Finite Group Databasebeta

← C₃×C₄×C₅² #64478

Subgroups — C₃×C₄×C₅² #64478

Order 300.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅²	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅²	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₃×C₄×C₅	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₅²	C₂×C₃	1	1	yes	characteristic
6	C₂×C₃×C₅	C₂×C₅	1	6	yes	—
7	C₅²	C₃×C₄	1	1	yes	characteristic, direct product factor, Sylow
8	C₄×C₅	C₃×C₅	1	6	yes	direct product factor
9	C₃×C₅	C₄×C₅	1	6	yes	direct product factor
10	C₃×C₄	C₅²	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₅	C₂×C₃×C₅	1	6	yes	—
12	C₂×C₃	C₂×C₅²	1	1	yes	characteristic
13	C₅	C₃×C₄×C₅	1	6	yes	direct product factor
14	C₄	C₃×C₅²	1	1	yes	characteristic, direct product factor, Sylow
15	C₃	C₄×C₅²	1	1	yes	characteristic, direct product factor, Sylow
16	C₂	C₂×C₃×C₅²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃×C₅²
socle, prime radical	C₂×C₃×C₅²	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅²	C₃	direct (N × H)
3. C₃×C₅²	C₄	direct (N × H)
4. C₃×C₄×C₅	C₅	direct (N × H)
7. C₅²	C₃×C₄	direct (N × H)
8. C₄×C₅	C₃×C₅	direct (N × H)
9. C₃×C₅	C₄×C₅	direct (N × H)
10. C₃×C₄	C₅²	direct (N × H)
13. C₅	C₃×C₄×C₅	direct (N × H)
14. C₄	C₃×C₅²	direct (N × H)
15. C₃	C₄×C₅²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅²	2
C₄×C₅²	3
C₃×C₄×C₅	5