Subgroups — C₃×Dic₃×C₁₉×C₅₃ #49136262

Order 36252.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₁₉×C₅₃	C₂	1	1	yes	maximal, characteristic
2	Dic₃×C₁₉×C₅₃	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₄×C₁₉×C₅₃	—	3	1	no	maximal
4	C₃²×C₁₉×C₅₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₁₉×C₅₃	—	2	1	no	—
6	C₂×C₃×C₁₉×C₅₃	D₃	1	1	yes	characteristic
7	C₂×C₃×C₁₉×C₅₃	C₂×C₃	1	1	yes	characteristic
8	C₄×C₁₉×C₅₃	—	3	1	no	Hall
9	C₃×C₁₉×C₅₃	—	2	1	no	—
10	C₃×C₁₉×C₅₃	Dic₃	1	1	yes	characteristic, direct product factor
11	C₃×C₁₉×C₅₃	C₃×C₄	1	1	yes	characteristic, complemented
12	C₂×C₁₉×C₅₃	C₃×D₃	1	1	yes	characteristic
13	C₃×Dic₃×C₅₃	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₁₉×C₅₃	C₃×Dic₃	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₃²×C₅₃	C₂×C₁₉	1	1	yes	characteristic
16	C₃×Dic₃×C₁₉	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₃×C₄×C₅₃	—	3	1	no	—
18	Dic₃×C₅₃	C₃×C₁₉	1	1	yes	characteristic, direct product factor
19	C₃²×C₅₃	C₄×C₁₉	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃²×C₁₉	C₂×C₅₃	1	1	yes	characteristic
21	C₂×C₃×C₅₃	—	2	1	no	—
22	C₂×C₃×C₅₃	D₃×C₁₉	1	1	yes	characteristic
23	C₂×C₃×C₅₃	C₂×C₃×C₁₉	1	1	yes	characteristic
24	C₃×C₄×C₁₉	—	3	1	no	—
25	Dic₃×C₁₉	C₃×C₅₃	1	1	yes	characteristic, direct product factor
26	C₄×C₅₃	—	3	1	no	Hall
27	C₃²×C₁₉	C₄×C₅₃	1	1	yes	characteristic, complemented, Hall
28	C₃×C₅₃	—	2	1	no	—
29	C₃×C₅₃	Dic₃×C₁₉	1	1	yes	characteristic, direct product factor
30	C₃×C₅₃	C₃×C₄×C₁₉	1	1	yes	characteristic, complemented
31	C₂×C₃×C₁₉	—	2	1	no	—
32	C₂×C₃×C₁₉	D₃×C₅₃	1	1	yes	characteristic
33	C₂×C₃×C₁₉	C₂×C₃×C₅₃	1	1	yes	characteristic
34	C₂×C₅₃	C₃×D₃×C₁₉	1	1	yes	characteristic
35	C₄×C₁₉	—	3	1	no	Hall
36	C₃×C₁₉	—	2	1	no	—
37	C₃×C₁₉	Dic₃×C₅₃	1	1	yes	characteristic, direct product factor
38	C₃×C₁₉	C₃×C₄×C₅₃	1	1	yes	characteristic, complemented
39	C₅₃	C₃×Dic₃×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₁₉	C₃×D₃×C₅₃	1	1	yes	characteristic
41	C₃×Dic₃	C₁₉×C₅₃	1	1	yes	characteristic, direct product factor, Hall
42	C₁₉	C₃×Dic₃×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃²	C₂×C₁₉×C₅₃	1	1	yes	characteristic
44	C₃×C₄	—	3	1	no	—
45	Dic₃	C₃×C₁₉×C₅₃	1	1	yes	characteristic, direct product factor
46	C₃²	C₄×C₁₉×C₅₃	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	C₂×C₃×C₁₉×C₅₃	1	1	yes	characteristic
49	C₂×C₃	D₃×C₁₉×C₅₃	1	1	yes	characteristic
50	C₄	—	3	1	no	Sylow
51	C₃	—	2	1	no	—
52	C₃	C₃×C₄×C₁₉×C₅₃	1	1	yes	characteristic, complemented
53	C₃	Dic₃×C₁₉×C₅₃	1	1	yes	characteristic, direct product factor
54	C₂	C₃×D₃×C₁₉×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₉×C₅₃	D₃
commutator, hypocenter	C₃	C₃×C₄×C₁₉×C₅₃
Frattini	C₂	C₃×D₃×C₁₉×C₅₃
Fitting, socle, prime radical	C₂×C₃²×C₁₉×C₅₃	C₂
cosocle	C₂×C₃	C₂×C₃×C₁₉×C₅₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₁₉×C₅₃

Complements

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

N	H ≅ G/N	type
2. Dic₃×C₁₉×C₅₃	C₃	direct (N × H)
4. C₃²×C₁₉×C₅₃	C₄	semidirect
10. C₃×C₁₉×C₅₃	Dic₃	direct (N × H)
11. C₃×C₁₉×C₅₃	C₃×C₄	semidirect
13. C₃×Dic₃×C₅₃	C₁₉	direct (N × H)
14. C₁₉×C₅₃	C₃×Dic₃	direct (N × H)
16. C₃×Dic₃×C₁₉	C₅₃	direct (N × H)
18. Dic₃×C₅₃	C₃×C₁₉	direct (N × H)
19. C₃²×C₅₃	C₄×C₁₉	semidirect
25. Dic₃×C₁₉	C₃×C₅₃	direct (N × H)
27. C₃²×C₁₉	C₄×C₅₃	semidirect
29. C₃×C₅₃	Dic₃×C₁₉	direct (N × H)
30. C₃×C₅₃	C₃×C₄×C₁₉	semidirect
37. C₃×C₁₉	Dic₃×C₅₃	direct (N × H)
38. C₃×C₁₉	C₃×C₄×C₅₃	semidirect
39. C₅₃	C₃×Dic₃×C₁₉	direct (N × H)
41. C₃×Dic₃	C₁₉×C₅₃	direct (N × H)
42. C₁₉	C₃×Dic₃×C₅₃	direct (N × H)
45. Dic₃	C₃×C₁₉×C₅₃	direct (N × H)
46. C₃²	C₄×C₁₉×C₅₃	semidirect
52. C₃	C₃×C₄×C₁₉×C₅₃	semidirect
53. C₃	Dic₃×C₁₉×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₁₉×C₅₃	2
Dic₃×C₁₉×C₅₃	3
C₃×C₄×C₁₉×C₅₃	3
C₃×Dic₃×C₅₃	19
C₃×Dic₃×C₁₉	53

Subgroups — C3×Dic3×C19×C53 #49136262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₃×C₁₉×C₅₃ #49136262