Subgroups — C₃×C₅×C₃⋊(C₃⋊D₃) #11717702

Order 810.

380 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₅	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₃⋊(C₃⋊D₃)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅×C₃⋊D₃	—	3	13	no	maximal
4	C₃×C₃⋊(C₃⋊D₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃³×C₅	—	2	13	no	—
6	C₃³×C₅	D₃	1	13	yes	complemented
7	C₃³×C₅	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅×C₃⋊D₃	—	3	13	no	—
9	C₃×C₅×D₃	—	9	13	no	—
10	C₃⁴	C₂×C₅	1	1	yes	characteristic, complemented, Sylow
11	C₃⋊(C₃⋊D₃)	C₃×C₅	1	1	yes	characteristic, direct product factor
12	C₃×C₃⋊D₃	—	3	13	no	—
13	C₃²×C₅	—	2	52	no	—
14	C₃²×C₅	C₃×D₃	1	13	yes	complemented
15	C₃²×C₅	C₃⋊D₃	1	13	yes	complemented
16	C₂×C₃×C₅	—	27	1	no	—
17	C₅×D₃	—	9	13	no	—
18	C₃³	—	2	13	no	—
19	C₃³	C₅×D₃	1	13	yes	complemented
20	C₃³	C₂×C₃×C₅	1	1	yes	characteristic, complemented
21	C₃⋊D₃	—	3	13	no	—
22	C₃×D₃	—	9	13	no	—
23	C₃×C₅	—	2	13	no	—
24	C₃×C₅	C₃×C₃⋊D₃	1	13	yes	complemented
25	C₃×C₅	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
26	C₂×C₅	—	27	1	no	Hall
27	C₃²	—	2	52	no	—
28	C₃²	C₃×C₅×D₃	1	13	yes	complemented
29	C₃²	C₅×C₃⋊D₃	1	13	yes	complemented
30	C₂×C₃	—	27	1	no	—
31	D₃	—	9	13	no	—
32	C₅	C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	—	2	13	no	—
34	C₃	C₃×C₅×C₃⋊D₃	1	13	yes	complemented
35	C₃	C₅×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
36	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃⁴×C₅	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃⁴×C₅	C₂	semidirect
2. C₅×C₃⋊(C₃⋊D₃)	C₃	direct (N × H)
4. C₃×C₃⋊(C₃⋊D₃)	C₅	direct (N × H)
6. C₃³×C₅	D₃	semidirect
7. C₃³×C₅	C₂×C₃	semidirect
10. C₃⁴	C₂×C₅	semidirect
11. C₃⋊(C₃⋊D₃)	C₃×C₅	direct (N × H)
14. C₃²×C₅	C₃×D₃	semidirect
15. C₃²×C₅	C₃⋊D₃	semidirect
19. C₃³	C₅×D₃	semidirect
20. C₃³	C₂×C₃×C₅	semidirect
24. C₃×C₅	C₃×C₃⋊D₃	semidirect
25. C₃×C₅	C₃⋊(C₃⋊D₃)	direct (N × H)
28. C₃²	C₃×C₅×D₃	semidirect
29. C₃²	C₅×C₃⋊D₃	semidirect
32. C₅	C₃×C₃⋊(C₃⋊D₃)	direct (N × H)
34. C₃	C₃×C₅×C₃⋊D₃	semidirect
35. C₃	C₅×C₃⋊(C₃⋊D₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₅	2
C₅×C₃⋊(C₃⋊D₃)	3
C₃×C₅×C₃⋊D₃	3
C₃×C₃⋊(C₃⋊D₃)	5

Subgroups — C3×C5×C3⋊(C3⋊D3) #11717702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₃⋊(C₃⋊D₃) #11717702