Finite Group Databasebeta

← C₅×C₄₃×H(3,𝔽₅) #41001840

Subgroups — C₅×C₄₃×H(3,𝔽₅) #41001840

Order 26875.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅³×C₄₃	C₅	1	6	yes	maximal, complemented
2	C₄₃×H(3,𝔽₅)	C₅	1	25	yes	maximal, direct product factor
3	C₅²×C₄₃	—	5	30	no	—
4	C₅²×C₄₃	C₅²	1	1	yes	characteristic
5	C₅²×C₄₃	C₅²	1	30	yes	complemented
6	C₅×H(3,𝔽₅)	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₅×C₄₃	—	5	30	no	—
8	C₅×C₄₃	H(3,𝔽₅)	1	5	yes	direct product factor
9	C₅×C₄₃	C₅³	1	1	yes	characteristic
10	C₅³	C₅×C₄₃	1	6	yes	complemented
11	H(3,𝔽₅)	C₅×C₄₃	1	25	yes	direct product factor
12	C₄₃	C₅×H(3,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
13	C₅²	—	5	30	no	—
14	C₅²	C₅²×C₄₃	1	1	yes	characteristic
15	C₅²	C₅²×C₄₃	1	30	yes	complemented
16	C₅	—	5	30	no	—
17	C₅	C₄₃×H(3,𝔽₅)	1	5	yes	direct product factor
18	C₅	C₅³×C₄₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₅²×C₄₃	C₅²
commutator, Frattini, cosocle	C₅	C₅³×C₄₃
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅³×C₄₃	C₅	semidirect
2. C₄₃×H(3,𝔽₅)	C₅	direct (N × H)
5. C₅²×C₄₃	C₅²	semidirect
6. C₅×H(3,𝔽₅)	C₄₃	direct (N × H)
8. C₅×C₄₃	H(3,𝔽₅)	direct (N × H)
10. C₅³	C₅×C₄₃	semidirect
11. H(3,𝔽₅)	C₅×C₄₃	direct (N × H)
12. C₄₃	C₅×H(3,𝔽₅)	direct (N × H)
15. C₅²	C₅²×C₄₃	semidirect
17. C₅	C₄₃×H(3,𝔽₅)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅³×C₄₃	5
C₄₃×H(3,𝔽₅)	5
C₅×H(3,𝔽₅)	43