Group #27277012 — subgroups

Subgroups — #27277012

Order 13568.

486 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20274707	C₂	1	2	yes	maximal
2	#20273471	C₂	1	2	yes	maximal, complemented
3	#20272590	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810647	C₂²	1	4	yes	—
6	C₂×#13133134	—	2	2	no	—
7	#16810392	—	2	2	no	—
8	#16810374	—	2	2	no	—
9	#16810374	—	2	2	no	—
10	C₂³×C₈×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₈×D₄×C₅₃	—	2	2	no	—
13	C₈×D₄×C₅₃	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719917	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	—	2	4	no	—
20	#13133134	—	4	4	no	—
21	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	4	4	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×D₄×C₅₃	—	2	2	no	—
30	C₄×D₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₂²⋊C₈×C₅₃	—	2	4	no	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	D₄	1	2	yes	—
38	C₂⁴×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂×D₄×C₅₃	—	2	2	no	—
40	C₂×D₄×C₅₃	—	2	2	no	—
41	C₂×D₄×C₅₃	SD₁₆	1	4	yes	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	4	1	no	—
53	C₂×C₈×C₅₃	—	4	4	no	—
54	C₂×C₈×C₅₃	—	4	4	no	—
55	C₂×C₈×C₅₃	—	4	2	no	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₂²⋊C₄×C₅₃	—	2	4	no	—
61	C₄²×C₅₃	C₂×D₄	1	2	yes	—
62	C₄²×C₅₃	#41	1	2	yes	—
63	#11719917	—	4	4	no	—
64	#11719917	—	4	4	no	—
65	C₂×#88322	—	4	4	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	—	4	2	no	—
69	C₂³×C₅₃	—	4	2	no	—
70	C₂³×C₅₃	#106	1	4	yes	—
71	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	D₄×C₅₃	—	2	8	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	4	1	no	—
77	C₂×C₄×C₅₃	—	4	4	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#121	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₅₃	#135	1	2	yes	—
87	C₈×C₅₃	—	2	1	no	—
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	—	4	2	no	—
90	C₈×C₅₃	#100	1	2	yes	complemented
91	#88322	—	8	4	no	—
92	#8614	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	4	no	—
96	C₂²×C₅₃	—	4	4	no	—
97	C₂²×C₅₃	—	4	4	no	—
98	C₂²×C₅₃	—	4	2	no	—
99	C₂²×C₅₃	—	4	2	no	—
100	C₂²×C₅₃	#380	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
102	C₂²×C₅₃	#427	1	1	yes	characteristic
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	4	2	no	—
106	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
107	C₄×C₅₃	#413	1	1	yes	characteristic
108	C₄×C₅₃	#465	1	4	yes	—
109	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
110	#1800	—	53	2	no	—
111	#1566	—	53	2	no	—
112	#1292	—	53	2	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	—	4	4	no	—
116	C₂×C₅₃	#1800	1	1	yes	characteristic
117	C₂×C₅₃	#1807	1	1	yes	characteristic
118	C₂×C₅₃	#1813	1	1	yes	characteristic
119	C₂³×C₈	Dic₅₃	1	2	yes	—
120	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
121	C₈×D₄	—	2	2	no	—
122	C₈×D₄	—	2	2	no	—
123	#423	—	53	4	no	—
124	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#100	—	106	2	no	—
126	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
127	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
128	#343	—	106	2	no	—
129	#334	—	106	2	no	—
130	#334	—	106	2	no	—
131	C₅₃	#8614	1	1	yes	characteristic, complemented, Sylow
132	C₂²×D₄	D₂₁₂	1	1	yes	characteristic
133	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
134	C₂×M₁₆	—	106	4	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	4	4	no	—
138	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
140	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
141	C₄×D₄	—	2	2	no	—
142	C₄×D₄	—	2	2	no	—
143	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
145	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	106	4	no	—
148	C₄⋊C₈	—	106	4	no	—
149	#100	—	212	4	no	—
150	C₂²⋊C₈	—	2	4	no	—
151	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
152	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
153	C₂⁴	C₄⋊Dic₅₃	1	2	yes	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	#11719913	1	4	yes	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
162	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
164	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
165	M₁₆	—	212	4	no	—
166	M₁₆	—	212	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	4	no	—
171	C₂×C₈	—	4	4	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	212	4	no	—
174	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
175	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₄²	#11719945	1	2	yes	—
180	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	#13133149	1	4	yes	—
186	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
187	D₄	—	2	8	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
197	C₂×C₄	#13133176	1	1	yes	characteristic
198	C₂×C₄	#13133191	1	2	yes	—
199	C₂×C₄	#13133196	1	2	yes	—
200	C₂×C₄	C₂×#11719913	1	2	yes	—
201	C₂×C₄	#13133239	1	2	yes	—
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	424	4	no	—
206	C₈	#13133134	1	2	yes	complemented
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	#16810483	1	1	yes	characteristic
215	C₂²	C₂×#13133149	1	1	yes	characteristic
216	C₂²	#16810661	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	2	no	—
220	C₄	C₂×#13133134	1	1	yes	characteristic
221	C₄	#16810607	1	1	yes	characteristic
222	C₄	#16810852	1	4	yes	—
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	4	4	no	—
226	C₂	#20274707	1	1	yes	characteristic
227	C₂	#20274740	1	1	yes	characteristic
228	C₂	#20274765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274707

C₂

yes

maximal

#20273471

C₂

yes

maximal, complemented

#20272590

C₂

yes

maximal, complemented

C₂×C₈×D₄×C₅₃

C₂

yes

maximal, characteristic

#16810647

C₂²

yes

—

—

—

—

—

C₂³×C₈×C₅₃

C₄

yes

—

C₂×C₄×D₄×C₅₃

C₂²

yes

characteristic

C₈×D₄×C₅₃

—

C₈×D₄×C₅₃

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

—

C₂²×D₄×C₅₃

D₄

yes

characteristic

C₂³×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂⁴×C₅₃

C₄⋊C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

SD₁₆

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#106

yes

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

D₄×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×C₂²⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

100

C₂²×C₅₃

#380

yes

characteristic

101

C₂²×C₅₃

C₂×#106

yes

characteristic

102

C₂²×C₅₃

#427

yes

characteristic

103

C₄×C₅₃

—

104

C₄×C₅₃

—

105

C₄×C₅₃

—

106

C₄×C₅₃

C₂×#100

yes

characteristic

107

C₄×C₅₃

#413

yes

characteristic

108

C₄×C₅₃

#465

yes

—

109

C₂×C₈×D₄

D₅₃

yes

characteristic

110

#1800

—

111

#1566

—

112

#1292

—

113

C₂×C₅₃

—

114

C₂×C₅₃

—

115

C₂×C₅₃

—

116

C₂×C₅₃

#1800

yes

characteristic

117

C₂×C₅₃

#1807

yes

characteristic

118

C₂×C₅₃

#1813

yes

characteristic

119

C₂³×C₈

Dic₅₃

yes

—

120

C₂×C₄×D₄

C₂×D₅₃

yes

characteristic

121

C₈×D₄

—

122

C₈×D₄

—

123

#423

—

124

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

125

C₂×#100

—

106

—

126

C₂×C₂²⋊C₈

Dic₅₃

yes

—

127

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

128

#343

—

106

—

129

#334

—

106

—

130

#334

—

106

—

131

C₅₃

#8614

yes

characteristic, complemented, Sylow

132

C₂²×D₄

D₂₁₂

yes

characteristic

133

C₂³×C₄

C₂×Dic₅₃

yes

—

134

C₂×M₁₆

—

106

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

138

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

139

C₂²×C₈

C₂×Dic₅₃

yes

—

140

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

141

C₄×D₄

—

142

C₄×D₄

—

143

C₂×C₄⋊C₄

D₂₁₂

yes

characteristic

144

C₂×C₂²⋊C₄

Dic₁₀₆

yes

—

145

C₂×C₄²

C₂²×D₅₃

yes

characteristic

146

C₄⋊C₈

—

147

C₄⋊C₈

—

106

—

148

C₄⋊C₈

—

106

—

149

#100

—

212

—

150

C₂²⋊C₈

—

151

C₄×C₈

C₂²⋊D₅₃

yes

—

152

C₄×C₈

C₂²⋊D₅₃

yes

—

153

C₂⁴

C₄⋊Dic₅₃

yes

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

#11719913

yes

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

C₄⋊Dic₅₃

yes

—

162

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

163

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

164

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

165

M₁₆

—

212

—

166

M₁₆

—

212

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

212

—

174

C₂×C₈

C₂²⋊Dic₅₃

yes

—

175

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

176

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

177

C₄⋊C₄

—

178

C₂²⋊C₄

—

179

C₄²

#11719945

yes

—

180

C₄²

C₂×C₂²⋊D₅₃

yes

—

181

C₂³

—

182

C₂³

—

183

C₂³

—

184

C₂³

—

185

C₂³

#13133149

yes

—

186

C₂³

C₂×C₄⋊Dic₅₃

yes

characteristic

187

D₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

197

C₂×C₄

#13133176

yes

characteristic

198

C₂×C₄

#13133191

yes

—

199

C₂×C₄

#13133196

yes

—

200

C₂×C₄

C₂×#11719913

yes

—

201

C₂×C₄

#13133239

yes

—

202

C₈

—

203

C₈

—

204

C₈

—

205

C₈

—

424

—

206

C₈

#13133134

yes

complemented

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

214

C₂²

#16810483

yes

characteristic

215

C₂²

C₂×#13133149

yes

characteristic

216

C₂²

#16810661

yes

characteristic

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

C₂×#13133134

yes

characteristic

221

C₄

#16810607

yes

characteristic

222

C₄

#16810852

yes

—

223

C₂

—

224

C₂

—

225

C₂

—

226

C₂

#20274707

yes

characteristic

227

C₂

#20274740

yes

characteristic

228

C₂

#20274765

yes

characteristic

	subgroup	G/H
center	C₂²	#16810483
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#8614
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄

subgroup

G/H

center

C₂²

#16810483

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₈×D₄×C₅₃

C₂

socle

C₂²×C₅₃

#380

prime radical

C₂²×D₄×C₅₃

D₄

cosocle

C₂×C₄²×C₅₃

C₂³

hypercenter

C₂×C₈×D₄

D₅₃

hypocenter

C₅₃

#8614

LCS

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄

N	H ≅ G/N	type
2. #20273471	C₂	semidirect
3. #20272590	C₂	semidirect
90. C₈×C₅₃	#100	semidirect
131. C₅₃	#8614	semidirect
206. C₈	#13133134	semidirect

H ≅ G/N

type

2. #20273471

C₂

semidirect

3. #20272590

C₂

semidirect

90. C₈×C₅₃

#100

semidirect

131. C₅₃

#8614

semidirect

206. C₈

#13133134

semidirect

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

subgroup	index
#20274707	2
#20273471	2
#20272590	2
C₂×C₈×D₄×C₅₃	2
#8614	53

subgroup

index

C₂×C₈×D₄×C₅₃

Subgroups — #27277012

Conjugacy classes of subgroups

Complements

Maximal subgroups