Finite Group Databasebeta

← C₈₉×(C₅⋊D₅)⋊C₃ #27211168

Subgroups — C₈₉×(C₅⋊D₅)⋊C₃ #27211168

Order 13350.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²⋊C₃×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅⋊D₅×C₈₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×C₈₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	D₅×C₈₉	—	15	2	no	—
5	C₂×C₃×C₈₉	—	25	1	no	maximal, Hall
6	C₅×C₈₉	—	3	2	no	—
7	C₃×C₈₉	—	25	1	no	Hall
8	C₂×C₈₉	—	25	1	no	Hall
9	(C₅⋊D₅)⋊C₃	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₈₉	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
11	C₅²⋊C₃	C₂×C₈₉	1	1	yes	characteristic, complemented, Hall
12	C₅⋊D₅	C₃×C₈₉	1	1	yes	characteristic, complemented, Hall
13	C₅²	C₂×C₃×C₈₉	1	1	yes	characteristic, complemented, Sylow
14	D₅	—	15	2	no	—
15	C₂×C₃	—	25	1	no	Hall
16	C₅	—	3	2	no	—
17	C₃	—	25	1	no	Sylow
18	C₂	—	25	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₈₉	(C₅⋊D₅)⋊C₃
commutator, cosocle, hypocenter	C₅²	C₂×C₃×C₈₉
Frattini	C₁	G
Fitting, socle	C₅²×C₈₉	C₂×C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₈₉

Complements

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

N	H ≅ G/N	type
1. C₅²⋊C₃×C₈₉	C₂	semidirect
2. C₅⋊D₅×C₈₉	C₃	semidirect
3. C₅²×C₈₉	C₂×C₃	semidirect
9. (C₅⋊D₅)⋊C₃	C₈₉	direct (N × H)
10. C₈₉	(C₅⋊D₅)⋊C₃	direct (N × H)
11. C₅²⋊C₃	C₂×C₈₉	semidirect
12. C₅⋊D₅	C₃×C₈₉	semidirect
13. C₅²	C₂×C₃×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₅²⋊C₃×C₈₉	2
C₅⋊D₅×C₈₉	3
C₂×C₃×C₈₉	25
(C₅⋊D₅)⋊C₃	89