Finite Group Databasebeta

← C₃×C₅×C₆₃₁⋊C₃ #43249167

Subgroups — C₃×C₅×C₆₃₁⋊C₃ #43249167

Order 28395.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₆₃₁	C₃	1	1	yes	maximal, characteristic, complemented
2	C₅×C₆₃₁⋊C₃	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₆₃₁⋊C₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₅×C₆₃₁	C₃²	1	1	yes	characteristic, complemented, Hall
5	C₃×C₆₃₁	C₃×C₅	1	1	yes	characteristic, complemented
6	C₆₃₁⋊C₃	C₃×C₅	1	3	yes	direct product factor
7	C₆₃₁	C₃²×C₅	1	1	yes	characteristic, complemented, Sylow
8	C₃²×C₅	—	631	1	no	maximal, Hall
9	C₃×C₅	—	631	3	no	—
10	C₃×C₅	C₆₃₁⋊C₃	1	1	yes	characteristic, direct product factor
11	C₃²	—	631	1	no	Sylow
12	C₅	C₃×C₆₃₁⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
13	C₃	—	631	3	no	—
14	C₃	C₅×C₆₃₁⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₃×C₅	C₆₃₁⋊C₃
commutator, cosocle, hypocenter	C₆₃₁	C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₆₃₁	C₃
LCS	G ▷ C₆₃₁
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₆₃₁	C₃	semidirect
2. C₅×C₆₃₁⋊C₃	C₃	direct (N × H)
3. C₃×C₆₃₁⋊C₃	C₅	direct (N × H)
4. C₅×C₆₃₁	C₃²	semidirect
5. C₃×C₆₃₁	C₃×C₅	semidirect
6. C₆₃₁⋊C₃	C₃×C₅	direct (N × H)
7. C₆₃₁	C₃²×C₅	semidirect
10. C₃×C₅	C₆₃₁⋊C₃	direct (N × H)
12. C₅	C₃×C₆₃₁⋊C₃	direct (N × H)
14. C₃	C₅×C₆₃₁⋊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×C₆₃₁	3
C₅×C₆₃₁⋊C₃	3
C₃×C₆₃₁⋊C₃	5
C₃²×C₅	631