Group #27301547 — subgroups

Subgroups — #27301547

Order 13568.

458 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20287818	C₂	1	1	yes	maximal, characteristic
2	#20276435	C₂	1	2	yes	maximal, complemented
3	#20274761	C₂	1	1	yes	maximal, characteristic
4	#20272816	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3059	C₂	1	1	yes	maximal, characteristic
6	#16811040	—	2	4	no	—
7	C₂×#13133157	—	2	2	no	—
8	C₂×#13133157	C₂²	1	2	yes	—
9	#16810661	C₂²	1	2	yes	—
10	#16810597	—	2	1	no	—
11	#16810427	—	2	4	no	—
12	#16810380	—	2	1	no	—
13	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#536	—	2	1	no	—
16	C₅₃×#470	C₄	1	4	yes	complemented
17	C₅₃×#467	—	2	2	no	—
18	C₅₃×#427	C₂²	1	1	yes	characteristic
19	C₅₃×#419	—	2	1	no	—
20	#13133176	—	2	2	no	—
21	#13133157	—	2	4	no	—
22	#13133157	—	2	2	no	—
23	#13133157	—	2	2	no	—
24	C₂×#11719908	—	2	4	no	—
25	C₂×#11719908	D₄	1	4	yes	—
26	#13133151	—	2	2	no	—
27	#13133149	—	2	2	no	—
28	#13133132	—	4	2	no	—
29	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×Q₈×C₅₃	—	2	1	no	—
31	C₂×Q₁₆×C₅₃	—	2	1	no	—
32	C₂×Q₁₆×C₅₃	—	2	2	no	—
33	C₂×Q₁₆×C₅₃	—	2	2	no	—
34	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	#131×C₅₃	—	2	2	no	—
36	#131×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	—	4	1	no	—
42	#124×C₅₃	—	4	1	no	—
43	#123×C₅₃	—	4	2	no	—
44	C₂²⋊Q₈×C₅₃	—	2	4	no	—
45	#117×C₅₃	—	2	1	no	—
46	#107×C₅₃	—	2	1	no	—
47	#106×C₅₃	C₂×C₄	1	2	yes	—
48	#103×C₅₃	—	2	4	no	—
49	#41×C₅₃	—	2	2	no	—
50	#41×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	—	4	1	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂×Q₈×C₅₃	SD₁₆	1	1	yes	characteristic
56	C₂×D₄×C₅₃	D₈	1	2	yes	—
57	C₂×D₄×C₅₃	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₅₃	—	2	4	no	—
62	Q₁₆×C₅₃	—	4	1	no	—
63	Q₁₆×C₅₃	—	4	1	no	—
64	M₁₆×C₅₃	—	2	2	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	2	yes	—
73	C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₄⋊C₄×C₅₃	—	4	4	no	—
75	C₂²⋊C₄×C₅₃	—	4	4	no	—
76	C₂²⋊C₄×C₅₃	—	4	1	no	—
77	C₄²×C₅₃	—	4	1	no	—
78	C₂²⋊Dic₅₃	—	4	2	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄×Dic₅₃	—	4	2	no	—
81	#11719908	—	2	4	no	—
82	C₂³×C₅₃	#102	1	2	yes	—
83	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
84	Q₈×C₅₃	—	2	2	no	—
85	Q₈×C₅₃	—	4	2	no	—
86	Q₈×C₅₃	—	4	2	no	—
87	Q₈×C₅₃	—	8	2	no	—
88	D₄×C₅₃	—	2	4	no	—
89	D₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	8	2	no	—
95	C₂×C₄×C₅₃	#102	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#121	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#135	1	2	yes	—
100	C₂×Dic₅₃	—	8	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	C₄×D₄	1	2	yes	—
103	C₈×C₅₃	#120	1	1	yes	characteristic
104	C₈×C₅₃	#123	1	1	yes	characteristic
105	#13075	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	—	4	2	no	—
109	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
110	C₂²×C₅₃	#466	1	2	yes	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	8	2	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	#386	1	1	yes	characteristic
115	C₄×C₅₃	#416	1	1	yes	characteristic
116	C₄×C₅₃	#465	1	2	yes	—
117	Dic₅₃	—	16	2	no	—
118	#3407	—	53	1	no	—
119	#3059	D₅₃	1	1	yes	characteristic
120	#2141	—	53	2	no	—
121	#1812	—	53	1	no	—
122	#1384	—	53	2	no	—
123	C₂×C₅₃	—	2	2	no	—
124	C₂×C₅₃	—	2	1	no	—
125	C₂×C₅₃	—	4	2	no	—
126	C₂×C₅₃	#1893	1	1	yes	characteristic
127	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
128	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
129	#536	—	2	1	no	—
130	#504	—	106	4	no	—
131	C₂×M₃₂	—	53	2	no	—
132	C₂×M₃₂	—	106	2	no	—
133	#470	Dic₅₃	1	4	yes	complemented
134	#467	—	2	2	no	—
135	#427	—	53	2	no	—
136	#427	C₂×D₅₃	1	1	yes	characteristic
137	#419	—	2	1	no	—
138	#410	—	106	1	no	—
139	#358	—	106	4	no	—
140	H(3,ℤ/4)	—	106	1	no	—
141	C₅₃	#13075	1	1	yes	characteristic, complemented, Sylow
142	C₂×#41	C₂²⋊D₅₃	1	1	yes	characteristic
143	C₂²×Q₈	—	2	1	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	1	no	—
147	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
148	#131	—	2	2	no	—
149	#131	—	2	2	no	—
150	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
151	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
153	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
154	C₄⋊Q₈	—	4	1	no	—
155	#124	—	4	1	no	—
156	#123	—	4	2	no	—
157	C₂²⋊Q₈	—	2	4	no	—
158	#117	—	2	1	no	—
159	#117	—	106	2	no	—
160	M₃₂	—	106	4	no	—
161	M₃₂	—	106	2	no	—
162	M₃₂	—	106	2	no	—
163	C₂×C₁₆	—	53	4	no	—
164	C₂×C₁₆	—	106	4	no	—
165	#107	—	2	1	no	—
166	#107	—	106	2	no	—
167	#106	—	106	2	no	—
168	#106	C₂×Dic₅₃	1	2	yes	—
169	#103	—	2	4	no	—
170	#99	—	212	2	no	—
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	#11719923	1	1	yes	characteristic
178	C₂×D₄	#11719922	1	2	yes	—
179	C₂×D₄	#11719923	1	1	yes	characteristic
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
182	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
183	Q₁₆	—	2	4	no	—
184	Q₁₆	—	4	1	no	—
185	Q₁₆	—	4	1	no	—
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
191	C₂×C₈	D₄×D₅₃	1	2	yes	—
192	C₂×C₈	#11719947	1	2	yes	—
193	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
194	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	212	2	no	—
198	C₂²⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	—	4	1	no	—
200	C₂²⋊C₄	—	212	2	no	—
201	C₄²	—	4	1	no	—
202	C₄²	—	212	2	no	—
203	C₁₆	—	106	4	no	—
204	C₂³	#13133138	1	2	yes	—
205	C₂³	C₂×#11719922	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	4	2	no	—
208	Q₈	—	4	2	no	—
209	Q₈	—	8	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	424	2	no	—
218	C₂×C₄	#13133138	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
220	C₂×C₄	#13133189	1	2	yes	—
221	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
222	C₂×C₄	#13133237	1	2	yes	—
223	C₈	—	2	2	no	—
224	C₈	D₄×Dic₅₃	1	2	yes	—
225	C₈	#13133185	1	1	yes	characteristic
226	C₈	#13133198	1	1	yes	characteristic
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	C₂×#13133138	1	1	yes	characteristic
231	C₂²	#16810857	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	8	2	no	—
234	C₄	—	8	2	no	—
235	C₄	—	848	2	no	—
236	C₄	#16810503	1	1	yes	characteristic
237	C₄	#16810619	1	1	yes	characteristic
238	C₄	#16810850	1	2	yes	—
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	2	no	—
242	C₂	#20275144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20287818

C₂

yes

maximal, characteristic

#20276435

C₂

yes

maximal, complemented

#20274761

C₂

yes

maximal, characteristic

#20272816

C₂

yes

maximal, complemented

C₅₃×#3059

C₂

yes

maximal, characteristic

#16811040

—

C₂×#13133157

—

C₂×#13133157

C₂²

yes

—

#16810661

C₂²

yes

—

#16810597

—

#16810427

—

#16810380

—

C₂²×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×#131×C₅₃

C₂²

yes

characteristic

C₅₃×#536

—

C₅₃×#470

C₄

yes

complemented

C₅₃×#467

—

C₅₃×#427

C₂²

yes

characteristic

C₅₃×#419

—

—

—

—

—

—

D₄

yes

—

#13133151

—

#13133149

—

#13133132

—

C₂×#41×C₅₃

D₄

yes

characteristic

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂×C₄

yes

—

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊Q₈×C₅₃

—

#124×C₅₃

—

#123×C₅₃

—

C₂²⋊Q₈×C₅₃

—

#117×C₅₃

—

#107×C₅₃

—

#106×C₅₃

C₂×C₄

yes

—

#103×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

SD₁₆

yes

characteristic

C₂×D₄×C₅₃

D₈

yes

—

C₂×D₄×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719908

—

C₂³×C₅₃

#102

yes

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

100

C₂×Dic₅₃

—

101

C₈×C₅₃

—

102

C₈×C₅₃

C₄×D₄

yes

—

103

C₈×C₅₃

#120

yes

characteristic

104

C₈×C₅₃

yes

105

—

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

—

109

C₂²×C₅₃

C₂×#102

yes

characteristic

110

C₂²×C₅₃

#466

yes

—

111

C₄×C₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

—

114

C₄×C₅₃

#386

yes

characteristic

115

C₄×C₅₃

#416

yes

characteristic

116

C₄×C₅₃

#465

yes

—

117

Dic₅₃

—

118

#3407

—

119

#3059

D₅₃

yes

characteristic

120

#2141

—

121

#1812

—

122

#1384

—

123

C₂×C₅₃

—

124

C₂×C₅₃

—

125

C₂×C₅₃

—

126

C₂×C₅₃

#1893

yes

characteristic

127

C₂²×Q₁₆

C₂×D₅₃

yes

characteristic

128

C₂×#131

C₂×D₅₃

yes

characteristic

129

#536

—

130

#504

—

106

—

131

C₂×M₃₂

—

132

C₂×M₃₂

—

106

—

133

#470

Dic₅₃

yes

complemented

134

#467

—

135

#427

—

136

#427

C₂×D₅₃

yes

characteristic

137

#419

—

138

#410

—

106

—

139

#358

—

106

—

140

H(3,ℤ/4)

—

106

—

141

C₅₃

#13075

yes

characteristic, complemented, Sylow

142

C₂×#41

C₂²⋊D₅₃

yes

characteristic

143

C₂²×Q₈

—

144

C₂×Q₁₆

—

145

C₂×Q₁₆

—

146

C₂×Q₁₆

—

147

C₂×Q₁₆

C₂×Dic₅₃

yes

—

148

#131

—

149

#131

—

150

C₂×M₁₆

C₂×Dic₅₃

yes

—

151

C₂×M₁₆

C₂²⋊D₅₃

yes

characteristic

152

C₂²×C₈

C₂²⋊D₅₃

yes

—

153

C₂²×C₈

C₂²×D₅₃

yes

characteristic

154

C₄⋊Q₈

—

155

#124

—

156

#123

—

157

C₂²⋊Q₈

—

158

#117

—

159

#117

—

106

—

160

M₃₂

—

106

—

161

M₃₂

—

106

—

162

M₃₂

—

106

—

163

C₂×C₁₆

—

164

C₂×C₁₆

—

106

—

165

#107

—

166

#107

—

106

—

167

#106

—

106

—

168

#106

C₂×Dic₅₃

yes

—

169

#103

—

170

#99

—

212

—

171

#41

—

172

#41

—

173

C₂×Q₈

—

174

C₂×Q₈

—

175

C₂×Q₈

—

176

C₂×Q₈

—

177

C₂×Q₈

#11719923

yes

characteristic

178

C₂×D₄

#11719922

yes

—

179

C₂×D₄

#11719923

yes

characteristic

180

C₂²×C₄

—

181

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

182

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

183

Q₁₆

—

184

Q₁₆

—

185

Q₁₆

—

186

M₁₆

—

187

M₁₆

—

188

C₂×C₈

—

189

C₂×C₈

—

190

C₂×C₈

C₂²⋊Dic₅₃

yes

—

191

C₂×C₈

D₄×D₅₃

yes

—

192

C₂×C₈

#11719947

yes

—

193

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

194

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

195

C₄⋊C₄

—

196

C₄⋊C₄

—

197

C₄⋊C₄

—

212

—

198

C₂²⋊C₄

—

199

C₂²⋊C₄

—

200

C₂²⋊C₄

—

212

—

201

C₄²

—

202

C₄²

—

212

—

203

C₁₆

—

106

—

204

C₂³

#13133138

yes

—

205

C₂³

C₂×#11719922

yes

characteristic

206

Q₈

—

207

Q₈

—

208

Q₈

—

209

Q₈

—

210

D₄

—

211

D₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

424

—

218

C₂×C₄

#13133138

yes

—

219

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

220

C₂×C₄

#13133189

yes

—

221

C₂×C₄

C₂×#11719923

yes

characteristic

222

C₂×C₄

#13133237

yes

—

223

C₈

—

224

C₈

D₄×Dic₅₃

yes

—

225

C₈

#13133185

yes

characteristic

226

C₈

#13133198

yes

characteristic

227

C₂²

—

228

C₂²

—

229

C₂²

—

230

C₂²

C₂×#13133138

yes

characteristic

231

C₂²

#16810857

yes

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

848

—

236

C₄

#16810503

yes

characteristic

237

C₄

#16810619

yes

characteristic

238

C₄

#16810850

yes

—

239

C₂

—

240

C₂

—

241

C₂

—

242

C₂

#20275144

yes

characteristic

	subgroup	G/H
center	C₂	#20275144
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#3059	C₂
socle	C₂×C₅₃	#1893
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3059	D₅₃
hypocenter	C₅₃	#13075
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3059

subgroup

G/H

center

C₂

#20275144

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#3059

C₂

socle

C₂×C₅₃

#1893

prime radical

C₂×#41×C₅₃

D₄

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3059

N	H ≅ G/N	type
2. #20276435	C₂	semidirect
4. #20272816	C₂	semidirect
16. C₅₃×#470	C₄	semidirect
133. #470	Dic₅₃	semidirect
141. C₅₃	#13075	semidirect

H ≅ G/N

type

2. #20276435

C₂

semidirect

4. #20272816

C₂

semidirect

16. C₅₃×#470

C₄

semidirect

133. #470

Dic₅₃

semidirect

141. C₅₃

#13075

semidirect

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

subgroup	index
#20287818	2
#20276435	2
#20274761	2
#20272816	2
C₅₃×#3059	2
#13075	53

subgroup

index

C₅₃×#3059

Subgroups — #27301547

Conjugacy classes of subgroups

Complements

Maximal subgroups