Finite Group Databasebeta

← C₃×D₃×C₁₇ #64561

Subgroups — C₃×D₃×C₁₇ #64561

Order 306.

18 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×C₃×C₁₇	—	3	1	no	maximal
3	D₃×C₁₇	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×C₁₇	—	2	1	no	—
5	C₃×C₁₇	D₃	1	1	yes	characteristic, direct product factor
6	C₃×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₁₇	—	3	1	no	Hall
8	C₃×D₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₁₇	C₃×D₃	1	1	yes	characteristic, direct product factor, Sylow
10	C₃²	C₂×C₁₇	1	1	yes	characteristic, complemented, Sylow
11	C₂×C₃	—	3	1	no	—
12	D₃	C₃×C₁₇	1	1	yes	characteristic, direct product factor
13	C₃	—	2	1	no	—
14	C₃	D₃×C₁₇	1	1	yes	characteristic, direct product factor
15	C₃	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
16	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₁₇	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₃²×C₁₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₃×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₃²×C₁₇	C₂	semidirect
3. D₃×C₁₇	C₃	direct (N × H)
5. C₃×C₁₇	D₃	direct (N × H)
6. C₃×C₁₇	C₂×C₃	semidirect
8. C₃×D₃	C₁₇	direct (N × H)
9. C₁₇	C₃×D₃	direct (N × H)
10. C₃²	C₂×C₁₇	semidirect
12. D₃	C₃×C₁₇	direct (N × H)
14. C₃	D₃×C₁₇	direct (N × H)
15. C₃	C₂×C₃×C₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₁₇	2
C₂×C₃×C₁₇	3
D₃×C₁₇	3
C₃×D₃	17