Group #27277322 — subgroups

Subgroups — #27277322

Order 13568.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20283649	C₂	1	1	yes	maximal, characteristic
2	#20274846	C₂	1	1	yes	maximal, characteristic
3	#20273280	C₂	1	2	yes	maximal
4	#20272574	C₂	1	2	yes	maximal
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	1	yes	characteristic
7	#16810753	—	2	1	no	—
8	#16810753	C₄	1	2	yes	complemented
9	C₈×Dic₁₀₆	—	2	1	no	—
10	C₈×Dic₁₀₆	C₄	1	2	yes	complemented
11	#16810649	—	2	1	no	—
12	#16810647	C₂²	1	1	yes	characteristic
13	#16810578	—	2	1	no	—
14	#16810489	C₂²	1	2	yes	—
15	#16810386	—	2	1	no	—
16	#16810386	—	2	1	no	—
17	#16810361	—	2	4	no	—
18	C₅₃×#431	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#335	—	2	1	no	—
21	C₅₃×#334	—	2	1	no	—
22	C₂×#11719917	—	2	1	no	—
23	#13133217	Q₈	1	1	yes	characteristic
24	#13133217	Q₈	1	1	yes	characteristic
25	#13133209	—	2	1	no	—
26	#13133196	—	2	1	no	—
27	#13133196	D₄	1	2	yes	—
28	C₄×Dic₁₀₆	—	2	1	no	—
29	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
30	#13133176	—	2	1	no	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	C₂×C₄	1	2	yes	—
34	#13133146	—	2	2	no	—
35	#13133146	—	2	2	no	—
36	#13133127	—	2	2	no	—
37	C₈×Dic₅₃	—	2	2	no	—
38	#13133124	—	2	2	no	—
39	#13133124	—	2	2	no	—
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₂²⋊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	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂²×Dic₅₃	—	4	2	no	—
54	C₂×Dic₁₀₆	—	2	1	no	—
55	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₅₃	#41	1	1	yes	characteristic
67	C₂²⋊Dic₅₃	—	2	2	no	—
68	C₄⋊Dic₅₃	—	2	1	no	—
69	C₄⋊Dic₅₃	—	2	1	no	—
70	C₄⋊Dic₅₃	—	2	1	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
74	#11719919	—	2	2	no	—
75	#11719919	—	2	2	no	—
76	C₄×Dic₅₃	—	2	2	no	—
77	#11719917	—	4	2	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂³×C₅₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	4	1	no	—
81	C₂×C₄×C₅₃	—	4	1	no	—
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#135	1	2	yes	—
89	C₂×C₄×C₅₃	#137	1	2	yes	—
90	C₂×Dic₅₃	—	2	2	no	—
91	C₂×Dic₅₃	—	2	2	no	—
92	C₂×Dic₅₃	—	8	2	no	—
93	C₂×Dic₅₃	—	8	2	no	—
94	C₂×Dic₅₃	—	8	2	no	—
95	Dic₁₀₆	—	4	2	no	—
96	Dic₁₀₆	—	4	2	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	1	no	—
101	C₈×C₅₃	—	8	2	no	—
102	#88322	—	4	2	no	—
103	#8679	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	#380	1	1	yes	characteristic
107	C₂²×C₅₃	#464	1	2	yes	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	#416	1	1	yes	characteristic
113	C₄×C₅₃	#419	1	1	yes	characteristic
114	C₄×C₅₃	#479	1	2	yes	—
115	Dic₅₃	—	4	2	no	—
116	Dic₅₃	—	4	2	no	—
117	Dic₅₃	—	16	2	no	—
118	#2960	—	53	1	no	—
119	#1835	—	53	1	no	—
120	#1564	D₅₃	1	1	yes	characteristic
121	#1518	—	53	2	no	—
122	#1286	—	53	2	no	—
123	C₂×C₅₃	—	2	2	no	—
124	C₂×C₅₃	#1805	1	1	yes	characteristic
125	C₂×C₅₃	#1812	1	1	yes	characteristic
126	C₂×C₅₃	#1814	1	1	yes	characteristic
127	#533	—	53	1	no	—
128	#446	—	53	2	no	—
129	#446	—	106	1	no	—
130	C₈×Q₈	—	53	2	no	—
131	C₈×Q₈	—	106	1	no	—
132	#431	C₂×D₅₃	1	2	yes	—
133	#424	—	106	1	no	—
134	#423	—	53	1	no	—
135	#405	—	106	1	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	#382	—	53	2	no	—
138	#340	—	106	1	no	—
139	#340	—	106	1	no	—
140	#335	—	2	1	no	—
141	#334	—	2	1	no	—
142	#328	—	106	4	no	—
143	C₅₃	#8679	1	1	yes	characteristic, complemented, Sylow
144	C₂×M₁₆	—	106	1	no	—
145	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
146	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
147	C₄⋊Q₈	—	53	1	no	—
148	C₄⋊Q₈	—	53	1	no	—
149	#125	—	106	1	no	—
150	C₂²⋊Q₈	—	53	2	no	—
151	C₂²⋊Q₈	—	106	1	no	—
152	C₄×Q₈	—	53	2	no	—
153	C₄×Q₈	—	106	1	no	—
154	#117	—	106	1	no	—
155	C₂×C₄⋊C₄	—	106	2	no	—
156	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	53	2	no	—
159	C₄⋊C₈	—	106	2	no	—
160	C₄⋊C₈	—	106	2	no	—
161	C₄⋊C₈	—	106	1	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	#97	—	106	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	106	2	no	—
166	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
167	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
170	#95	—	106	2	no	—
171	#95	—	106	2	no	—
172	C₂×Q₈	—	53	2	no	—
173	C₂×Q₈	—	106	1	no	—
174	C₂²×C₄	—	212	2	no	—
175	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
176	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
177	C₂²×C₄	#11719949	1	1	yes	characteristic
178	M₁₆	—	212	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	106	2	no	—
183	C₂×C₈	#11719921	1	1	yes	characteristic
184	C₂×C₈	#11719921	1	1	yes	characteristic
185	C₂×C₈	#11719921	1	1	yes	characteristic
186	C₂×C₈	#11719921	1	1	yes	characteristic
187	C₄⋊C₄	—	53	2	no	—
188	C₄⋊C₄	—	106	2	no	—
189	C₄⋊C₄	—	106	2	no	—
190	C₄⋊C₄	—	106	1	no	—
191	C₄⋊C₄	—	106	1	no	—
192	C₄⋊C₄	—	106	1	no	—
193	C₄⋊C₄	—	212	2	no	—
194	C₄⋊C₄	—	212	2	no	—
195	C₂²⋊C₄	—	106	2	no	—
196	C₄²	—	106	2	no	—
197	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₄²	#11719947	1	1	yes	characteristic
199	C₄²	Q₈×D₅₃	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
201	C₂³	#13133192	1	1	yes	characteristic
202	Q₈	—	212	2	no	—
203	Q₈	—	212	2	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	106	2	no	—
207	C₂×C₄	—	106	2	no	—
208	C₂×C₄	—	424	2	no	—
209	C₂×C₄	—	424	2	no	—
210	C₂×C₄	—	424	2	no	—
211	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
212	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₄	#13133175	1	1	yes	characteristic
214	C₂×C₄	#13133188	1	1	yes	characteristic
215	C₂×C₄	#13133193	1	1	yes	characteristic
216	C₂×C₄	#13133197	1	1	yes	characteristic
217	C₂×C₄	#13133239	1	2	yes	—
218	C₂×C₄	#13133250	1	2	yes	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	8	2	no	—
224	C₈	—	212	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#16810480	1	1	yes	characteristic
228	C₂²	#16810846	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	848	2	no	—
234	C₄	—	212	2	no	—
235	C₄	—	212	2	no	—
236	C₄	#16810620	1	1	yes	characteristic
237	C₄	#16810633	1	1	yes	characteristic
238	C₄	#16810931	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	#20274731	1	1	yes	characteristic
241	C₂	#20274758	1	1	yes	characteristic
242	C₂	#20274770	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283649

C₂

yes

maximal, characteristic

#20274846

C₂

yes

maximal, characteristic

#20273280

C₂

yes

maximal

#20272574

C₂

yes

maximal

C₅₃×#1564

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

characteristic

#16810753

—

#16810753

C₄

yes

complemented

C₈×Dic₁₀₆

—

C₈×Dic₁₀₆

C₄

yes

complemented

#16810649

—

#16810647

C₂²

yes

characteristic

#16810578

—

#16810489

C₂²

yes

—

#16810386

—

#16810386

—

#16810361

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₅₃×#334

—

C₂×#11719917

—

#13133217

Q₈

yes

characteristic

#13133217

Q₈

yes

—

—

D₄

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂×C₄

yes

—

#13133176

—

C₂×C₄⋊Dic₅₃

—

C₄⋊#88322

—

C₄⋊#88322

C₂×C₄

yes

—

#13133146

—

#13133146

—

#13133127

—

C₈×Dic₅₃

—

#13133124

—

#13133124

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₄⋊C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

100

C₈×C₅₃

—

101

C₈×C₅₃

—

102

#88322

—

103

#8679

—

maximal, Sylow

104

C₂²×C₅₃

—

105

C₂²×C₅₃

—

106

C₂²×C₅₃

#380

yes

characteristic

107

C₂²×C₅₃

#464

yes

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

#416

yes

characteristic

113

C₄×C₅₃

#419

yes

characteristic

114

C₄×C₅₃

#479

yes

—

115

Dic₅₃

—

116

Dic₅₃

—

117

Dic₅₃

—

118

#2960

—

119

#1835

—

120

#1564

D₅₃

yes

characteristic

121

#1518

—

122

#1286

—

123

C₂×C₅₃

—

124

C₂×C₅₃

#1805

yes

characteristic

125

C₂×C₅₃

#1812

yes

characteristic

126

C₂×C₅₃

#1814

yes

characteristic

127

#533

—

128

#446

—

129

#446

—

106

—

130

C₈×Q₈

—

131

C₈×Q₈

—

106

—

132

#431

C₂×D₅₃

yes

—

133

#424

—

106

—

134

#423

—

135

#405

—

106

—

136

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

137

#382

—

138

#340

—

106

—

139

#340

—

106

—

140

#335

—

141

#334

—

142

#328

—

106

—

143

C₅₃

#8679

yes

characteristic, complemented, Sylow

144

C₂×M₁₆

—

106

—

145

C₂²×C₈

D₂₁₂

yes

characteristic

146

C₂²×C₈

D₂₁₂

yes

characteristic

147

C₄⋊Q₈

—

148

C₄⋊Q₈

—

149

#125

—

106

—

150

C₂²⋊Q₈

—

151

C₂²⋊Q₈

—

106

—

152

C₄×Q₈

—

153

C₄×Q₈

—

106

—

154

#117

—

106

—

155

C₂×C₄⋊C₄

—

106

—

156

C₂×C₄²

C₂²×D₅₃

yes

characteristic

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

106

—

160

C₄⋊C₈

—

106

—

161

C₄⋊C₈

—

106

—

162

C₂²⋊C₈

—

163

#97

—

106

—

164

C₄×C₈

—

165

C₄×C₈

—

106

—

166

C₄×C₈

C₄×D₅₃

yes

characteristic

167

C₄×C₈

C₄×D₅₃

yes

characteristic

168

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

169

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

170

#95

—

106

—

171

#95

—

106

—

172

C₂×Q₈

—

173

C₂×Q₈

—

106

—

174

C₂²×C₄

—

212

—

175

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

176

C₂²×C₄

D₄×D₅₃

yes

characteristic

177

C₂²×C₄

#11719949

yes

characteristic

178

M₁₆

—

212

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

106

—

183

C₂×C₈

#11719921

yes

characteristic

184

C₂×C₈

#11719921

yes

characteristic

185

C₂×C₈

#11719921

yes

characteristic

186

C₂×C₈

#11719921

yes

characteristic

187

C₄⋊C₄

—

188

C₄⋊C₄

—

106

—

189

C₄⋊C₄

—

106

—

190

C₄⋊C₄

—

106

—

191

C₄⋊C₄

—

106

—

192

C₄⋊C₄

—

106

—

193

C₄⋊C₄

—

212

—

194

C₄⋊C₄

—

212

—

195

C₂²⋊C₄

—

106

—

196

C₄²

—

106

—

197

C₄²

C₂×C₄×D₅₃

yes

characteristic

198

C₄²

#11719947

yes

characteristic

199

C₄²

Q₈×D₅₃

yes

characteristic

200

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

201

C₂³

#13133192

yes

characteristic

202

Q₈

—

212

—

203

Q₈

—

212

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

106

—

207

C₂×C₄

—

106

—

208

C₂×C₄

—

424

—

209

C₂×C₄

—

424

—

210

C₂×C₄

—

424

—

211

C₂×C₄

C₂×#11719921

yes

characteristic

212

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

213

C₂×C₄

#13133175

yes

characteristic

214

C₂×C₄

#13133188

yes

characteristic

215

C₂×C₄

#13133193

yes

characteristic

216

C₂×C₄

#13133197

yes

characteristic

217

C₂×C₄

#13133239

yes

—

218

C₂×C₄

#13133250

yes

—

219

C₈

—

220

C₈

—

221

C₈

—

222

C₈

—

223

C₈

—

224

C₈

—

212

—

225

C₂²

—

226

C₂²

—

227

C₂²

#16810480

yes

characteristic

228

C₂²

#16810846

yes

—

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

—

848

—

234

C₄

—

212

—

235

C₄

—

212

—

236

C₄

#16810620

yes

characteristic

237

C₄

#16810633

yes

characteristic

238

C₄

#16810931

yes

—

239

C₂

—

240

C₂

#20274731

yes

characteristic

241

C₂

#20274758

yes

characteristic

242

C₂

#20274770

yes

characteristic

	subgroup	G/H
center	C₂²	#16810480
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#8679
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

subgroup

G/H

center

C₂²

#16810480

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1564

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

N	H ≅ G/N	type
8. #16810753	C₄	semidirect
10. C₈×Dic₁₀₆	C₄	semidirect
143. C₅₃	#8679	semidirect

H ≅ G/N

type

8. #16810753

C₄

semidirect

10. C₈×Dic₁₀₆

C₄

semidirect

143. C₅₃

#8679

semidirect

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

subgroup	index
#20283649	2
#20274846	2
#20273280	2
#20272574	2
C₅₃×#1564	2
#8679	53

subgroup

index

C₅₃×#1564

Subgroups — #27277322

Conjugacy classes of subgroups

Complements

Maximal subgroups