Subgroups — C₃×C₃⋊(C₃⋊(C₅⋊D₅)) #13062549

Order 1350.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₅²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃⋊(C₃⋊(C₅⋊D₅))	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₃⋊(C₅⋊D₅)	—	3	4	no	maximal
4	C₃×C₃⋊D₁₅	—	5	6	no	maximal
5	C₃²×C₅²	—	2	4	no	—
6	C₃²×C₅²	D₃	1	4	yes	complemented
7	C₃²×C₅²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃⋊(C₅⋊D₅)	—	3	4	no	—
9	C₃×C₅⋊D₅	—	9	1	no	—
10	C₃³×C₅	D₅	1	6	yes	complemented
11	C₃⋊D₁₅	—	5	6	no	—
12	C₃×D₁₅	—	15	24	no	—
13	C₃×C₅²	—	2	4	no	—
14	C₃×C₅²	C₃×D₃	1	4	yes	complemented
15	C₃×C₅²	C₃⋊D₃	1	1	yes	characteristic, complemented
16	C₃×C₃⋊D₃	—	25	1	no	Hall
17	C₅⋊D₅	—	9	1	no	Hall
18	C₃²×C₅	—	2	24	no	—
19	C₃²×C₅	C₃×D₅	1	6	yes	complemented
20	C₃²×C₅	D₁₅	1	24	yes	complemented
21	D₁₅	—	15	24	no	—
22	C₃×D₅	—	45	6	no	—
23	C₃³	C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
24	C₅²	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
25	C₃⋊D₃	—	25	1	no	—
26	C₃×D₃	—	75	4	no	—
27	C₃×C₅	—	2	24	no	—
28	C₃×C₅	C₃×D₁₅	1	24	yes	complemented
29	C₃×C₅	C₃⋊D₁₅	1	6	yes	complemented
30	D₅	—	45	6	no	—
31	C₃²	—	2	4	no	—
32	C₃²	C₃×C₅⋊D₅	1	1	yes	characteristic, complemented
33	C₃²	C₃⋊(C₅⋊D₅)	1	4	yes	complemented
34	C₂×C₃	—	225	1	no	—
35	D₃	—	75	4	no	—
36	C₅	C₃×C₃⋊D₁₅	1	6	yes	complemented
37	C₃	—	2	4	no	—
38	C₃	C₃×C₃⋊(C₅⋊D₅)	1	4	yes	complemented
39	C₃	C₃⋊(C₃⋊(C₅⋊D₅))	1	1	yes	characteristic, direct product factor
40	C₂	—	225	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	C₃⋊(C₃⋊(C₅⋊D₅))
commutator, cosocle, hypocenter	C₃²×C₅²	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃³×C₅²	C₂
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃³×C₅²	C₂	semidirect
2. C₃⋊(C₃⋊(C₅⋊D₅))	C₃	direct (N × H)
6. C₃²×C₅²	D₃	semidirect
7. C₃²×C₅²	C₂×C₃	semidirect
10. C₃³×C₅	D₅	semidirect
14. C₃×C₅²	C₃×D₃	semidirect
15. C₃×C₅²	C₃⋊D₃	semidirect
19. C₃²×C₅	C₃×D₅	semidirect
20. C₃²×C₅	D₁₅	semidirect
23. C₃³	C₅⋊D₅	semidirect
24. C₅²	C₃×C₃⋊D₃	semidirect
28. C₃×C₅	C₃×D₁₅	semidirect
29. C₃×C₅	C₃⋊D₁₅	semidirect
32. C₃²	C₃×C₅⋊D₅	semidirect
33. C₃²	C₃⋊(C₅⋊D₅)	semidirect
36. C₅	C₃×C₃⋊D₁₅	semidirect
38. C₃	C₃×C₃⋊(C₅⋊D₅)	semidirect
39. C₃	C₃⋊(C₃⋊(C₅⋊D₅))	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₅²	2
C₃⋊(C₃⋊(C₅⋊D₅))	3
C₃×C₃⋊(C₅⋊D₅)	3
C₃×C₃⋊D₁₅	5

Subgroups — C3×C3⋊(C3⋊(C5⋊D5)) #13062549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊(C₃⋊(C₅⋊D₅)) #13062549