Group #27283995 — subgroups

Subgroups — #27283995

Order 13568.

530 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₄×#13133127	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273165	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1566	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×#13133127	—	2	2	no	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
16	#16810350	—	2	4	no	—
17	C₅₃×#423	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#334	—	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	#13133127	—	2	8	no	—
37	C₄×#88322	—	2	1	no	—
38	C₄×#88322	D₄	1	2	yes	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂×C₂²⋊D₅₃	—	4	2	no	—
51	C₂²×Dic₅₃	—	2	2	no	—
52	#11719945	—	4	4	no	—
53	#11719945	—	4	4	no	—
54	C₂×D₂₁₂	—	2	2	no	—
55	C₂×D₂₁₂	—	2	2	no	—
56	C₂×D₂₁₂	—	2	2	no	—
57	C₂×C₄×D₅₃	—	4	2	no	—
58	C₂×Dic₁₀₆	—	2	2	no	—
59	M₁₆×C₅₃	—	4	4	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	#41	1	1	yes	characteristic
68	#11719921	—	4	2	no	—
69	#11719921	—	4	2	no	—
70	C₄⋊Dic₅₃	—	2	2	no	—
71	C₄⋊Dic₅₃	—	2	2	no	—
72	C₄×Dic₅₃	—	2	4	no	—
73	C₄×Dic₅₃	—	2	4	no	—
74	C₄×Dic₅₃	—	4	2	no	—
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	—	2	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	AGL(1,ℤ/8)	1	4	yes	—
89	C₂²⋊D₅₃	—	8	4	no	—
90	C₂×Dic₅₃	—	2	4	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	—	4	2	no	—
94	D₂₁₂	—	4	4	no	—
95	D₂₁₂	—	4	4	no	—
96	D₂₁₂	—	4	4	no	—
97	C₄×D₅₃	—	8	2	no	—
98	C₄×D₅₃	—	8	2	no	—
99	Dic₁₀₆	—	4	4	no	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	8	2	no	—
102	C₈×C₅₃	#104	1	4	yes	complemented
103	#88322	—	2	4	no	—
104	#9849	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	#390	1	1	yes	characteristic
108	C₂²×C₅₃	#469	1	2	yes	—
109	C₂×D₅₃	—	8	2	no	—
110	C₂×D₅₃	—	8	2	no	—
111	C₂×D₅₃	—	8	2	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	C₂×#104	1	2	yes	—
115	C₄×C₅₃	#442	1	2	yes	—
116	C₄×C₅₃	#444	1	2	yes	—
117	C₄×C₅₃	#496	1	2	yes	—
118	Dic₅₃	—	4	4	no	—
119	Dic₅₃	—	8	2	no	—
120	#3169	—	53	1	no	—
121	#1962	—	53	2	no	—
122	C₄×#97	—	53	1	no	—
123	#1566	D₅₃	1	1	yes	characteristic, complemented
124	#1492	—	53	2	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	#1962	1	1	yes	characteristic
127	C₂×C₅₃	#1972	1	1	yes	characteristic
128	C₂×C₅₃	#1975	1	1	yes	characteristic
129	D₅₃	—	16	2	no	—
130	#532	—	53	2	no	—
131	#500	—	53	1	no	—
132	#493	—	53	1	no	—
133	#486	—	53	2	no	—
134	#466	—	106	2	no	—
135	C₄×D₈	—	106	2	no	—
136	#423	C₂×D₅₃	1	2	yes	—
137	C₂×#104	—	106	4	no	—
138	C₂×#97	—	106	2	no	—
139	C₂×C₄×C₈	—	53	1	no	—
140	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₄³	—	53	1	no	—
142	#334	—	2	2	no	—
143	D₄⋊C₈	—	106	4	no	—
144	C₅₃	#9849	1	1	yes	characteristic, complemented, Sylow
145	C₂×#41	—	106	2	no	—
146	C₂×D₈	—	106	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	—	106	1	no	—
149	C₂²×C₈	D₂₁₂	1	2	yes	—
150	C₄⋊Q₈	—	53	2	no	—
151	#127	—	53	2	no	—
152	#124	—	106	2	no	—
153	#121	—	106	2	no	—
154	#121	—	106	2	no	—
155	C₄×D₄	—	106	2	no	—
156	C₄×D₄	—	106	2	no	—
157	C₂×C₄²	—	53	2	no	—
158	C₂×C₄²	—	106	1	no	—
159	C₂×C₄²	—	106	1	no	—
160	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
161	#107	—	106	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	#104	—	212	8	no	—
165	#102	—	106	4	no	—
166	#97	—	106	8	no	—
167	C₄×C₈	—	53	2	no	—
168	C₄×C₈	—	106	1	no	—
169	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
170	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
172	#41	—	212	4	no	—
173	#41	—	212	4	no	—
174	C₂×Q₈	—	106	2	no	—
175	C₂×D₄	—	106	2	no	—
176	C₂×D₄	—	106	2	no	—
177	C₂×D₄	—	106	2	no	—
178	C₂×D₄	—	212	2	no	—
179	C₂²×C₄	—	106	2	no	—
180	C₂²×C₄	—	212	2	no	—
181	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
183	C₂²×C₄	#11719949	1	1	yes	characteristic
184	D₈	—	212	4	no	—
185	M₁₆	—	4	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	106	4	no	—
189	C₂×C₈	—	106	2	no	—
190	C₂×C₈	#11719921	1	2	yes	—
191	C₂×C₈	#11719921	1	2	yes	—
192	C₄⋊C₄	—	106	2	no	—
193	C₄⋊C₄	—	106	2	no	—
194	C₂²⋊C₄	—	212	2	no	—
195	C₂²⋊C₄	—	212	2	no	—
196	C₄²	—	106	4	no	—
197	C₄²	—	106	4	no	—
198	C₄²	—	212	2	no	—
199	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₄²	D₄×D₅₃	1	1	yes	characteristic
201	C₄²	#11719949	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂³	—	212	2	no	—
204	C₂³	#13133192	1	1	yes	characteristic
205	Q₈	—	212	4	no	—
206	D₄	—	424	4	no	—
207	D₄	—	212	4	no	—
208	D₄	—	212	4	no	—
209	D₄	—	212	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	106	4	no	—
213	C₂×C₄	—	424	2	no	—
214	C₂×C₄	—	424	2	no	—
215	C₂×C₄	—	212	2	no	—
216	C₂×C₄	—	212	2	no	—
217	C₂×C₄	—	212	2	no	—
218	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
219	C₂×C₄	#13133179	1	1	yes	characteristic
220	C₂×C₄	#13133179	1	1	yes	characteristic
221	C₂×C₄	#13133192	1	1	yes	characteristic
222	C₂×C₄	#13133206	1	1	yes	characteristic
223	C₂×C₄	#13133213	1	1	yes	characteristic
224	C₂×C₄	#13133244	1	4	yes	—
225	C₈	—	2	2	no	—
226	C₈	—	8	2	no	—
227	C₈	—	106	4	no	—
228	C₈	#13133143	1	4	yes	complemented
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	424	2	no	—
232	C₂²	—	424	2	no	—
233	C₂²	—	424	2	no	—
234	C₂²	#16810524	1	1	yes	characteristic
235	C₂²	#16810877	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	424	2	no	—
239	C₄	—	212	4	no	—
240	C₄	C₂×#13133143	1	2	yes	—
241	C₄	#16810734	1	2	yes	—
242	C₄	#16810744	1	2	yes	—
243	C₄	#16811009	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	848	2	no	—
246	C₂	#20275542	1	1	yes	characteristic
247	C₂	#20275589	1	1	yes	characteristic
248	C₂	#20275606	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20285294

C₂

yes

maximal, characteristic

#20275542

C₂

yes

maximal, complemented

C₄×#13133127

C₂

yes

maximal, characteristic, complemented

#20273165

C₂

yes

maximal, complemented

C₅₃×#1566

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×#13133127

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₄²×Dic₅₃

C₂²

yes

characteristic

#16810350

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

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

—

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

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

100

C₈×C₅₃

—

101

C₈×C₅₃

—

102

C₈×C₅₃

#104

yes

complemented

103

#88322

—

104

#9849

—

maximal, Sylow

105

C₂²×C₅₃

—

106

C₂²×C₅₃

—

107

C₂²×C₅₃

#390

yes

characteristic

108

C₂²×C₅₃

#469

yes

—

109

C₂×D₅₃

—

110

C₂×D₅₃

—

111

C₂×D₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

—

114

C₄×C₅₃

C₂×#104

yes

—

115

C₄×C₅₃

#442

yes

—

116

C₄×C₅₃

#444

yes

—

117

C₄×C₅₃

#496

yes

—

118

Dic₅₃

—

119

Dic₅₃

—

120

#3169

—

121

#1962

—

122

C₄×#97

—

123

#1566

D₅₃

yes

characteristic, complemented

124

#1492

—

125

C₂×C₅₃

—

126

C₂×C₅₃

#1962

yes

characteristic

127

C₂×C₅₃

#1972

yes

characteristic

128

C₂×C₅₃

#1975

yes

characteristic

129

D₅₃

—

130

#532

—

131

#500

—

132

#493

—

133

#486

—

134

#466

—

106

—

135

C₄×D₈

—

106

—

136

#423

C₂×D₅₃

yes

—

137

C₂×#104

—

106

—

138

C₂×#97

—

106

—

139

C₂×C₄×C₈

—

140

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

141

C₄³

—

142

#334

—

143

D₄⋊C₈

—

106

—

144

C₅₃

#9849

yes

characteristic, complemented, Sylow

145

C₂×#41

—

106

—

146

C₂×D₈

—

106

—

147

C₂×M₁₆

—

148

C₂²×C₈

—

106

—

149

C₂²×C₈

D₂₁₂

yes

—

150

C₄⋊Q₈

—

151

#127

—

152

#124

—

106

—

153

#121

—

106

—

154

#121

—

106

—

155

C₄×D₄

—

106

—

156

C₄×D₄

—

106

—

157

C₂×C₄²

—

158

C₂×C₄²

—

106

—

159

C₂×C₄²

—

106

—

160

C₂×C₄²

C₂²×D₅₃

yes

characteristic

161

#107

—

106

—

162

C₄⋊C₈

—

163

C₄⋊C₈

—

164

#104

—

212

—

165

#102

—

106

—

166

#97

—

106

—

167

C₄×C₈

—

168

C₄×C₈

—

106

—

169

C₄×C₈

C₄×D₅₃

yes

characteristic

170

C₄×C₈

C₄×D₅₃

yes

characteristic

171

C₄×C₈

C₂²⋊D₅₃

yes

—

172

#41

—

212

—

173

#41

—

212

—

174

C₂×Q₈

—

106

—

175

C₂×D₄

—

106

—

176

C₂×D₄

—

106

—

177

C₂×D₄

—

106

—

178

C₂×D₄

—

212

—

179

C₂²×C₄

—

106

—

180

C₂²×C₄

—

212

—

181

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

182

C₂²×C₄

D₄×D₅₃

yes

characteristic

183

C₂²×C₄

#11719949

yes

characteristic

184

D₈

—

212

—

185

M₁₆

—

186

C₂×C₈

—

187

C₂×C₈

—

188

C₂×C₈

—

106

—

189

C₂×C₈

—

106

—

190

C₂×C₈

#11719921

yes

—

191

C₂×C₈

#11719921

yes

—

192

C₄⋊C₄

—

106

—

193

C₄⋊C₄

—

106

—

194

C₂²⋊C₄

—

212

—

195

C₂²⋊C₄

—

212

—

196

C₄²

—

106

—

197

C₄²

—

106

—

198

C₄²

—

212

—

199

C₄²

C₂×C₄×D₅₃

yes

characteristic

200

C₄²

D₄×D₅₃

yes

characteristic

201

C₄²

#11719949

yes

characteristic

202

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

203

C₂³

—

212

—

204

C₂³

#13133192

yes

characteristic

205

Q₈

—

212

—

206

D₄

—

424

—

207

D₄

—

212

—

208

D₄

—

212

—

209

D₄

—

212

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

106

—

213

C₂×C₄

—

424

—

214

C₂×C₄

—

424

—

215

C₂×C₄

—

212

—

216

C₂×C₄

—

212

—

217

C₂×C₄

—

212

—

218

C₂×C₄

C₂×#11719921

yes

characteristic

219

C₂×C₄

#13133179

yes

characteristic

220

C₂×C₄

#13133179

yes

characteristic

221

C₂×C₄

#13133192

yes

characteristic

222

C₂×C₄

#13133206

yes

characteristic

223

C₂×C₄

#13133213

yes

characteristic

224

C₂×C₄

#13133244

yes

—

225

C₈

—

226

C₈

—

227

C₈

—

106

—

228

C₈

#13133143

yes

complemented

229

C₂²

—

230

C₂²

—

231

C₂²

—

424

—

232

C₂²

—

424

—

233

C₂²

—

424

—

234

C₂²

#16810524

yes

characteristic

235

C₂²

#16810877

yes

—

236

C₄

—

237

C₄

—

238

C₄

—

424

—

239

C₄

—

212

—

240

C₄

C₂×#13133143

yes

—

241

C₄

#16810734

yes

—

242

C₄

#16810744

yes

—

243

C₄

#16811009

yes

—

244

C₂

—

245

C₂

—

848

—

246

C₂

#20275542

yes

characteristic

247

C₂

#20275589

yes

characteristic

248

C₂

#20275606

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1566	C₂
socle	C₂²×C₅₃	#390
prime radical	#20285294	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1566	D₅₃
hypocenter	C₅₃	#9849
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1566

C₂

C₂²×C₅₃

C₂

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
2. #20275542	C₂	semidirect
3. C₄×#13133127	C₂	semidirect
4. #20273165	C₂	semidirect
5. C₅₃×#1566	C₂	semidirect
7. #16811028	C₄	semidirect
8. #16811000	C₄	semidirect
9. #16810970	C₄	semidirect
102. C₈×C₅₃	#104	semidirect
123. #1566	D₅₃	semidirect
144. C₅₃	#9849	semidirect
228. C₈	#13133143	semidirect

H ≅ G/N

type

2. #20275542

C₂

semidirect

3. C₄×#13133127

C₂

semidirect

4. #20273165

C₂

semidirect

5. C₅₃×#1566

C₂

semidirect

7. #16811028

C₄

semidirect

8. #16811000

C₄

semidirect

9. #16810970

C₄

semidirect

102. C₈×C₅₃

#104

semidirect

123. #1566

D₅₃

semidirect

144. C₅₃

#9849

semidirect

228. C₈

#13133143

semidirect

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

subgroup	index
#20285294	2
#20275542	2
C₄×#13133127	2
#20273165	2
C₅₃×#1566	2
#9849	53

subgroup

index

C₅₃×#1566

Subgroups — #27283995

Conjugacy classes of subgroups

Complements

Maximal subgroups