Finite Group Databasebeta

← C₅×C₂²⋊(C₃₁₃⋊C₃) #34037460

Subgroups — C₅×C₂²⋊(C₃₁₃⋊C₃) #34037460

Order 18780.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₃₁₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₃₁₃⋊C₃	—	4	1	no	maximal, Hall
3	C₂²⋊(C₃₁₃⋊C₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₅×C₃₁₃	—	3	1	no	—
5	C₅×C₃₁₃	A₄	1	1	yes	characteristic, complemented, Hall
6	C₂²×C₃₁₃	C₃×C₅	1	1	yes	characteristic, complemented, Hall
7	C₃₁₃⋊C₃	—	4	1	no	Hall
8	C₂×C₃₁₃	—	3	1	no	—
9	C₃₁₃	C₅×A₄	1	1	yes	characteristic, complemented, Sylow
10	C₅×A₄	—	313	1	no	maximal, Hall
11	C₂²×C₅	C₃₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
12	C₃×C₅	—	1252	1	no	Hall
13	A₄	—	313	1	no	Hall
14	C₂×C₅	—	3	1	no	—
15	C₅	C₂²⋊(C₃₁₃⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
16	C₂²	C₅×C₃₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	1252	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅	C₂²⋊(C₃₁₃⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₃₁₃	C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₃₁₃	C₃
LCS	G ▷ C₂²×C₃₁₃
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₃₁₃	C₃	semidirect
3. C₂²⋊(C₃₁₃⋊C₃)	C₅	direct (N × H)
5. C₅×C₃₁₃	A₄	semidirect
6. C₂²×C₃₁₃	C₃×C₅	semidirect
9. C₃₁₃	C₅×A₄	semidirect
11. C₂²×C₅	C₃₁₃⋊C₃	semidirect
15. C₅	C₂²⋊(C₃₁₃⋊C₃)	direct (N × H)
16. C₂²	C₅×C₃₁₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₃₁₃	3
C₅×C₃₁₃⋊C₃	4
C₂²⋊(C₃₁₃⋊C₃)	5
C₅×A₄	313