Finite Group Databasebeta

← C₃²×C₅⁵ #72205113

Subgroups — C₃²×C₅⁵ #72205113

Order 28125.

253056 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅⁴	C₅	1	781	yes	maximal, direct product factor
2	C₃×C₅⁵	C₃	1	4	yes	maximal, direct product factor
3	C₅⁵	C₃²	1	1	yes	characteristic, direct product factor, Sylow
4	C₃×C₅⁴	C₃×C₅	1	3124	yes	direct product factor
5	C₃²×C₅³	C₅²	1	20306	yes	direct product factor
6	C₅⁴	C₃²×C₅	1	781	yes	direct product factor
7	C₃×C₅³	C₃×C₅²	1	81224	yes	direct product factor
8	C₃²×C₅²	C₅³	1	20306	yes	direct product factor
9	C₅³	C₃²×C₅²	1	20306	yes	direct product factor
10	C₃×C₅²	C₃×C₅³	1	81224	yes	direct product factor
11	C₃²×C₅	C₅⁴	1	781	yes	direct product factor
12	C₅²	C₃²×C₅³	1	20306	yes	direct product factor
13	C₃×C₅	C₃×C₅⁴	1	3124	yes	direct product factor
14	C₃²	C₅⁵	1	1	yes	characteristic, direct product factor, Sylow
15	C₅	C₃²×C₅⁴	1	781	yes	direct product factor
16	C₃	C₃×C₅⁵	1	4	yes	direct product factor

	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. C₃²×C₅⁴	C₅	direct (N × H)
2. C₃×C₅⁵	C₃	direct (N × H)
3. C₅⁵	C₃²	direct (N × H)
4. C₃×C₅⁴	C₃×C₅	direct (N × H)
5. C₃²×C₅³	C₅²	direct (N × H)
6. C₅⁴	C₃²×C₅	direct (N × H)
7. C₃×C₅³	C₃×C₅²	direct (N × H)
8. C₃²×C₅²	C₅³	direct (N × H)
9. C₅³	C₃²×C₅²	direct (N × H)
10. C₃×C₅²	C₃×C₅³	direct (N × H)
11. C₃²×C₅	C₅⁴	direct (N × H)
12. C₅²	C₃²×C₅³	direct (N × H)
13. C₃×C₅	C₃×C₅⁴	direct (N × H)
14. C₃²	C₅⁵	direct (N × H)
15. C₅	C₃²×C₅⁴	direct (N × H)
16. C₃	C₃×C₅⁵	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅⁴	5
C₃×C₅⁵	3