Subgroups — #27277823

Order 13568.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272587	C₂	1	3	yes	maximal
2	#20272580	C₂	1	3	yes	maximal
3	C₅₃×#1455	C₂	1	1	yes	maximal, characteristic
4	#16810680	C₂²	1	3	yes	—
5	#16810491	C₂²	1	1	yes	characteristic
6	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
7	C₅₃×#322	C₄	1	3	yes	complemented
8	C₈²×C₅₃	C₄	1	1	yes	characteristic, complemented
9	C₄⋊#88322	—	2	3	no	—
10	C₂²⋊#88322	—	2	3	no	—
11	#13133128	—	2	3	no	—
12	#13133124	—	2	3	no	—
13	C₂×M₁₆×C₅₃	—	2	3	no	—
14	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
15	#97×C₅₃	—	2	3	no	—
16	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
17	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
18	C₂²×C₄×C₅₃	#41	1	3	yes	—
19	C₂²×Dic₅₃	—	4	1	no	—
20	M₁₆×C₅₃	—	4	6	no	—
21	C₂×C₈×C₅₃	—	2	3	no	—
22	C₂×C₈×C₅₃	—	2	3	no	—
23	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₄²×C₅₃	#41	1	3	yes	—
25	C₂×#88322	—	4	3	no	—
26	C₂³×C₅₃	#126	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	4	3	no	—
28	C₂×C₄×C₅₃	#117	1	3	yes	—
29	C₂×C₄×C₅₃	#126	1	3	yes	—
30	C₂×Dic₅₃	—	8	3	no	—
31	C₈×C₅₃	—	2	6	no	—
32	C₈×C₅₃	—	4	3	no	—
33	#88322	—	8	3	no	—
34	#8803	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	—	2	3	no	—
36	C₂²×C₅₃	#383	1	1	yes	characteristic
37	C₄×C₅₃	—	2	3	no	—
38	C₄×C₅₃	—	2	3	no	—
39	Dic₅₃	—	16	1	no	—
40	#1455	D₅₃	1	1	yes	characteristic
41	#1291	—	53	3	no	—
42	#1288	—	53	3	no	—
43	C₂×C₅₃	—	4	1	no	—
44	C₂×C₅₃	#1841	1	3	yes	—
45	#432	—	53	3	no	—
46	C₄×M₁₆	C₂×D₅₃	1	3	yes	—
47	#383	—	53	1	no	—
48	#322	Dic₅₃	1	3	yes	complemented
49	C₈²	Dic₅₃	1	1	yes	characteristic, complemented
50	C₅₃	#8803	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₁₆	—	2	3	no	—
52	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
53	C₄⋊C₈	—	106	3	no	—
54	C₂²⋊C₈	—	106	3	no	—
55	#97	—	2	3	no	—
56	#97	—	106	3	no	—
57	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
58	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
59	#95	—	106	3	no	—
60	C₂²×C₄	—	212	1	no	—
61	C₂²×C₄	#11719945	1	3	yes	—
62	M₁₆	—	4	6	no	—
63	C₂×C₈	—	2	3	no	—
64	C₂×C₈	—	2	3	no	—
65	C₂×C₈	—	212	3	no	—
66	C₄²	#11719945	1	3	yes	—
67	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
68	C₂³	#13133211	1	1	yes	characteristic
69	C₂×C₄	—	4	3	no	—
70	C₂×C₄	—	424	3	no	—
71	C₂×C₄	#13133176	1	3	yes	—
72	C₂×C₄	#13133211	1	3	yes	—
73	C₈	—	2	6	no	—
74	C₈	—	4	3	no	—
75	C₈	—	424	3	no	—
76	C₂²	—	2	3	no	—
77	C₂²	#16810491	1	1	yes	characteristic
78	C₄	—	2	3	no	—
79	C₄	—	2	3	no	—
80	C₄	—	848	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#20274864	1	3	yes	—

	subgroup	G/H
center	C₂²	#16810491
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1455	C₂
socle	C₂²×C₅₃	#383
prime radical	C₂³×C₅₃	#126
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1455	D₅₃
hypocenter	C₅₃	#8803
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
7. C₅₃×#322	C₄	semidirect
8. C₈²×C₅₃	C₄	semidirect
48. #322	Dic₅₃	semidirect
49. C₈²	Dic₅₃	semidirect
50. C₅₃	#8803	semidirect

Maximal subgroups

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

subgroup	index
#20272587	2
#20272580	2
C₅₃×#1455	2
#8803	53