Subgroups — C₃²×C₅×D₅×C₅₃ #37558724

Order 23850.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅×D₅×C₅₃	C₃	1	4	yes	maximal, direct product factor
3	C₃²×D₅×C₅₃	C₅	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃²×C₅×C₅₃	—	5	1	no	maximal
5	C₃×C₅²×C₅₃	C₂×C₃	1	4	yes	complemented
6	C₅×D₅×C₅₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×C₅×C₅₃	—	2	2	no	—
8	C₃²×C₅×C₅₃	D₅	1	1	yes	characteristic, direct product factor
9	C₃²×C₅×C₅₃	C₂×C₅	1	1	yes	characteristic, complemented
10	C₂×C₃×C₅×C₅₃	—	5	4	no	—
11	C₃×D₅×C₅₃	C₃×C₅	1	4	yes	direct product factor
12	C₅²×C₅₃	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃²×C₅₃	—	5	1	no	Hall
14	C₃×C₅×C₅₃	—	2	8	no	—
15	C₃×C₅×C₅₃	C₃×D₅	1	4	yes	direct product factor
16	C₃×C₅×C₅₃	C₂×C₃×C₅	1	4	yes	complemented
17	C₂×C₅×C₅₃	—	5	1	no	—
18	D₅×C₅₃	C₃²×C₅	1	1	yes	characteristic, direct product factor
19	C₃²×C₅₃	C₅×D₅	1	1	yes	characteristic, direct product factor, Hall
20	C₃²×C₅×D₅	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₃×C₅₃	—	5	4	no	—
22	C₅×C₅₃	—	2	2	no	—
23	C₅×C₅₃	C₃²×D₅	1	1	yes	characteristic, direct product factor
24	C₅×C₅₃	C₂×C₃²×C₅	1	1	yes	characteristic, complemented
25	C₃²×C₅²	C₂×C₅₃	1	1	yes	characteristic, complemented, Hall
26	C₃×C₅₃	C₃×C₅×D₅	1	4	yes	direct product factor
27	C₃×C₅×D₅	C₃×C₅₃	1	4	yes	direct product factor
28	C₂×C₅₃	—	5	1	no	Hall
29	C₂×C₃²×C₅	—	5	1	no	—
30	C₃²×D₅	C₅×C₅₃	1	1	yes	characteristic, direct product factor
31	C₃×C₅²	C₂×C₃×C₅₃	1	4	yes	complemented
32	C₅₃	C₃²×C₅×D₅	1	1	yes	characteristic, direct product factor, Sylow
33	C₅×D₅	C₃²×C₅₃	1	1	yes	characteristic, direct product factor, Hall
34	C₃²×C₅	—	2	2	no	—
35	C₃²×C₅	D₅×C₅₃	1	1	yes	characteristic, direct product factor
36	C₃²×C₅	C₂×C₅×C₅₃	1	1	yes	characteristic, complemented
37	C₂×C₃×C₅	—	5	4	no	—
38	C₃×D₅	C₃×C₅×C₅₃	1	4	yes	direct product factor
39	C₅²	C₂×C₃²×C₅₃	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃²	—	5	1	no	Hall
41	C₃×C₅	—	2	8	no	—
42	C₃×C₅	C₃×D₅×C₅₃	1	4	yes	direct product factor
43	C₃×C₅	C₂×C₃×C₅×C₅₃	1	4	yes	complemented
44	C₂×C₅	—	5	1	no	—
45	D₅	C₃²×C₅×C₅₃	1	1	yes	characteristic, direct product factor
46	C₃²	C₅×D₅×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×C₃	—	5	4	no	—
48	C₅	—	2	2	no	—
49	C₅	C₂×C₃²×C₅×C₅₃	1	1	yes	characteristic, complemented
50	C₅	C₃²×D₅×C₅₃	1	1	yes	characteristic, direct product factor
51	C₃	C₃×C₅×D₅×C₅₃	1	4	yes	direct product factor
52	C₂	—	5	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²×C₅×C₅₃	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃²×C₅×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₅₃	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²×C₅×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₅₃	C₂	semidirect
2. C₃×C₅×D₅×C₅₃	C₃	direct (N × H)
3. C₃²×D₅×C₅₃	C₅	direct (N × H)
5. C₃×C₅²×C₅₃	C₂×C₃	semidirect
6. C₅×D₅×C₅₃	C₃²	direct (N × H)
8. C₃²×C₅×C₅₃	D₅	direct (N × H)
9. C₃²×C₅×C₅₃	C₂×C₅	semidirect
11. C₃×D₅×C₅₃	C₃×C₅	direct (N × H)
12. C₅²×C₅₃	C₂×C₃²	semidirect
15. C₃×C₅×C₅₃	C₃×D₅	direct (N × H)
16. C₃×C₅×C₅₃	C₂×C₃×C₅	semidirect
18. D₅×C₅₃	C₃²×C₅	direct (N × H)
19. C₃²×C₅₃	C₅×D₅	direct (N × H)
20. C₃²×C₅×D₅	C₅₃	direct (N × H)
23. C₅×C₅₃	C₃²×D₅	direct (N × H)
24. C₅×C₅₃	C₂×C₃²×C₅	semidirect
25. C₃²×C₅²	C₂×C₅₃	semidirect
26. C₃×C₅₃	C₃×C₅×D₅	direct (N × H)
27. C₃×C₅×D₅	C₃×C₅₃	direct (N × H)
30. C₃²×D₅	C₅×C₅₃	direct (N × H)
31. C₃×C₅²	C₂×C₃×C₅₃	semidirect
32. C₅₃	C₃²×C₅×D₅	direct (N × H)
33. C₅×D₅	C₃²×C₅₃	direct (N × H)
35. C₃²×C₅	D₅×C₅₃	direct (N × H)
36. C₃²×C₅	C₂×C₅×C₅₃	semidirect
38. C₃×D₅	C₃×C₅×C₅₃	direct (N × H)
39. C₅²	C₂×C₃²×C₅₃	semidirect
42. C₃×C₅	C₃×D₅×C₅₃	direct (N × H)
43. C₃×C₅	C₂×C₃×C₅×C₅₃	semidirect
45. D₅	C₃²×C₅×C₅₃	direct (N × H)
46. C₃²	C₅×D₅×C₅₃	direct (N × H)
49. C₅	C₂×C₃²×C₅×C₅₃	semidirect
50. C₅	C₃²×D₅×C₅₃	direct (N × H)
51. C₃	C₃×C₅×D₅×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
C₃²×C₅²×C₅₃	2
C₃×C₅×D₅×C₅₃	3
C₃²×D₅×C₅₃	5
C₂×C₃²×C₅×C₅₃	5
C₃²×C₅×D₅	53

Subgroups — C32×C5×D5×C53 #37558724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×D₅×C₅₃ #37558724