Subgroups — D₅₃×#2253 #27539570

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499×D₅₃	C₂	1	2	yes	maximal
2	#20276994	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276993	C₂	1	2	yes	maximal, complemented
4	#20276992	C₂	1	2	yes	maximal, complemented
5	#20276991	C₂	1	2	yes	maximal, complemented
6	#367×D₅₃	C₂	1	2	yes	maximal
7	#366×D₅₃	C₂	1	2	yes	maximal
8	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
10	#125×D₅₃	—	2	2	no	—
11	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
13	#16811025	C₂²	1	2	yes	—
14	#16811024	C₂²	1	4	yes	—
15	#16811023	—	2	2	no	—
16	#16811022	—	2	2	no	—
17	#107×D₅₃	C₂²	1	4	yes	—
18	#106×D₅₃	—	2	2	no	—
19	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	#16810451	C₂²	1	2	yes	—
21	#16810450	C₂²	1	4	yes	—
22	#16810449	C₂²	1	2	yes	—
23	#16810448	C₂²	1	4	yes	—
24	C₄×#11719908	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#499	C₂²	1	2	yes	—
28	C₅₃×#367	C₂²	1	2	yes	—
29	C₅₃×#366	C₂²	1	2	yes	—
30	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
32	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
33	#13133209	—	2	2	no	—
34	#13133208	—	2	4	no	—
35	#13133207	—	4	4	no	—
36	C₄⋊C₄×D₅₃	—	2	4	no	—
37	C₄⋊C₄×D₅₃	—	4	4	no	—
38	C₄²×D₅₃	D₄	1	1	yes	characteristic
39	C₂×#11719908	C₂³	1	1	yes	characteristic
40	C₂×#11719908	C₂³	1	1	yes	characteristic
41	C₁₆×D₅₃	Q₈	1	4	yes	—
42	C₁₆×D₅₃	Q₈	1	4	yes	—
43	#13133151	C₂³	1	4	yes	—
44	#13133150	—	2	4	no	—
45	#13133149	—	2	2	no	—
46	#13133148	—	2	4	no	—
47	C₄×#88322	D₄	1	1	yes	characteristic
48	C₈×Dic₅₃	D₄	1	1	yes	characteristic
49	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
50	#125×C₅₃	—	2	2	no	—
51	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
53	#107×C₅₃	C₂³	1	4	yes	—
54	#106×C₅₃	—	2	2	no	—
55	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₅₃	—	4	4	no	—
57	C₂×C₄×D₅₃	D₈	1	2	yes	—
58	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	4	4	no	—
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Dic₅₃	—	2	4	no	—
65	C₄⋊Dic₅₃	—	4	4	no	—
66	#11719919	—	4	4	no	—
67	#11719919	—	4	4	no	—
68	#11719919	—	4	4	no	—
69	C₄×Dic₅₃	D₈	1	2	yes	—
70	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×#88322	C₂×D₄	1	1	yes	characteristic
72	C₂×#88322	C₂×D₄	1	1	yes	characteristic
73	C₈×D₅₃	Q₁₆	1	4	yes	—
74	C₈×D₅₃	C₂×Q₈	1	2	yes	—
75	C₈×D₅₃	C₂×Q₈	1	2	yes	—
76	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
77	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
78	#11719908	C₂×Q₈	1	2	yes	—
79	#11719908	C₂×Q₈	1	2	yes	—
80	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₅₃	—	4	4	no	—
85	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
86	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
87	C₄×D₅₃	—	2	4	no	—
88	C₄×D₅₃	—	8	8	no	—
89	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
90	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
91	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
92	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
94	#88322	C₄⋊Q₈	1	2	yes	—
95	#88322	C₂×Q₁₆	1	2	yes	—
96	C₂×#2253	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
98	C₂×D₅₃	#500	1	2	yes	—
99	C₂×D₅₃	#508	1	4	yes	—
100	C₄×C₅₃	—	2	2	no	—
101	C₄×C₅₃	—	8	4	no	—
102	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₅₃	—	2	2	no	—
105	Dic₅₃	—	8	4	no	—
106	Dic₅₃	#500	1	2	yes	—
107	C₂×#499	—	53	2	no	—
108	#2253	—	53	2	no	—
109	#2253	—	53	2	no	—
110	#2253	—	53	2	no	—
111	#2253	—	53	1	no	—
112	#2253	D₅₃	1	1	yes	characteristic, direct product factor
113	C₂×#367	—	53	2	no	—
114	C₂×#366	—	53	2	no	—
115	C₂×C₄×C₁₆	—	53	1	no	—
116	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#508	1	2	yes	—
118	D₅₃	#2253	1	4	yes	direct product factor
119	C₂×#125	—	106	2	no	—
120	C₂²×C₁₆	—	53	1	no	—
121	C₂²×C₁₆	—	53	1	no	—
122	#499	—	53	4	no	—
123	#499	—	53	2	no	—
124	#499	—	106	2	no	—
125	#499	—	106	2	no	—
126	#499	C₂×D₅₃	1	2	yes	—
127	C₂×#107	—	53	4	no	—
128	C₂×#106	—	106	2	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	#367	—	53	2	no	—
131	#367	—	53	4	no	—
132	#367	C₂×D₅₃	1	2	yes	—
133	#366	—	53	2	no	—
134	#366	—	53	4	no	—
135	#366	C₂×D₅₃	1	2	yes	—
136	C₄×C₁₆	—	53	1	no	—
137	C₄×C₁₆	—	53	1	no	—
138	C₄×C₁₆	—	53	1	no	—
139	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
140	C₅₃	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₈	—	53	1	no	—
142	C₂²×C₈	—	53	1	no	—
143	#125	—	2	2	no	—
144	#125	—	106	4	no	—
145	#125	—	106	2	no	—
146	#125	—	212	4	no	—
147	C₂×C₄⋊C₄	—	106	4	no	—
148	C₂×C₄⋊C₄	—	212	4	no	—
149	C₂×C₄²	—	53	1	no	—
150	C₂×C₁₆	—	53	1	no	—
151	C₂×C₁₆	—	53	1	no	—
152	C₂×C₁₆	—	53	4	no	—
153	C₂×C₁₆	—	53	4	no	—
154	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
156	#107	—	53	4	no	—
157	#107	—	106	4	no	—
158	#107	C₂²×D₅₃	1	4	yes	—
159	#106	—	2	2	no	—
160	#106	—	106	4	no	—
161	#106	—	106	2	no	—
162	C₄×C₈	—	53	1	no	—
163	C₄×C₈	—	53	1	no	—
164	C₄×C₈	—	53	1	no	—
165	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	2	no	—
168	C₂²×C₄	—	212	4	no	—
169	C₂×C₈	—	53	1	no	—
170	C₂×C₈	—	53	1	no	—
171	C₂×C₈	—	53	4	no	—
172	C₂×C₈	—	53	2	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	106	4	no	—
179	C₄⋊C₄	—	212	4	no	—
180	C₄⋊C₄	—	212	4	no	—
181	C₄⋊C₄	—	212	4	no	—
182	C₄⋊C₄	—	212	4	no	—
183	C₄²	—	53	2	no	—
184	C₄²	—	53	1	no	—
185	C₄²	D₄×D₅₃	1	1	yes	characteristic
186	C₁₆	—	53	2	no	—
187	C₁₆	—	53	2	no	—
188	C₁₆	Q₈×D₅₃	1	2	yes	—
189	C₁₆	Q₈×D₅₃	1	2	yes	—
190	C₂³	—	53	1	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	106	4	no	—
197	C₂×C₄	—	424	8	no	—
198	C₂×C₄	—	212	4	no	—
199	C₂×C₄	D₈×D₅₃	1	2	yes	—
200	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
201	C₈	—	53	2	no	—
202	C₈	—	53	2	no	—
203	C₈	Q₁₆×D₅₃	1	2	yes	—
204	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
205	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
206	C₂²	—	53	2	no	—
207	C₂²	—	53	4	no	—
208	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	8	4	no	—
211	C₄	—	53	2	no	—
212	C₄	—	106	2	no	—
213	C₄	—	424	4	no	—
214	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
215	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
216	C₂	—	53	4	no	—
217	C₂	#500×D₅₃	1	1	yes	characteristic
218	C₂	#508×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2253	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	#2253	D₅₃
hypocenter	C₅₃	C₂×#2253
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #20276994	C₂	semidirect
3. #20276993	C₂	semidirect
4. #20276992	C₂	semidirect
5. #20276991	C₂	semidirect
9. C₅₃×#2253	C₂	semidirect
112. #2253	D₅₃	direct (N × H)
118. D₅₃	#2253	direct (N × H)
140. C₅₃	C₂×#2253	semidirect

Maximal subgroups

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

subgroup	index
#499×D₅₃	2
#20276994	2
#20276993	2
#20276992	2
#20276991	2
#367×D₅₃	2
#366×D₅₃	2
C₄×C₁₆×D₅₃	2
C₅₃×#2253	2
C₂×#2253	53

Subgroups — D53×#2253 #27539570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2253 #27539570