Finite Group Databasebeta

← C₃×C₅²×C₃₁₁ #37552256

Subgroups — C₃×C₅²×C₃₁₁ #37552256

Order 23325.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₃₁₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₃×C₅×C₃₁₁	C₅	1	6	yes	maximal, direct product factor
3	C₅×C₃₁₁	C₃×C₅	1	6	yes	direct product factor
4	C₃×C₃₁₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
5	C₃₁₁	C₃×C₅²	1	1	yes	characteristic, direct product factor, Sylow
6	C₃×C₅²	C₃₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅²	C₃×C₃₁₁	1	1	yes	characteristic, direct product factor, Sylow
8	C₃×C₅	C₅×C₃₁₁	1	6	yes	direct product factor
9	C₅	C₃×C₅×C₃₁₁	1	6	yes	direct product factor
10	C₃	C₅²×C₃₁₁	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅²×C₃₁₁	C₃	direct (N × H)
2. C₃×C₅×C₃₁₁	C₅	direct (N × H)
3. C₅×C₃₁₁	C₃×C₅	direct (N × H)
4. C₃×C₃₁₁	C₅²	direct (N × H)
5. C₃₁₁	C₃×C₅²	direct (N × H)
6. C₃×C₅²	C₃₁₁	direct (N × H)
7. C₅²	C₃×C₃₁₁	direct (N × H)
8. C₃×C₅	C₅×C₃₁₁	direct (N × H)
9. C₅	C₃×C₅×C₃₁₁	direct (N × H)
10. C₃	C₅²×C₃₁₁	direct (N × H)

Maximal subgroups

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

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