Finite Group Databasebeta

← C₃×C₅×D₁₂₁₃ #49138596

Subgroups — C₃×C₅×D₁₂₁₃ #49138596

Order 36390.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₁₂₁₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×D₁₂₁₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×D₁₂₁₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₅×C₁₂₁₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₃×C₁₂₁₃	C₂×C₅	1	1	yes	characteristic, complemented, Hall
6	D₁₂₁₃	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
7	C₁₂₁₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented, Sylow
8	C₂×C₃×C₅	—	1213	1	no	maximal, Hall
9	C₃×C₅	D₁₂₁₃	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₅	—	1213	1	no	Hall
11	C₂×C₃	—	1213	1	no	Hall
12	C₅	C₃×D₁₂₁₃	1	1	yes	characteristic, direct product factor, Sylow
13	C₃	C₅×D₁₂₁₃	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	1213	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅	D₁₂₁₃
commutator, cosocle, hypocenter	C₁₂₁₃	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₂₁₃	C₂
LCS	G ▷ C₁₂₁₃
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₁₂₁₃	C₂	semidirect
2. C₅×D₁₂₁₃	C₃	direct (N × H)
3. C₃×D₁₂₁₃	C₅	direct (N × H)
4. C₅×C₁₂₁₃	C₂×C₃	semidirect
5. C₃×C₁₂₁₃	C₂×C₅	semidirect
6. D₁₂₁₃	C₃×C₅	direct (N × H)
7. C₁₂₁₃	C₂×C₃×C₅	semidirect
9. C₃×C₅	D₁₂₁₃	direct (N × H)
12. C₅	C₃×D₁₂₁₃	direct (N × H)
13. C₃	C₅×D₁₂₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₁₂₁₃	2
C₅×D₁₂₁₃	3
C₃×D₁₂₁₃	5
C₂×C₃×C₅	1213