Group #27274772 — subgroups

Subgroups — #27274772

Order 13568.

514 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
2	#20273095	C₂	1	2	yes	maximal
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272551	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133223	C₄	1	2	yes	—
7	C₄×#11719945	C₂²	1	1	yes	characteristic
8	#16810578	C₄	1	2	yes	—
9	#16810577	—	2	2	no	—
10	C₄×#11719917	C₂²	1	1	yes	characteristic
11	C₄×#11719912	—	2	1	no	—
12	C₂×C₄×#88322	C₂²	1	2	yes	—
13	C₄×C₈×D₅₃	—	2	1	no	—
14	#16810353	C₄	1	4	yes	—
15	#16810350	C₄	1	4	yes	—
16	#16810344	—	2	2	no	—
17	C₅₃×#431	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#322	—	2	1	no	—
20	C₈²×C₅₃	—	2	1	no	—
21	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
22	#13133223	—	2	4	no	—
23	#13133223	C₈	1	8	yes	complemented
24	C₂×#11719917	C₂×C₄	1	2	yes	—
25	C₂×#11719912	—	2	2	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
29	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
30	C₄×C₂²⋊D₅₃	—	2	2	no	—
31	#13133176	C₂×C₄	1	1	yes	characteristic
32	#13133174	—	2	1	no	—
33	C₄²×D₅₃	—	2	1	no	—
34	#13133128	—	2	1	no	—
35	#13133127	—	2	2	no	—
36	C₄×#88322	—	2	1	no	—
37	C₄×#88322	—	2	2	no	—
38	C₄×#88322	C₂×C₄	1	4	yes	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₂²⋊C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	2	yes	—
50	C₂×C₂²⋊D₅₃	—	2	1	no	—
51	#11719945	—	2	2	no	—
52	#11719945	C₂×C₈	1	4	yes	—
53	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×C₄×D₅₃	—	2	2	no	—
56	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²⋊Dic₅₃	—	2	1	no	—
66	#11719921	—	2	2	no	—
67	C₄⋊Dic₅₃	C₄²	1	2	yes	—
68	#11719919	—	2	2	no	—
69	C₄×Dic₅₃	—	2	1	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	#11719917	—	2	2	no	—
72	#11719917	C₂×C₈	1	4	yes	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	—	2	4	no	—
75	C₂×#88322	—	4	4	no	—
76	#11719912	—	2	8	no	—
77	C₈×D₅₃	—	2	8	no	—
78	C₂³×C₅₃	#117	1	1	yes	characteristic
79	C₂²×D₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
87	C₂²⋊D₅₃	—	2	4	no	—
88	C₂×Dic₅₃	—	2	1	no	—
89	C₂×Dic₅₃	—	2	2	no	—
90	D₂₁₂	—	2	1	no	—
91	D₂₁₂	C₄×C₈	1	2	yes	—
92	C₄×D₅₃	—	2	4	no	—
93	C₄×D₅₃	—	4	4	no	—
94	Dic₁₀₆	—	2	1	no	—
95	Dic₁₀₆	C₄×C₈	1	2	yes	—
96	C₈×C₅₃	—	4	4	no	—
97	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
98	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
99	#88322	—	2	4	no	—
100	#88322	—	4	4	no	—
101	#8185	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₅₃	#431	1	2	yes	—
105	C₂×D₅₃	—	2	2	no	—
106	C₂×D₅₃	—	4	2	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
109	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
110	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
111	C₄×C₅₃	#444	1	4	yes	—
112	Dic₅₃	—	2	2	no	—
113	Dic₅₃	—	4	2	no	—
114	C₄×#131	—	53	1	no	—
115	#1476	—	53	2	no	—
116	#1450	D₅₃	1	1	yes	characteristic, complemented
117	C₂×#322	—	53	1	no	—
118	#1278	—	53	2	no	—
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
121	C₂×C₅₃	#1765	1	2	yes	—
122	D₅₃	—	4	2	no	—
123	C₂×#131	—	53	2	no	—
124	C₄×#41	—	53	1	no	—
125	#431	C₂×D₅₃	1	2	yes	—
126	#405	—	53	2	no	—
127	#405	—	106	2	no	—
128	C₄×M₁₆	—	53	1	no	—
129	C₄×M₁₆	—	106	1	no	—
130	C₂×C₄×C₈	—	53	2	no	—
131	C₂×C₄×C₈	—	106	1	no	—
132	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
133	Q₈⋊C₈	—	53	4	no	—
134	D₄⋊C₈	—	53	4	no	—
135	#322	—	2	1	no	—
136	#322	—	106	2	no	—
137	C₈²	—	2	1	no	—
138	C₅₃	#8185	1	1	yes	characteristic, complemented, Sylow
139	C₂×#41	—	53	1	no	—
140	#131	—	53	8	no	—
141	#131	—	106	4	no	—
142	C₂×M₁₆	—	53	2	no	—
143	C₂×M₁₆	—	106	2	no	—
144	C₂²×C₈	—	106	2	no	—
145	C₂²×C₈	—	106	2	no	—
146	C₂²×C₈	C₄×D₅₃	1	2	yes	—
147	C₄×Q₈	—	53	2	no	—
148	C₄×D₄	—	53	2	no	—
149	C₄×D₄	—	106	2	no	—
150	#117	—	53	1	no	—
151	#117	—	106	1	no	—
152	C₂×C₄²	—	106	1	no	—
153	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
154	C₄⋊C₈	C₄×D₅₃	1	4	yes	—
155	C₂²⋊C₈	—	2	2	no	—
156	#97	—	106	2	no	—
157	#97	—	106	1	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	53	4	no	—
160	C₄×C₈	—	106	2	no	—
161	C₄×C₈	—	106	2	no	—
162	C₄×C₈	—	106	1	no	—
163	C₄×C₈	D₂₁₂	1	2	yes	—
164	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
166	#41	—	53	4	no	—
167	#41	—	106	2	no	—
168	C₂×Q₈	—	53	1	no	—
169	C₂×D₄	—	53	1	no	—
170	C₂×D₄	—	106	1	no	—
171	C₂²×C₄	—	106	1	no	—
172	C₂²×C₄	—	106	2	no	—
173	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
174	C₂²×C₄	#11719945	1	2	yes	—
175	M₁₆	—	53	4	no	—
176	M₁₆	—	106	2	no	—
177	M₁₆	—	106	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	106	2	no	—
180	C₂×C₈	—	106	8	no	—
181	C₂×C₈	—	106	4	no	—
182	C₂×C₈	—	212	4	no	—
183	C₂×C₈	C₈×D₅₃	1	4	yes	—
184	C₂×C₈	#11719912	1	4	yes	—
185	C₂×C₈	#11719921	1	2	yes	—
186	C₂×C₈	#11719921	1	2	yes	—
187	C₄⋊C₄	—	53	2	no	—
188	C₄⋊C₄	—	106	2	no	—
189	C₂²⋊C₄	—	106	1	no	—
190	C₂²⋊C₄	—	106	2	no	—
191	C₄²	—	106	1	no	—
192	C₄²	—	106	2	no	—
193	C₄²	C₂×C₄×D₅₃	1	2	yes	—
194	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂³	—	106	1	no	—
197	C₂³	#13133174	1	1	yes	characteristic
198	Q₈	—	53	2	no	—
199	Q₈	—	106	1	no	—
200	D₄	—	53	2	no	—
201	D₄	—	106	4	no	—
202	D₄	—	106	1	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	106	1	no	—
205	C₂×C₄	—	106	4	no	—
206	C₂×C₄	—	106	2	no	—
207	C₂×C₄	—	212	4	no	—
208	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
210	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
211	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
212	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
213	C₂×C₄	C₂×#11719912	1	2	yes	—
214	C₈	—	4	4	no	—
215	C₈	—	106	4	no	—
216	C₈	—	212	4	no	—
217	C₈	#13133129	1	4	yes	—
218	C₈	#13133129	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	106	2	no	—
221	C₂²	—	212	2	no	—
222	C₂²	C₄×#11719921	1	1	yes	characteristic
223	C₂²	#16810674	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	106	2	no	—
226	C₄	—	212	2	no	—
227	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
228	C₄	C₄×#11719912	1	1	yes	characteristic
229	C₄	C₂×#13133129	1	2	yes	—
230	C₄	#16810743	1	4	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	212	2	no	—
233	C₂	C₄×#13133129	1	1	yes	characteristic
234	C₂	#20274561	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133223

C₂

yes

maximal, characteristic

#20273095

C₂

yes

maximal

C₂×#16810344

C₂

yes

maximal, characteristic, complemented

#20272551

C₂

yes

maximal, complemented

C₅₃×#1450

C₂

yes

maximal, characteristic, complemented

C₂×#13133223

C₄

yes

—

C₄×#11719945

C₂²

yes

characteristic

#16810578

C₄

yes

—

#16810577

—

C₄×#11719917

C₂²

yes

characteristic

C₄×#11719912

—

C₂×C₄×#88322

C₂²

yes

—

C₄×C₈×D₅₃

—

#16810353

C₄

yes

—

#16810350

C₄

yes

—

#16810344

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

—

C₈²×C₅₃

—

C₂×#11719945

C₂×C₄

yes

characteristic

#13133223

—

#13133223

C₈

yes

complemented

C₂×#11719917

C₂×C₄

yes

—

C₂×#11719912

—

C₂²×#88322

—

C₂×C₈×D₅₃

—

C₄×Dic₁₀₆

C₂×C₄

yes

—

C₄×D₂₁₂

C₂×C₄

yes

—

C₄×C₂²⋊D₅₃

—

#13133176

C₂×C₄

yes

characteristic

#13133174

—

C₄²×D₅₃

—

—

—

—

—

C₂×C₄

yes

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

C₂×C₈

yes

—

C₂×D₂₁₂

C₄²

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₄²

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊Dic₅₃

—

#11719921

—

C₄⋊Dic₅₃

C₄²

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

C₂×C₈

yes

—

—

—

—

—

C₈×D₅₃

—

C₂³×C₅₃

#117

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

C₄×C₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

C₄×C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₈

yes

—

C₈×C₅₃

C₂²⋊C₈

yes

—

#88322

—

100

#88322

—

101

#8185

—

maximal, Sylow

102

C₂²×C₅₃

—

103

C₂²×C₅₃

C₄×C₂²⋊C₄

yes

characteristic

104

C₂²×C₅₃

#431

yes

—

105

C₂×D₅₃

—

106

C₂×D₅₃

—

107

C₄×C₅₃

—

108

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

109

C₄×C₅₃

C₄×M₁₆

yes

characteristic

110

C₄×C₅₃

C₂×C₂²⋊C₈

yes

—

111

C₄×C₅₃

#444

yes

—

112

Dic₅₃

—

113

Dic₅₃

—

114

C₄×#131

—

115

#1476

—

116

#1450

D₅₃

yes

characteristic, complemented

117

C₂×#322

—

118

#1278

—

119

C₂×C₅₃

—

120

C₂×C₅₃

C₄×C₂²⋊C₈

yes

characteristic

121

C₂×C₅₃

#1765

yes

—

122

D₅₃

—

123

C₂×#131

—

124

C₄×#41

—

125

#431

C₂×D₅₃

yes

—

126

#405

—

127

#405

—

106

—

128

C₄×M₁₆

—

129

C₄×M₁₆

—

106

—

130

C₂×C₄×C₈

—

131

C₂×C₄×C₈

—

106

—

132

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

133

Q₈⋊C₈

—

134

D₄⋊C₈

—

135

#322

—

136

#322

—

106

—

137

C₈²

—

138

C₅₃

#8185

yes

characteristic, complemented, Sylow

139

C₂×#41

—

140

#131

—

141

#131

—

106

—

142

C₂×M₁₆

—

143

C₂×M₁₆

—

106

—

144

C₂²×C₈

—

106

—

145

C₂²×C₈

—

106

—

146

C₂²×C₈

C₄×D₅₃

yes

—

147

C₄×Q₈

—

148

C₄×D₄

—

149

C₄×D₄

—

106

—

150

#117

—

151

#117

—

106

—

152

C₂×C₄²

—

106

—

153

C₂×C₄²

C₂²×D₅₃

yes

characteristic

154

C₄⋊C₈

C₄×D₅₃

yes

—

155

C₂²⋊C₈

—

156

#97

—

106

—

157

#97

—

106

—

158

C₄×C₈

—

159

C₄×C₈

—

160

C₄×C₈

—

106

—

161

C₄×C₈

—

106

—

162

C₄×C₈

—

106

—

163

C₄×C₈

D₂₁₂

yes

—

164

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

165

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

166

#41

—

167

#41

—

106

—

168

C₂×Q₈

—

169

C₂×D₄

—

170

C₂×D₄

—

106

—

171

C₂²×C₄

—

106

—

172

C₂²×C₄

—

106

—

173

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

174

C₂²×C₄

#11719945

yes

—

175

M₁₆

—

176

M₁₆

—

106

—

177

M₁₆

—

106

—

178

C₂×C₈

—

179

C₂×C₈

—

106

—

180

C₂×C₈

—

106

—

181

C₂×C₈

—

106

—

182

C₂×C₈

—

212

—

183

C₂×C₈

C₈×D₅₃

yes

—

184

C₂×C₈

#11719912

yes

—

185

C₂×C₈

#11719921

yes

—

186

C₂×C₈

#11719921

yes

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

106

—

189

C₂²⋊C₄

—

106

—

190

C₂²⋊C₄

—

106

—

191

C₄²

—

106

—

192

C₄²

—

106

—

193

C₄²

C₂×C₄×D₅₃

yes

—

194

C₄²

C₂×D₂₁₂

yes

characteristic

195

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

196

C₂³

—

106

—

197

C₂³

#13133174

yes

characteristic

198

Q₈

—

199

Q₈

—

106

—

200

D₄

—

201

D₄

—

106

—

202

D₄

—

106

—

203

C₂×C₄

—

204

C₂×C₄

—

106

—

205

C₂×C₄

—

106

—

206

C₂×C₄

—

106

—

207

C₂×C₄

—

212

—

208

C₂×C₄

C₄²×D₅₃

yes

characteristic

209

C₂×C₄

C₂×#11719921

yes

characteristic

210

C₂×C₄

C₄×C₂²⋊D₅₃

yes

—

211

C₂×C₄

C₄×D₂₁₂

yes

—

212

C₂×C₄

C₂×C₈×D₅₃

yes

—

213

C₂×C₄

C₂×#11719912

yes

—

214

C₈

—

215

C₈

—

106

—

216

C₈

—

212

—

217

C₈

#13133129

yes

—

218

C₈

#13133129

yes

—

219

C₂²

—

220

C₂²

—

106

—

221

C₂²

—

212

—

222

C₂²

C₄×#11719921

yes

characteristic

223

C₂²

#16810674

yes

—

224

C₄

—

225

C₄

—

106

—

226

C₄

—

212

—

227

C₄

C₄×C₈×D₅₃

yes

characteristic

228

C₄

C₄×#11719912

yes

characteristic

229

C₄

C₂×#13133129

yes

—

230

C₄

#16810743

yes

—

231

C₂

—

232

C₂

—

212

—

233

C₂

C₄×#13133129

yes

characteristic

234

C₂

#20274561

yes

—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#8185
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1450

subgroup

G/H

center

C₄²

C₂×C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1450

C₂

socle

C₂²×C₅₃

C₄×C₂²⋊C₄

prime radical

C₂×#11719945

C₂×C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1450

N	H ≅ G/N	type
3. C₂×#16810344	C₂	semidirect
4. #20272551	C₂	semidirect
5. C₅₃×#1450	C₂	semidirect
23. #13133223	C₈	semidirect
116. #1450	D₅₃	semidirect
138. C₅₃	#8185	semidirect

H ≅ G/N

type

3. C₂×#16810344

C₂

semidirect

4. #20272551

C₂

semidirect

5. C₅₃×#1450

C₂

semidirect

23. #13133223

C₈

semidirect

116. #1450

D₅₃

semidirect

138. C₅₃

#8185

semidirect

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

subgroup	index
C₄×#13133223	2
#20273095	2
C₂×#16810344	2
#20272551	2
C₅₃×#1450	2
#8185	53

subgroup

index

C₅₃×#1450

Subgroups — #27274772

Conjugacy classes of subgroups

Complements

Maximal subgroups