Finite Group Databasebeta

← C₃×C₅×C₆₁₃⋊C₃ #42119902

Subgroups — C₃×C₅×C₆₁₃⋊C₃ #42119902

Order 27585.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₆₁₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₅×C₆₁₃⋊C₃	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₆₁₃⋊C₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₅×C₆₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
5	C₃×C₆₁₃	C₃×C₅	1	1	yes	characteristic, complemented
6	C₆₁₃⋊C₃	C₃×C₅	1	3	yes	direct product factor
7	C₆₁₃	C₃²×C₅	1	1	yes	characteristic, complemented, Sylow
8	C₃²×C₅	—	613	1	no	maximal, Hall
9	C₃×C₅	—	613	3	no	—
10	C₃×C₅	C₆₁₃⋊C₃	1	1	yes	characteristic, direct product factor
11	C₃²	—	613	1	no	Sylow
12	C₅	C₃×C₆₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
13	C₃	—	613	3	no	—
14	C₃	C₅×C₆₁₃⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₃×C₅	C₆₁₃⋊C₃
commutator, cosocle, hypocenter	C₆₁₃	C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₆₁₃	C₃
LCS	G ▷ C₆₁₃
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₆₁₃	C₃	semidirect
2. C₅×C₆₁₃⋊C₃	C₃	direct (N × H)
3. C₃×C₆₁₃⋊C₃	C₅	direct (N × H)
4. C₅×C₆₁₃	C₃²	semidirect
5. C₃×C₆₁₃	C₃×C₅	semidirect
6. C₆₁₃⋊C₃	C₃×C₅	direct (N × H)
7. C₆₁₃	C₃²×C₅	semidirect
10. C₃×C₅	C₆₁₃⋊C₃	direct (N × H)
12. C₅	C₃×C₆₁₃⋊C₃	direct (N × H)
14. C₃	C₅×C₆₁₃⋊C₃	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×C₆₁₃	3
C₅×C₆₁₃⋊C₃	3
C₃×C₆₁₃⋊C₃	5
C₃²×C₅	613