Finite Group Databasebeta

← C₅×D₃×C₁₂₁₇ #49141011

Subgroups — C₅×D₃×C₁₂₁₇ #49141011

Order 36510.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₁₂₁₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×C₅×C₁₂₁₇	—	3	1	no	maximal, Hall
3	D₃×C₁₂₁₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₅×C₁₂₁₇	D₃	1	1	yes	characteristic, direct product factor, Hall
5	C₃×C₁₂₁₇	C₂×C₅	1	1	yes	characteristic, complemented, Hall
6	C₂×C₁₂₁₇	—	3	1	no	Hall
7	C₁₂₁₇	C₅×D₃	1	1	yes	characteristic, direct product factor, Sylow
8	C₅×D₃	C₁₂₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅	C₂×C₁₂₁₇	1	1	yes	characteristic, complemented, Hall
10	C₂×C₅	—	3	1	no	Hall
11	D₃	C₅×C₁₂₁₇	1	1	yes	characteristic, direct product factor, Hall
12	C₅	D₃×C₁₂₁₇	1	1	yes	characteristic, direct product factor, Sylow
13	C₃	C₂×C₅×C₁₂₁₇	1	1	yes	characteristic, complemented, Sylow
14	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₁₂₁₇	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₅×C₁₂₁₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₂₁₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₅×C₁₂₁₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₁₂₁₇	C₂	semidirect
3. D₃×C₁₂₁₇	C₅	direct (N × H)
4. C₅×C₁₂₁₇	D₃	direct (N × H)
5. C₃×C₁₂₁₇	C₂×C₅	semidirect
7. C₁₂₁₇	C₅×D₃	direct (N × H)
8. C₅×D₃	C₁₂₁₇	direct (N × H)
9. C₃×C₅	C₂×C₁₂₁₇	semidirect
11. D₃	C₅×C₁₂₁₇	direct (N × H)
12. C₅	D₃×C₁₂₁₇	direct (N × H)
13. C₃	C₂×C₅×C₁₂₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₁₂₁₇	2
C₂×C₅×C₁₂₁₇	3
D₃×C₁₂₁₇	5
C₅×D₃	1217