	212	2	no	—
197	C₄²	—	53	4	no	—
198	C₄²	—	106	2	no	—
199	C₄²	—	212	2	no	—
200	C₄²	—	212	2	no	—
201	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₄²	D₄×D₅₃	1	1	yes	characteristic
203	C₄²	#11719949	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂³	—	212	2	no	—
206	C₂³	#13133192	1	1	yes	characteristic
207	Q₈	—	212	4	no	—
208	D₄	—	424	4	no	—
209	D₄	—	212	4	no	—
210	D₄	—	212	4	no	—
211	D₄	—	212	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	106	4	no	—
215	C₂×C₄	—	424	2	no	—
216	C₂×C₄	—	424	2	no	—
217	C₂×C₄	—	212	2	no	—
218	C₂×C₄	—	212	2	no	—
219	C₂×C₄	—	212	2	no	—
220	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
221	C₂×C₄	#13133179	1	1	yes	characteristic
222	C₂×C₄	#13133179	1	1	yes	characteristic
223	C₂×C₄	#13133192	1	1	yes	characteristic
224	C₂×C₄	#13133206	1	1	yes	characteristic
225	C₂×C₄	#13133213	1	1	yes	characteristic
226	C₂×C₄	D₈×D₅₃	1	2	yes	—
227	C₂×C₄	#13133240	1	2	yes	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	8	2	no	—
231	C₈	—	212	2	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	424	2	no	—
235	C₂²	—	424	2	no	—
236	C₂²	—	424	2	no	—
237	C₂²	#16810524	1	1	yes	characteristic
238	C₂²	#16810860	1	2	yes	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	424	2	no	—
242	C₄	—	212	4	no	—
243	C₄	#16810639	1	2	yes	—
244	C₄	#16810705	1	2	yes	—
245	C₄	#16810744	1	2	yes	—
246	C₄	#16810998	1	1	yes	characteristic
247	C₄	#16811008	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	848	2	no	—
250	C₂	#20275546	1	1	yes	characteristic
251	C₂	#20275579	1	1	yes	characteristic
252	C₂	#20275606	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20285294

C₂

yes

maximal, characteristic

#20275542

C₂

yes

maximal, complemented

#20274460

C₂

yes

maximal, characteristic, complemented

#20273165

C₂

yes

maximal, complemented

C₅₃×#1567

C₂

yes

maximal, characteristic, complemented

#16811173

C₂²

yes

—

#16811028

C₄

yes

characteristic, complemented

#16811000

C₄

yes

characteristic, complemented

#16810970

C₄

yes

complemented

#16810859

—

C₄×D₄₂₄

—

C₂×#13133143

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₄²×Dic₅₃

C₂²

yes

characteristic

#16810377

—

#16810350

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₂×#11719945

—

C₂×D₄₂₄

—

C₂²×#88322

—

#13133217

C₂×C₄

yes

—

#13133214

C₂×C₄

yes

—

#13133205

—

#13133191

—

#13133191

—

C₄×D₂₁₂

—

C₄×D₂₁₂

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

D₄

yes

—

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

—

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₈

yes

complemented

C₂×#88322

—

C₂×#88322

—

D₄₂₄

—

C₂³×C₅₃

#121

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

100

Dic₁₀₆

—

101

C₈×C₅₃

—

102

C₈×C₅₃

—

103

C₈×C₅₃

—

104

#88322

—

105

#9847

—

maximal, Sylow

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

#390

yes

characteristic

109

C₂²×C₅₃

#466

yes

—

110

C₂×D₅₃

—

111

C₂×D₅₃

—

112

C₂×D₅₃

—

113

C₄×C₅₃

—

114

C₄×C₅₃

—

115

C₄×C₅₃

#421

yes

—

116

C₄×C₅₃

C₄×D₈

yes

—

117

C₄×C₅₃

#444

yes

—

118

C₄×C₅₃

#493

yes

characteristic

119

C₄×C₅₃

#495

yes

characteristic

120

Dic₅₃

—

121

Dic₅₃

—

122

#3169

—

123

#1962

—

124

#1744

—

125

#1567

D₅₃

yes

characteristic, complemented

126

#1492

—

127

C₂×C₅₃

—

128

C₂×C₅₃

#1963

yes

characteristic

129

C₂×C₅₃

#1970

yes

characteristic

130

C₂×C₅₃

#1975

yes

characteristic

131

D₅₃

—

132

#532

—

133

#500

—

134

#493

—

135

#486

—

136

#466

—

106

—

137

C₄×D₈

—

106

—

138

#423

C₂×D₅₃

yes

—

139

C₂×#104

—

106

—

140

C₂×C₄×C₈

—

141

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

142

C₄³

—

143

#336

—

106

—

144

#335

—

145

D₄⋊C₈

—

106

—

146

C₅₃

#9847

yes

characteristic, complemented, Sylow

147

C₂×#41

—

106

—

148

C₂×D₈

—

106

—

149

C₂×M₁₆

—

150

C₂²×C₈

—

106

—

151

C₂²×C₈

D₂₁₂

yes

—

152

C₄⋊Q₈

—

153

#127

—

154

#124

—

106

—

155

#121

—

106

—

156

#121

—

106

—

157

C₄×D₄

—

106

—

158

C₄×D₄

—

106

—

159

C₂×C₄²

—

160

C₂×C₄²

—

106

—

161

C₂×C₄²

—

106

—

162

C₂×C₄²

C₂²×D₅₃

yes

characteristic

163

#107

—

106

—

164

C₄⋊C₈

—

165

C₄⋊C₈

—

166

#104

—

212

—

167

#102

—

106

—

168

C₄×C₈

—

169

C₄×C₈

—

106

—

170

C₄×C₈

C₄×D₅₃

yes

characteristic

171

C₄×C₈

C₄×D₅₃

yes

characteristic

172

C₄×C₈

C₂²⋊D₅₃

yes

—

173

#41

—

212

—

174

#41

—

212

—

175

C₂×Q₈

—

106

—

176

C₂×D₄

—

106

—

177

C₂×D₄

—

106

—

178

C₂×D₄

—

106

—

179

C₂×D₄

—

212

—

180

C₂²×C₄

—

106

—

181

C₂²×C₄

—

212

—

182

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

183

C₂²×C₄

D₄×D₅₃

yes

characteristic

184

C₂²×C₄

#11719949

yes

characteristic

185

D₈

—

212

—

186

M₁₆

—

187

C₂×C₈

—

188

C₂×C₈

—

189

C₂×C₈

—

106

—

190

C₂×C₈

—

212

—

191

C₂×C₈

#11719921

yes

—

192

C₂×C₈

#11719921

yes

—

193

C₄⋊C₄

—

106

—

194

C₄⋊C₄

—

106

—

195

C₂²⋊C₄

—

212

—

196

C₂²⋊C₄

—

212

—

197

C₄²

—

198

C₄²

—

106

—

199

C₄²

—

212

—

200

C₄²

—

212

—

201

C₄²

C₂×C₄×D₅₃

yes

characteristic

202

C₄²

D₄×D₅₃

yes

characteristic

203

C₄²

#11719949

yes

characteristic

204

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

205

C₂³

—

212

—

206

C₂³

#13133192

yes

characteristic

207

Q₈

—

212

—

208

D₄

—

424

—

209

D₄

—

212

—

210

D₄

—

212

—

211

D₄

—

212

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

106

—

215

C₂×C₄

—

424

—

216

C₂×C₄

—

424

—

217

C₂×C₄

—

212

—

218

C₂×C₄

—

212

—

219

C₂×C₄

—

212

—

220

C₂×C₄

C₂×#11719921

yes

characteristic

221

C₂×C₄

#13133179

yes

characteristic

222

C₂×C₄

#13133179

yes

characteristic

223

C₂×C₄

#13133192

yes

characteristic

224

C₂×C₄

#13133206

yes

characteristic

225

C₂×C₄

#13133213

yes

characteristic

226

C₂×C₄

D₈×D₅₃

yes

—

227

C₂×C₄

#13133240

yes

—

228

C₈

—

229

C₈

—

230

C₈

—

231

C₈

—

212

—

232

C₂²

—

233

C₂²

—

234

C₂²

—

424

—

235

C₂²

—

424

—

236

C₂²

—

424

—

237

C₂²

#16810524

yes

characteristic

238

C₂²

#16810860

yes

—

239

C₄

—

240

C₄

—

241

C₄

—

424

—

242

C₄

—

212

—

243

C₄

#16810639

yes

—

244

C₄

#16810705

yes

—

245

C₄

#16810744

yes

—

246

C₄

#16810998

yes

characteristic

247

C₄

#16811008

yes

characteristic

248

C₂

—

249

C₂

—

848

—

250

C₂

#20275546

yes

characteristic

251

C₂

#20275579

yes

characteristic

252

C₂

#20275606

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1567	C₂
socle	C₂²×C₅₃	#390
prime radical	#20285294	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1567	D₅₃
hypocenter	C₅₃	#9847
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1567

C₂

C₂²×C₅₃

C₂

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
2. #20275542	C₂	semidirect
3. #20274460	C₂	semidirect
4. #20273165	C₂	semidirect
5. C₅₃×#1567	C₂	semidirect
7. #16811028	C₄	semidirect
8. #16811000	C₄	semidirect
9. #16810970	C₄	semidirect
74. C₄×Dic₅₃	D₈	semidirect
125. #1567	D₅₃	semidirect
146. C₅₃	#9847	semidirect

H ≅ G/N

type

2. #20275542

C₂

semidirect

3. #20274460

C₂

semidirect

4. #20273165

C₂

semidirect

5. C₅₃×#1567

C₂

semidirect

7. #16811028

C₄

semidirect

8. #16811000

C₄

semidirect

9. #16810970

C₄

semidirect

74. C₄×Dic₅₃

D₈

semidirect

125. #1567

D₅₃

semidirect

146. C₅₃

#9847

semidirect

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

subgroup	index
#20285294	2
#20275542	2
#20274460	2
#20273165	2
C₅₃×#1567	2
#9847	53

subgroup

index

C₅₃×#1567

Subgroups — #27283985

Conjugacy classes of subgroups

Complements

Maximal subgroups