Finite Group Databasebeta

← C₅₃×D₂₁₅ #37503548

Subgroups — C₅₃×D₂₁₅ #37503548

Order 22790.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₄₃×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅₃×D₄₃	—	5	1	no	maximal, Hall
3	C₄₃×C₅₃	D₅	1	1	yes	characteristic, complemented, Hall
4	D₅×C₅₃	—	43	1	no	maximal, Hall
5	D₂₁₅	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×C₅₃	D₄₃	1	1	yes	characteristic, complemented, Hall
7	C₅×C₄₃	C₂×C₅₃	1	1	yes	characteristic, complemented, Hall
8	C₂×C₅₃	—	215	1	no	Hall
9	D₄₃	—	5	1	no	Hall
10	C₅₃	D₂₁₅	1	1	yes	characteristic, direct product factor, Sylow
11	C₄₃	D₅×C₅₃	1	1	yes	characteristic, complemented, Sylow
12	D₅	—	43	1	no	Hall
13	C₅	C₅₃×D₄₃	1	1	yes	characteristic, complemented, Sylow
14	C₂	—	215	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅₃	D₂₁₅
commutator, cosocle, hypocenter	C₅×C₄₃	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₅×C₄₃×C₅₃	C₂
LCS	G ▷ C₅×C₄₃
UCS	C₁ ◁ C₅₃

Complements

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

N	H ≅ G/N	type
1. C₅×C₄₃×C₅₃	C₂	semidirect
3. C₄₃×C₅₃	D₅	semidirect
5. D₂₁₅	C₅₃	direct (N × H)
6. C₅×C₅₃	D₄₃	semidirect
7. C₅×C₄₃	C₂×C₅₃	semidirect
10. C₅₃	D₂₁₅	direct (N × H)
11. C₄₃	D₅×C₅₃	semidirect
13. C₅	C₅₃×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₄₃×C₅₃	2
C₅₃×D₄₃	5
D₅×C₅₃	43
D₂₁₅	53