Group #27301566 — subgroups

Subgroups — #27301566

Order 13568.

486 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20284363	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810437	C₂	1	2	yes	maximal, complemented
3	#20274765	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272830	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
7	#16811187	—	2	1	no	—
8	#16810876	C₄	1	4	yes	—
9	#16810859	—	2	2	no	—
10	C₂×#13133153	C₂²	1	2	yes	—
11	#16810661	C₂²	1	1	yes	characteristic
12	#16810629	—	2	1	no	—
13	#16810607	—	2	1	no	—
14	#16810437	—	2	4	no	—
15	#16810381	—	2	1	no	—
16	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#504	—	2	4	no	—
18	C₂×M₃₂×C₅₃	—	2	2	no	—
19	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
20	C₂²×D₂₁₂	—	2	1	no	—
21	C₂×D₄₂₄	—	2	1	no	—
22	C₂×D₄₂₄	—	2	2	no	—
23	C₂×D₄₂₄	—	2	2	no	—
24	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
25	C₂×#11719917	—	2	2	no	—
26	#13133213	—	4	1	no	—
27	#13133206	—	4	1	no	—
28	#13133191	—	2	4	no	—
29	#13133185	—	4	2	no	—
30	#13133176	—	2	1	no	—
31	#13133153	—	2	2	no	—
32	#13133153	—	2	2	no	—
33	#13133151	—	2	1	no	—
34	#13133149	C₂×C₄	1	2	yes	—
35	#13133139	—	2	4	no	—
36	#13133136	—	4	1	no	—
37	#13133134	—	4	1	no	—
38	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
39	#131×C₅₃	—	2	2	no	—
40	#131×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	D₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	M₃₂×C₅₃	—	2	4	no	—
46	M₃₂×C₅₃	—	2	2	no	—
47	M₃₂×C₅₃	D₄	1	4	yes	complemented
48	C₂×C₁₆×C₅₃	—	2	4	no	—
49	C₂×C₁₆×C₅₃	—	2	2	no	—
50	C₂³×D₅₃	—	4	1	no	—
51	#41×C₅₃	—	2	2	no	—
52	#41×C₅₃	—	2	2	no	—
53	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
54	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₅₃	—	4	4	no	—
59	C₂×D₂₁₂	—	2	2	no	—
60	C₂×D₂₁₂	—	4	2	no	—
61	C₂×D₂₁₂	—	4	2	no	—
62	C₂×D₂₁₂	—	4	1	no	—
63	M₁₆×C₅₃	—	2	2	no	—
64	M₁₆×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	#41	1	2	yes	—
72	C₂²⋊Dic₅₃	—	4	1	no	—
73	#11719921	—	4	4	no	—
74	C₄⋊Dic₅₃	—	2	2	no	—
75	C₄×Dic₅₃	—	4	1	no	—
76	#11719917	—	4	2	no	—
77	#11719917	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	D₄₂₄	—	2	4	no	—
80	D₄₂₄	—	4	1	no	—
81	D₄₂₄	—	4	1	no	—
82	C₁₆×C₅₃	—	2	4	no	—
83	C₂³×C₅₃	#102	1	2	yes	—
84	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂²×D₅₃	—	4	2	no	—
86	C₂²×D₅₃	—	8	2	no	—
87	C₂²×D₅₃	—	16	2	no	—
88	Q₈×C₅₃	—	2	2	no	—
89	D₄×C₅₃	—	2	4	no	—
90	D₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	#102	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#121	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#135	1	2	yes	—
98	C₂²⋊D₅₃	—	16	4	no	—
99	C₂×Dic₅₃	—	4	2	no	—
100	D₂₁₂	—	4	2	no	—
101	D₂₁₂	—	4	2	no	—
102	D₂₁₂	—	8	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	C₄×D₄	1	2	yes	—
105	C₈×C₅₃	#120	1	1	yes	characteristic
106	C₈×C₅₃	#123	1	1	yes	characteristic
107	#88322	—	8	2	no	—
108	#13079	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	4	1	no	—
111	C₂²×C₅₃	—	4	2	no	—
112	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
113	C₂²×C₅₃	#465	1	2	yes	—
114	C₂×D₅₃	—	8	4	no	—
115	C₂×D₅₃	—	8	2	no	—
116	C₂×D₅₃	—	16	4	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	#386	1	1	yes	characteristic
119	C₄×C₅₃	#416	1	1	yes	characteristic
120	C₄×C₅₃	#466	1	2	yes	—
121	Dic₅₃	—	8	2	no	—
122	#3407	D₅₃	1	1	yes	characteristic, complemented
123	#3058	—	53	1	no	—
124	C₂×#361	—	53	2	no	—
125	#1813	—	53	1	no	—
126	#1389	—	53	2	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	—	2	2	no	—
129	C₂×C₅₃	—	4	2	no	—
130	C₂×C₅₃	#1893	1	1	yes	characteristic
131	D₅₃	—	16	2	no	—
132	C₂²×D₈	—	53	1	no	—
133	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
134	#535	—	106	1	no	—
135	#504	—	2	4	no	—
136	C₂×M₃₂	—	2	2	no	—
137	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
138	#469	—	53	4	no	—
139	#466	—	106	2	no	—
140	C₂×#108	—	53	2	no	—
141	#427	—	53	1	no	—
142	#418	—	106	1	no	—
143	#413	—	106	1	no	—
144	#361	—	106	4	no	—
145	#338	—	106	1	no	—
146	C₅₃	#13079	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	D₂₁₂	1	1	yes	characteristic
148	C₂²×D₄	—	106	1	no	—
149	C₂×D₈	—	53	2	no	—
150	C₂×D₈	—	106	2	no	—
151	C₂×D₈	—	106	2	no	—
152	C₂×D₈	—	106	1	no	—
153	#131	—	2	2	no	—
154	#131	—	2	2	no	—
155	C₂×M₁₆	—	106	2	no	—
156	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
157	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
159	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
160	#127	—	212	1	no	—
161	#124	—	212	1	no	—
162	#121	—	106	4	no	—
163	#120	—	212	2	no	—
164	#117	—	106	1	no	—
165	M₃₂	—	2	4	no	—
166	M₃₂	—	2	2	no	—
167	M₃₂	C₂²⋊D₅₃	1	4	yes	complemented
168	C₂×C₁₆	—	2	4	no	—
169	C₂×C₁₆	—	2	2	no	—
170	#108	—	106	2	no	—
171	#108	—	106	2	no	—
172	#107	—	106	1	no	—
173	#106	—	53	2	no	—
174	#102	—	106	4	no	—
175	#101	—	212	1	no	—
176	#100	—	212	1	no	—
177	C₂⁴	—	212	1	no	—
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	C₂×Q₈	D₄₂₄	1	1	yes	characteristic
181	C₂×D₄	—	106	2	no	—
182	C₂×D₄	—	212	4	no	—
183	C₂×D₄	—	212	2	no	—
184	C₂×D₄	—	212	1	no	—
185	C₂×D₄	—	212	2	no	—
186	C₂×D₄	#11719913	1	2	yes	—
187	C₂×D₄	D₄₂₄	1	1	yes	characteristic
188	C₂²×C₄	#11719921	1	2	yes	—
189	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
190	D₈	—	106	4	no	—
191	D₈	—	212	1	no	—
192	D₈	—	212	1	no	—
193	M₁₆	—	2	2	no	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	212	2	no	—
196	M₁₆	—	212	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	#11719921	1	2	yes	—
201	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₂×C₈	#11719945	1	2	yes	—
203	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
204	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₄⋊C₄	—	106	2	no	—
206	C₂²⋊C₄	—	212	1	no	—
207	C₂²⋊C₄	—	212	4	no	—
208	C₄²	—	212	1	no	—
209	C₁₆	—	2	4	no	—
210	C₂³	—	848	2	no	—
211	C₂³	—	424	2	no	—
212	C₂³	—	212	2	no	—
213	C₂³	#13133139	1	2	yes	—
214	C₂³	C₂×#11719913	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	D₄	—	2	4	no	—
217	D₄	—	2	2	no	—
218	D₄	—	848	4	no	—
219	D₄	—	424	2	no	—
220	D₄	—	212	2	no	—
221	D₄	—	212	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	212	2	no	—
225	C₂×C₄	#13133139	1	2	yes	—
226	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
227	C₂×C₄	#13133191	1	2	yes	—
228	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
229	C₂×C₄	#13133239	1	2	yes	—
230	C₈	—	2	2	no	—
231	C₈	—	424	2	no	—
232	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
233	C₈	#13133186	1	1	yes	characteristic
234	C₈	#13133200	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	848	4	no	—
239	C₂²	—	424	4	no	—
240	C₂²	—	424	2	no	—
241	C₂²	C₂×#13133139	1	1	yes	characteristic
242	C₂²	#16810852	1	2	yes	—
243	C₄	—	2	2	no	—
244	C₄	—	424	2	no	—
245	C₄	#16810502	1	1	yes	characteristic
246	C₄	#16810620	1	1	yes	characteristic
247	C₄	#16810859	1	2	yes	—
248	C₂	—	2	1	no	—
249	C₂	—	2	2	no	—
250	C₂	—	4	2	no	—
251	C₂	—	848	2	no	—
252	C₂	#20275143	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20284363

C₂

yes

maximal, characteristic

C₂×#16810437

C₂

yes

maximal, complemented

#20274765

C₂

yes

maximal, characteristic, complemented

#20272830

C₂

yes

maximal, complemented

C₅₃×#3407

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₂₄

C₂²

yes

characteristic

#16811187

—

#16810876

C₄

yes

—

#16810859

—

C₂×#13133153

C₂²

yes

—

#16810661

C₂²

yes

—

—

—

—

C₂×#131×C₅₃

C₂²

yes

characteristic

C₅₃×#504

—

C₂×M₃₂×C₅₃

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂²×D₂₁₂

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

#13133139

—

#13133136

—

#13133134

—

C₂×#41×C₅₃

D₄

yes

characteristic

#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

M₃₂×C₅₃

—

M₃₂×C₅₃

—

M₃₂×C₅₃

D₄

yes

complemented

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂³×D₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

characteristic

C₂×D₄×C₅₃

D₈

yes

characteristic

C₂×D₄×C₅₃

SD₁₆

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

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

—

C₂²⋊Dic₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

C₁₆×C₅₃

—

C₂³×C₅₃

#102

yes

—

C₂³×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×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×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

100

D₂₁₂

—

101

D₂₁₂

—

102

D₂₁₂

—

103

C₈×C₅₃

—

104

C₈×C₅₃

C₄×D₄

yes

—

105

C₈×C₅₃

#120

yes

characteristic

106

C₈×C₅₃

#123

yes

characteristic

107

#88322

—

108

#13079

—

maximal, Sylow

109

C₂²×C₅₃

—

110

C₂²×C₅₃

—

111

C₂²×C₅₃

—

112

C₂²×C₅₃

C₂×#102

yes

characteristic

113

C₂²×C₅₃

#465

yes

—

114

C₂×D₅₃

—

115

C₂×D₅₃

—

116

C₂×D₅₃

—

117

C₄×C₅₃

—

118

C₄×C₅₃

#386

yes

characteristic

119

C₄×C₅₃

#416

yes

characteristic

120

C₄×C₅₃

#466

yes

—

121

Dic₅₃

—

122

#3407

D₅₃

yes

characteristic, complemented

123

#3058

—

124

C₂×#361

—

125

#1813

—

126

#1389

—

127

C₂×C₅₃

—

128

C₂×C₅₃

—

129

C₂×C₅₃

—

130

C₂×C₅₃

#1893

yes

characteristic

131

D₅₃

—

132

C₂²×D₈

—

133

C₂×#131

C₂×D₅₃

yes

characteristic

134

#535

—

106

—

135

#504

—

136

C₂×M₃₂

—

137

C₂×M₃₂

C₂×D₅₃

yes

—

138

#469

—

139

#466

—

106

—

140

C₂×#108

—

141

#427

—

142

#418

—

106

—

143

#413

—

106

—

144

#361

—

106

—

145

#338

—

106

—

146

C₅₃

#13079

yes

characteristic, complemented, Sylow

147

C₂×#41

D₂₁₂

yes

characteristic

148

C₂²×D₄

—

106

—

149

C₂×D₈

—

150

C₂×D₈

—

106

—

151

C₂×D₈

—

106

—

152

C₂×D₈

—

106

—

153

#131

—

154

#131

—

155

C₂×M₁₆

—

106

—

156

C₂×M₁₆

C₄×D₅₃

yes

—

157

C₂×M₁₆

D₂₁₂

yes

characteristic

158

C₂²×C₈

C₂²⋊D₅₃

yes

—

159

C₂²×C₈

C₂²×D₅₃

yes

characteristic

160

#127

—

212

—

161

#124

—

212

—

162

#121

—

106

—

163

#120

—

212

—

164

#117

—

106

—

165

M₃₂

—

166

M₃₂

—

167

M₃₂

C₂²⋊D₅₃

yes

complemented

168

C₂×C₁₆

—

169

C₂×C₁₆

—

170

#108

—

106

—

171

#108

—

106

—

172

#107

—

106

—

173

#106

—

174

#102

—

106

—

175

#101

—

212

—

176

#100

—

212

—

177

C₂⁴

—

212

—

178

#41

—

179

#41

—

180

C₂×Q₈

D₄₂₄

yes

characteristic

181

C₂×D₄

—

106

—

182

C₂×D₄

—

212

—

183

C₂×D₄

—

212

—

184

C₂×D₄

—

212

—

185

C₂×D₄

—

212

—

186

C₂×D₄

#11719913

yes

—

187

C₂×D₄

D₄₂₄

yes

characteristic

188

C₂²×C₄

#11719921

yes

—

189

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

190

D₈

—

106

—

191

D₈

—

212

—

192

D₈

—

212

—

193

M₁₆

—

194

M₁₆

—

195

M₁₆

—

212

—

196

M₁₆

—

212

—

197

C₂×C₈

—

198

C₂×C₈

—

199

C₂×C₈

—

212

—

200

C₂×C₈

#11719921

yes

—

201

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

202

C₂×C₈

#11719945

yes

—

203

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

204

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

205

C₄⋊C₄

—

106

—

206

C₂²⋊C₄

—

212

—

207

C₂²⋊C₄

—

212

—

208

C₄²

—

212

—

209

C₁₆

—

210

C₂³

—

848

—

211

C₂³

—

424

—

212

C₂³

—

212

—

213

C₂³

#13133139

yes

—

214

C₂³

C₂×#11719913

yes

characteristic

215

Q₈

—

216

D₄

—

217

D₄

—

218

D₄

—

848

—

219

D₄

—

424

—

220

D₄

—

212

—

221

D₄

—

212

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

—

212

—

225

C₂×C₄

#13133139

yes

—

226

C₂×C₄

C₂×#11719921

yes

characteristic

227

C₂×C₄

#13133191

yes

—

228

C₂×C₄

C₂×D₄₂₄

yes

characteristic

229

C₂×C₄

#13133239

yes

—

230

C₈

—

231

C₈

—

424

—

232

C₈

C₄×C₂²⋊D₅₃

yes

—

233

C₈

#13133186

yes

characteristic

234

C₈

#13133200

yes

characteristic

235

C₂²

—

236

C₂²

—

237

C₂²

—

238

C₂²

—

848

—

239

C₂²

—

424

—

240

C₂²

—

424

—

241

C₂²

C₂×#13133139

yes

characteristic

242

C₂²

#16810852

yes

—

243

C₄

—

244

C₄

—

424

—

245

C₄

#16810502

yes

characteristic

246

C₄

#16810620

yes

characteristic

247

C₄

#16810859

yes

—

248

C₂

—

249

C₂

—

250

C₂

—

251

C₂

—

848

—

252

C₂

#20275143

yes

characteristic

	subgroup	G/H
center	C₂	#20275143
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#3407	C₂
socle	C₂×C₅₃	#1893
prime radical	#20284363	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3407	D₅₃
hypocenter	C₅₃	#13079
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3407

subgroup

G/H

center

C₂

#20275143

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#3407

C₂

C₂×C₅₃

C₂

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3407

N	H ≅ G/N	type
2. C₂×#16810437	C₂	semidirect
3. #20274765	C₂	semidirect
4. #20272830	C₂	semidirect
5. C₅₃×#3407	C₂	semidirect
47. M₃₂×C₅₃	D₄	semidirect
122. #3407	D₅₃	semidirect
146. C₅₃	#13079	semidirect
167. M₃₂	C₂²⋊D₅₃	semidirect

H ≅ G/N

type

2. C₂×#16810437

C₂

semidirect

3. #20274765

C₂

semidirect

4. #20272830

C₂

semidirect

5. C₅₃×#3407

C₂

semidirect

47. M₃₂×C₅₃

D₄

semidirect

122. #3407

D₅₃

semidirect

146. C₅₃

#13079

semidirect

167. M₃₂

C₂²⋊D₅₃

semidirect

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

subgroup	index
#20284363	2
C₂×#16810437	2
#20274765	2
#20272830	2
C₅₃×#3407	2
#13079	53

subgroup

index

C₅₃×#3407

Subgroups — #27301566

Conjugacy classes of subgroups

Complements

Maximal subgroups