Finite Group Databasebeta

← C₃×C₅×C₃₁₁⋊C₅ #37552257

Subgroups — C₃×C₅×C₃₁₁⋊C₅ #37552257

Order 23325.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃₁₁⋊C₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₃×C₅×C₃₁₁	C₅	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃₁₁⋊C₅	C₅	1	5	yes	maximal, direct product factor
4	C₅×C₃₁₁	C₃×C₅	1	1	yes	characteristic, complemented
5	C₃₁₁⋊C₅	C₃×C₅	1	5	yes	direct product factor
6	C₃×C₃₁₁	C₅²	1	1	yes	characteristic, complemented, Hall
7	C₃₁₁	C₃×C₅²	1	1	yes	characteristic, complemented, Sylow
8	C₃×C₅²	—	311	1	no	maximal, Hall
9	C₅²	—	311	1	no	Sylow
10	C₃×C₅	—	311	5	no	—
11	C₃×C₅	C₃₁₁⋊C₅	1	1	yes	characteristic, direct product factor
12	C₅	—	311	5	no	—
13	C₅	C₃×C₃₁₁⋊C₅	1	1	yes	characteristic, direct product factor
14	C₃	C₅×C₃₁₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅	C₃₁₁⋊C₅
commutator, cosocle, hypocenter	C₃₁₁	C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₃₁₁	C₅
LCS	G ▷ C₃₁₁
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₅×C₃₁₁⋊C₅	C₃	direct (N × H)
2. C₃×C₅×C₃₁₁	C₅	semidirect
3. C₃×C₃₁₁⋊C₅	C₅	direct (N × H)
4. C₅×C₃₁₁	C₃×C₅	semidirect
5. C₃₁₁⋊C₅	C₃×C₅	direct (N × H)
6. C₃×C₃₁₁	C₅²	semidirect
7. C₃₁₁	C₃×C₅²	semidirect
11. C₃×C₅	C₃₁₁⋊C₅	direct (N × H)
13. C₅	C₃×C₃₁₁⋊C₅	direct (N × H)
14. C₃	C₅×C₃₁₁⋊C₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₃₁₁⋊C₅	3
C₃×C₅×C₃₁₁	5
C₃×C₃₁₁⋊C₅	5
C₃×C₅²	311