Group #27277368 — subgroups

Subgroups — #27277368

Order 13568.

346 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	#20273406	C₂	1	2	yes	maximal
4	#20272589	C₂	1	2	yes	maximal
5	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	1	yes	characteristic
7	#16810753	—	2	1	no	—
8	#16810753	C₄	1	2	yes	—
9	C₈×Dic₁₀₆	—	2	1	no	—
10	C₈×Dic₁₀₆	C₄	1	2	yes	—
11	#16810649	—	2	1	no	—
12	#16810647	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	#16810578	—	2	1	no	—
15	#16810376	—	2	1	no	—
16	#16810376	—	2	1	no	—
17	#16810371	—	2	2	no	—
18	#16810368	—	2	2	no	—
19	C₅₃×#423	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#341	—	2	1	no	—
22	C₅₃×#341	—	2	1	no	—
23	C₂×#11719917	—	2	1	no	—
24	C₂²×#88322	—	2	2	no	—
25	#13133217	D₄	1	1	yes	characteristic
26	#13133217	D₄	1	1	yes	characteristic
27	#13133209	—	2	1	no	—
28	#13133196	—	2	1	no	—
29	#13133196	Q₈	1	2	yes	—
30	C₄×Dic₁₀₆	—	2	1	no	—
31	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
32	#13133176	—	2	1	no	—
33	C₄⋊#88322	—	2	1	no	—
34	C₄⋊#88322	—	2	2	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	C₂×C₄	1	2	yes	—
37	#13133146	—	2	2	no	—
38	#13133146	—	2	2	no	—
39	#13133127	—	2	2	no	—
40	C₈×Dic₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	2	no	—
46	C₄⋊C₈×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×Dic₁₀₆	—	2	1	no	—
55	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
56	M₁₆×C₅₃	—	4	2	no	—
57	M₁₆×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	#41	1	1	yes	characteristic
69	C₂²⋊Dic₅₃	—	2	2	no	—
70	C₄⋊Dic₅₃	—	2	1	no	—
71	C₄⋊Dic₅₃	—	2	1	no	—
72	C₄⋊Dic₅₃	—	2	1	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	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	8	2	no	—
81	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	4	1	no	—
85	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#135	1	2	yes	—
92	C₂×C₄×C₅₃	#137	1	2	yes	—
93	C₂×Dic₅₃	—	2	2	no	—
94	C₂×Dic₅₃	—	2	2	no	—
95	Dic₁₀₆	—	4	1	no	—
96	Dic₁₀₆	—	4	1	no	—
97	Dic₁₀₆	—	4	2	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	1	no	—
101	C₈×C₅₃	—	2	1	no	—
102	C₈×C₅₃	—	8	2	no	—
103	#88322	—	4	2	no	—
104	#88322	—	8	2	no	—
105	#8689	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	#380	1	1	yes	characteristic
109	C₂²×C₅₃	#479	1	2	yes	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	#416	1	1	yes	characteristic
114	C₄×C₅₃	#419	1	1	yes	characteristic
115	C₄×C₅₃	#464	1	2	yes	—
116	C₄×C₅₃	#472	1	1	yes	characteristic
117	C₄×C₅₃	#473	1	1	yes	characteristic
118	Dic₅₃	—	4	2	no	—
119	Dic₅₃	—	4	2	no	—
120	#2960	—	53	1	no	—
121	#1837	D₅₃	1	1	yes	characteristic
122	C₂×#335	—	53	1	no	—
123	#1550	—	53	2	no	—
124	#1292	—	53	2	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	#1805	1	1	yes	characteristic
127	C₂×C₅₃	#1813	1	1	yes	characteristic
128	C₂×C₅₃	#1816	1	1	yes	characteristic
129	#533	—	53	1	no	—
130	#446	—	53	2	no	—
131	#446	—	106	1	no	—
132	C₈×Q₈	—	53	2	no	—
133	C₈×Q₈	—	106	1	no	—
134	#424	—	106	1	no	—
135	#423	—	53	1	no	—
136	#423	C₂×D₅₃	1	2	yes	—
137	C₂×C₄⋊C₈	—	53	2	no	—
138	#405	—	106	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	#341	—	2	1	no	—
141	#341	—	2	1	no	—
142	#335	—	106	1	no	—
143	#335	—	106	1	no	—
144	#333	—	106	2	no	—
145	#332	—	106	2	no	—
146	C₅₃	#8689	1	1	yes	characteristic, complemented, Sylow
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	106	1	no	—
149	C₂²×C₈	—	106	2	no	—
150	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	53	1	no	—
153	C₄⋊Q₈	—	53	1	no	—
154	#125	—	106	1	no	—
155	C₂²⋊Q₈	—	53	2	no	—
156	C₂²⋊Q₈	—	106	1	no	—
157	C₄×Q₈	—	53	2	no	—
158	C₄×Q₈	—	106	1	no	—
159	#117	—	106	1	no	—
160	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	53	2	no	—
164	C₄⋊C₈	—	106	2	no	—
165	C₄⋊C₈	—	106	2	no	—
166	C₄⋊C₈	—	106	1	no	—
167	C₄⋊C₈	—	106	2	no	—
168	C₄⋊C₈	—	106	2	no	—
169	#97	—	106	2	no	—
170	C₄×C₈	—	106	2	no	—
171	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
172	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
173	C₄×C₈	D₂₁₂	1	1	yes	characteristic
174	C₄×C₈	D₂₁₂	1	1	yes	characteristic
175	C₂×Q₈	—	53	2	no	—
176	C₂×Q₈	—	106	1	no	—
177	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
178	C₂²×C₄	#11719949	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	4	2	no	—
182	M₁₆	—	212	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	106	2	no	—
187	C₂×C₈	—	424	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	#11719921	1	1	yes	characteristic
190	C₂×C₈	#11719921	1	1	yes	characteristic
191	C₂×C₈	#11719921	1	1	yes	characteristic
192	C₂×C₈	#11719921	1	1	yes	characteristic
193	C₄⋊C₄	—	53	2	no	—
194	C₄⋊C₄	—	106	2	no	—
195	C₄⋊C₄	—	106	1	no	—
196	C₄⋊C₄	—	106	1	no	—
197	C₄⋊C₄	—	106	1	no	—
198	C₄⋊C₄	—	106	2	no	—
199	C₂²⋊C₄	—	106	2	no	—
200	C₄²	—	106	2	no	—
201	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
203	C₄²	D₄×D₅₃	1	1	yes	characteristic
204	C₄²	#11719947	1	1	yes	characteristic
205	C₂³	#13133193	1	1	yes	characteristic
206	Q₈	—	212	2	no	—
207	Q₈	—	212	1	no	—
208	Q₈	—	212	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	106	2	no	—
213	C₂×C₄	—	106	2	no	—
214	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
215	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
216	C₂×C₄	#13133175	1	1	yes	characteristic
217	C₂×C₄	#13133188	1	1	yes	characteristic
218	C₂×C₄	#13133192	1	1	yes	characteristic
219	C₂×C₄	#13133197	1	1	yes	characteristic
220	C₂×C₄	#13133237	1	2	yes	—
221	C₂×C₄	#13133247	1	2	yes	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	8	2	no	—
227	C₈	—	424	2	no	—
228	C₈	—	212	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#16810480	1	1	yes	characteristic
232	C₂²	#16810927	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	212	2	no	—
237	C₄	—	212	2	no	—
238	C₄	#16810617	1	1	yes	characteristic
239	C₄	#16810631	1	1	yes	characteristic
240	C₄	#16810842	1	2	yes	—
241	C₄	#16810894	1	1	yes	characteristic
242	C₄	#16810899	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#20274727	1	1	yes	characteristic
245	C₂	#20274762	1	1	yes	characteristic
246	C₂	#20274776	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283649

C₂

yes

maximal, characteristic

C₂×#16810376

C₂

yes

maximal, characteristic

#20273406

C₂

yes

maximal

#20272589

C₂

yes

maximal

C₅₃×#1837

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

characteristic

#16810753

—

#16810753

C₄

yes

—

C₈×Dic₁₀₆

—

C₈×Dic₁₀₆

C₄

yes

—

#16810649

—

#16810647

C₂²

yes

characteristic

C₂×C₄⋊#88322

C₂²

yes

—

—

—

—

—

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#341

—

C₅₃×#341

—

C₂×#11719917

—

C₂²×#88322

—

#13133217

D₄

yes

characteristic

#13133217

D₄

yes

—

—

Q₈

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

#13133146

—

#13133146

—

#13133127

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

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

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₄⋊C₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

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×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₄⋊C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

—

—

—

—

C₂³×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

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

yes

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

100

C₈×C₅₃

—

101

C₈×C₅₃

—

102

C₈×C₅₃

—

103

#88322

—

104

#88322

—

105

#8689

—

maximal, Sylow

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

#380

yes

characteristic

109

C₂²×C₅₃

#479

yes

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

#416

yes

characteristic

114

C₄×C₅₃

#419

yes

characteristic

115

C₄×C₅₃

#464

yes

—

116

C₄×C₅₃

#472

yes

characteristic

117

C₄×C₅₃

#473

yes

characteristic

118

Dic₅₃

—

119

Dic₅₃

—

120

#2960

—

121

#1837

D₅₃

yes

characteristic

122

C₂×#335

—

123

#1550

—

124

#1292

—

125

C₂×C₅₃

—

126

C₂×C₅₃

#1805

yes

characteristic

127

C₂×C₅₃

#1813

yes

characteristic

128

C₂×C₅₃

#1816

yes

characteristic

129

#533

—

130

#446

—

131

#446

—

106

—

132

C₈×Q₈

—

133

C₈×Q₈

—

106

—

134

#424

—

106

—

135

#423

—

136

#423

C₂×D₅₃

yes

—

137

C₂×C₄⋊C₈

—

138

#405

—

106

—

139

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

140

#341

—

141

#341

—

142

#335

—

106

—

143

#335

—

106

—

144

#333

—

106

—

145

#332

—

106

—

146

C₅₃

#8689

yes

characteristic, complemented, Sylow

147

C₂×M₁₆

—

148

C₂×M₁₆

—

106

—

149

C₂²×C₈

—

106

—

150

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

151

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

152

C₄⋊Q₈

—

153

C₄⋊Q₈

—

154

#125

—

106

—

155

C₂²⋊Q₈

—

156

C₂²⋊Q₈

—

106

—

157

C₄×Q₈

—

158

C₄×Q₈

—

106

—

159

#117

—

106

—

160

C₂×C₄²

C₂²×D₅₃

yes

characteristic

161

C₄⋊C₈

—

162

C₄⋊C₈

—

163

C₄⋊C₈

—

164

C₄⋊C₈

—

106

—

165

C₄⋊C₈

—

106

—

166

C₄⋊C₈

—

106

—

167

C₄⋊C₈

—

106

—

168

C₄⋊C₈

—

106

—

169

#97

—

106

—

170

C₄×C₈

—

106

—

171

C₄×C₈

C₄×D₅₃

yes

characteristic

172

C₄×C₈

C₄×D₅₃

yes

characteristic

173

C₄×C₈

D₂₁₂

yes

characteristic

174

C₄×C₈

D₂₁₂

yes

characteristic

175

C₂×Q₈

—

176

C₂×Q₈

—

106

—

177

C₂²×C₄

Q₈×D₅₃

yes

characteristic

178

C₂²×C₄

#11719949

yes

characteristic

179

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

180

M₁₆

—

181

M₁₆

—

182

M₁₆

—

212

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

—

106

—

187

C₂×C₈

—

424

—

188

C₂×C₈

—

212

—

189

C₂×C₈

#11719921

yes

characteristic

190

C₂×C₈

#11719921

yes

characteristic

191

C₂×C₈

#11719921

yes

characteristic

192

C₂×C₈

#11719921

yes

characteristic

193

C₄⋊C₄

—

194

C₄⋊C₄

—

106

—

195

C₄⋊C₄

—

106

—

196

C₄⋊C₄

—

106

—

197

C₄⋊C₄

—

106

—

198

C₄⋊C₄

—

106

—

199

C₂²⋊C₄

—

106

—

200

C₄²

—

106

—

201

C₄²

C₂×C₄×D₅₃

yes

characteristic

202

C₄²

C₂×D₂₁₂

yes

characteristic

203

C₄²

D₄×D₅₃

yes

characteristic

204

C₄²

#11719947

yes

characteristic

205

C₂³

#13133193

yes

characteristic

206

Q₈

—

212

—

207

Q₈

—

212

—

208

Q₈

—

212

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

106

—

213

C₂×C₄

—

106

—

214

C₂×C₄

C₂×#11719921

yes

characteristic

215

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

216

C₂×C₄

#13133175

yes

characteristic

217

C₂×C₄

#13133188

yes

characteristic

218

C₂×C₄

#13133192

yes

characteristic

219

C₂×C₄

#13133197

yes

characteristic

220

C₂×C₄

#13133237

yes

—

221

C₂×C₄

#13133247

yes

—

222

C₈

—

223

C₈

—

224

C₈

—

225

C₈

—

226

C₈

—

227

C₈

—

424

—

228

C₈

—

212

—

229

C₂²

—

230

C₂²

—

231

C₂²

#16810480

yes

characteristic

232

C₂²

#16810927

yes

—

233

C₄

—

234

C₄

—

235

C₄

—

236

C₄

—

212

—

237

C₄

—

212

—

238

C₄

#16810617

yes

characteristic

239

C₄

#16810631

yes

characteristic

240

C₄

#16810842

yes

—

241

C₄

#16810894

yes

characteristic

242

C₄

#16810899

yes

characteristic

243

C₂

—

244

C₂

#20274727

yes

characteristic

245

C₂

#20274762

yes

characteristic

246

C₂

#20274776

yes

characteristic

	subgroup	G/H
center	C₂²	#16810480
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1837	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1837	D₅₃
hypocenter	C₅₃	#8689
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#16810480

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1837

C₂

socle

C₂²×C₅₃

#380

prime radical

C₂³×C₅₃

C₂²⋊Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
146. C₅₃	#8689	semidirect

H ≅ G/N

type

146. C₅₃

#8689

semidirect

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

subgroup	index
#20283649	2
C₂×#16810376	2
#20273406	2
#20272589	2
C₅₃×#1837	2
#8689	53

subgroup

index

C₅₃×#1837

Subgroups — #27277368

Conjugacy classes of subgroups

Complements

Maximal subgroups