Group #27281453 — subgroups

Subgroups — #27281453

Order 13568.

486 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 139 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	#20283796	C₂	1	1	yes	maximal, characteristic
2	#20274711	C₂	1	1	yes	maximal, characteristic
3	#20273396	C₂	1	2	yes	maximal
4	#20272618	C₂	1	2	yes	maximal
5	C₅₃×#2567	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88322	—	2	4	no	—
7	#16810676	—	2	2	no	—
8	#16810676	C₂²	1	2	yes	—
9	#16810647	C₂²	1	2	yes	—
10	C₂×#13133136	—	2	1	no	—
11	#16810392	—	2	1	no	—
12	#16810372	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#395	C₄	1	4	yes	—
16	C₅₃×#391	—	2	1	no	—
17	C₅₃×#389	—	2	2	no	—
18	C₅₃×#384	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	2	2	no	—
21	C₄⋊#88322	—	2	4	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	D₄	1	4	yes	—
25	#13133136	—	4	2	no	—
26	C₂²⋊#88322	—	2	2	no	—
27	C₄×#88322	—	2	4	no	—
28	C₄×#88322	—	2	2	no	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₄×Q₈×C₅₃	—	2	2	no	—
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	—	2	1	no	—
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#95×C₅₃	—	2	4	no	—
45	#95×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	2	4	no	—
49	C₂×Q₈×C₅₃	—	8	2	no	—
50	C₂×Q₈×C₅₃	SD₁₆	1	2	yes	—
51	C₂×Q₈×C₅₃	Q₁₆	1	2	yes	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	—	2	4	no	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	4	1	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	—	4	1	no	—
65	C₄⋊C₄×C₅₃	—	8	4	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	C₂×D₄	1	2	yes	—
70	C₄²×C₅₃	#41	1	2	yes	—
71	#11719917	—	4	2	no	—
72	#11719917	—	4	2	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Q₈×C₅₃	—	2	4	no	—
77	Q₈×C₅₃	—	2	4	no	—
78	Q₈×C₅₃	—	8	4	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	4	1	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	#103	1	4	yes	—
91	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#121	1	2	yes	—
94	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#135	1	2	yes	—
98	#88322	—	4	4	no	—
99	#88322	—	8	2	no	—
100	#9392	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	#386	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
105	C₂²×C₅₃	#416	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	4	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	4	4	no	—
110	C₄×C₅₃	—	4	2	no	—
111	C₄×C₅₃	—	8	2	no	—
112	C₄×C₅₃	#456	1	2	yes	—
113	C₄×C₅₃	#465	1	2	yes	—
114	C₄×C₅₃	#467	1	2	yes	—
115	#2982	—	53	1	no	—
116	#2567	D₅₃	1	1	yes	characteristic
117	#1801	—	53	1	no	—
118	#1548	—	53	2	no	—
119	#1303	—	53	2	no	—
120	C₂×C₅₃	—	2	1	no	—
121	C₂×C₅₃	—	2	1	no	—
122	C₂×C₅₃	#1886	1	1	yes	characteristic
123	C₂×C₅₃	#1892	1	1	yes	characteristic
124	C₂×C₅₃	#1914	1	1	yes	characteristic
125	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
127	C₈×Q₈	—	106	4	no	—
128	#431	—	53	2	no	—
129	#431	—	106	2	no	—
130	#423	—	53	2	no	—
131	C₂×#101	—	106	1	no	—
132	#395	Dic₅₃	1	4	yes	—
133	#391	—	2	1	no	—
134	#389	—	2	2	no	—
135	#384	—	2	1	no	—
136	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
137	#343	—	106	1	no	—
138	#333	—	106	4	no	—
139	C₅₃	#9392	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
141	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂×M₁₆	—	106	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	4	no	—
151	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
155	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
156	C₄⋊C₈	—	53	4	no	—
157	C₄⋊C₈	—	106	4	no	—
158	C₄⋊C₈	—	106	2	no	—
159	C₄⋊C₈	—	106	2	no	—
160	#101	—	212	2	no	—
161	C₂²⋊C₈	—	106	2	no	—
162	C₄×C₈	—	106	4	no	—
163	C₄×C₈	—	106	2	no	—
164	#95	—	2	4	no	—
165	#95	C₂×Dic₅₃	1	2	yes	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	8	2	no	—
170	C₂×Q₈	#11719924	1	2	yes	—
171	C₂×Q₈	#11719925	1	2	yes	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
175	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
179	M₁₆	—	212	2	no	—
180	M₁₆	—	212	2	no	—
181	C₂×C₈	—	106	4	no	—
182	C₂×C₈	—	212	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	8	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	4	2	no	—
193	C₄²	D₄×D₅₃	1	2	yes	—
194	C₄²	#11719947	1	2	yes	—
195	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
196	Q₈	—	2	4	no	—
197	Q₈	—	2	4	no	—
198	Q₈	—	8	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	#13133141	1	4	yes	—
211	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
212	C₂×C₄	#13133185	1	1	yes	characteristic
213	C₂×C₄	#13133189	1	2	yes	—
214	C₂×C₄	#13133198	1	1	yes	characteristic
215	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
217	C₂×C₄	#13133237	1	2	yes	—
218	C₈	—	424	2	no	—
219	C₈	—	212	4	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	#16810503	1	1	yes	characteristic
223	C₂²	C₂×#13133141	1	1	yes	characteristic
224	C₂²	#16810619	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	4	no	—
228	C₄	—	4	4	no	—
229	C₄	—	4	2	no	—
230	C₄	—	8	2	no	—
231	C₄	#16810805	1	2	yes	—
232	C₄	#16810854	1	2	yes	—
233	C₄	#16810862	1	2	yes	—
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	#20275109	1	1	yes	characteristic
237	C₂	#20275139	1	1	yes	characteristic
238	C₂	#20275263	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283796

C₂

yes

maximal, characteristic

#20274711

C₂

yes

maximal, characteristic

#20273396

C₂

yes

maximal

#20272618

C₂

yes

maximal

C₅₃×#2567

C₂

yes

maximal, characteristic

Q₈×#88322

—

#16810676

—

#16810676

C₂²

yes

—

#16810647

C₂²

yes

—

C₂×#13133136

—

#16810392

—

#16810372

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×#395

C₄

yes

—

C₅₃×#391

—

C₅₃×#389

—

C₅₃×#384

—

C₄×C₄⋊C₄×C₅₃

C₂²

yes

—

—

—

—

D₄

yes

—

—

—

—

—

C₂²×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×Q₈×C₅₃

D₄

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

SD₁₆

yes

—

C₂×Q₈×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

#88322

—

#88322

—

100

#9392

—

maximal, Sylow

101

C₂²×C₅₃

—

102

C₂²×C₅₃

—

103

C₂²×C₅₃

#386

yes

characteristic

104

C₂²×C₅₃

C₂×#103

yes

characteristic

105

C₂²×C₅₃

#416

yes

characteristic

106

C₄×C₅₃

—

107

C₄×C₅₃

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

#456

yes

—

113

C₄×C₅₃

#465

yes

—

114

C₄×C₅₃

#467

yes

—

115

#2982

—

116

#2567

D₅₃

yes

characteristic

117

#1801

—

118

#1548

—

119

#1303

—

120

C₂×C₅₃

—

121

C₂×C₅₃

—

122

C₂×C₅₃

#1886

yes

characteristic

123

C₂×C₅₃

#1892

yes

characteristic

124

C₂×C₅₃

#1914

yes

characteristic

125

C₂×C₄⋊Q₈

C₂×D₅₃

yes

characteristic

126

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

127

C₈×Q₈

—

106

—

128

#431

—

129

#431

—

106

—

130

#423

—

131

C₂×#101

—

106

—

132

#395

Dic₅₃

yes

—

133

#391

—

134

#389

—

135

#384

—

136

C₄×C₄⋊C₄

C₂×D₅₃

yes

characteristic

137

#343

—

106

—

138

#333

—

106

—

139

C₅₃

#9392

yes

characteristic, complemented, Sylow

140

C₂²×Q₈

C₂×Dic₅₃

yes

—

141

C₂²×Q₈

C₂²⋊D₅₃

yes

characteristic

142

C₂×M₁₆

—

106

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄×Q₈

—

146

C₄×Q₈

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

152

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

153

C₂×C₄²

—

154

C₂×C₄²

C₂²⋊D₅₃

yes

—

155

C₂×C₄²

C₂²×D₅₃

yes

characteristic

156

C₄⋊C₈

—

157

C₄⋊C₈

—

106

—

158

C₄⋊C₈

—

106

—

159

C₄⋊C₈

—

106

—

160

#101

—

212

—

161

C₂²⋊C₈

—

106

—

162

C₄×C₈

—

106

—

163

C₄×C₈

—

106

—

164

#95

—

165

#95

C₂×Dic₅₃

yes

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

#11719924

yes

—

171

C₂×Q₈

#11719925

yes

—

172

C₂²×C₄

—

173

C₂²×C₄

—

174

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

175

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

176

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

177

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

178

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

179

M₁₆

—

212

—

180

M₁₆

—

212

—

181

C₂×C₈

—

106

—

182

C₂×C₈

—

212

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄²

—

191

C₄²

—

192

C₄²

—

193

C₄²

D₄×D₅₃

yes

—

194

C₄²

#11719947

yes

—

195

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

196

Q₈

—

197

Q₈

—

198

Q₈

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

#13133141

yes

—

211

C₂×C₄

D₄×Dic₅₃

yes

—

212

C₂×C₄

#13133185

yes

characteristic

213

C₂×C₄

#13133189

yes

—

214

C₂×C₄

#13133198

yes

characteristic

215

C₂×C₄

C₂×#11719924

yes

characteristic

216

C₂×C₄

C₂×#11719925

yes

characteristic

217

C₂×C₄

#13133237

yes

—

218

C₈

—

424

—

219

C₈

—

212

—

220

C₂²

—

221

C₂²

—

222

C₂²

#16810503

yes

characteristic

223

C₂²

C₂×#13133141

yes

characteristic

224

C₂²

#16810619

yes

characteristic

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

#16810805

yes

—

232

C₄

#16810854

yes

—

233

C₄

#16810862

yes

—

234

C₂

—

235

C₂

—

236

C₂

#20275109

yes

characteristic

237

C₂

#20275139

yes

characteristic

238

C₂

#20275263

yes

characteristic

	subgroup	G/H
center	C₂²	#16810503
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2567	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2567	D₅₃
hypocenter	C₅₃	#9392
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2567

subgroup

G/H

center

C₂²

#16810503

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2567

C₂

socle

C₂²×C₅₃

#386

prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2567

N	H ≅ G/N	type
139. C₅₃	#9392	semidirect

H ≅ G/N

type

139. C₅₃

#9392

semidirect

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

subgroup	index
#20283796	2
#20274711	2
#20273396	2
#20272618	2
C₅₃×#2567	2
#9392	53

subgroup

index

C₅₃×#2567

Subgroups — #27281453

Conjugacy classes of subgroups

Complements

Maximal subgroups