Finite Group Databasebeta

← C₂³×C₃×C₅ #1237

Subgroups — C₂³×C₃×C₅ #1237

Order 120.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂×C₃×C₅	C₂²	1	7	yes	direct product factor
4	C₂³×C₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₅	C₂×C₃	1	7	yes	direct product factor
6	C₃×C₅	C₂³	1	1	yes	characteristic, direct product factor, Hall
7	C₂²×C₃	C₂×C₅	1	7	yes	direct product factor
8	C₂×C₅	C₂²×C₃	1	7	yes	direct product factor
9	C₂³	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
10	C₂×C₃	C₂²×C₅	1	7	yes	direct product factor
11	C₅	C₂³×C₃	1	1	yes	characteristic, direct product factor, Sylow
12	C₂²	C₂×C₃×C₅	1	7	yes	direct product factor
13	C₃	C₂³×C₅	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	C₂²×C₃×C₅	1	7	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. C₂²×C₃×C₅	C₂	direct (N × H)
2. C₂³×C₅	C₃	direct (N × H)
3. C₂×C₃×C₅	C₂²	direct (N × H)
4. C₂³×C₃	C₅	direct (N × H)
5. C₂²×C₅	C₂×C₃	direct (N × H)
6. C₃×C₅	C₂³	direct (N × H)
7. C₂²×C₃	C₂×C₅	direct (N × H)
8. C₂×C₅	C₂²×C₃	direct (N × H)
9. C₂³	C₃×C₅	direct (N × H)
10. C₂×C₃	C₂²×C₅	direct (N × H)
11. C₅	C₂³×C₃	direct (N × H)
12. C₂²	C₂×C₃×C₅	direct (N × H)
13. C₃	C₂³×C₅	direct (N × H)
14. C₂	C₂²×C₃×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
C₂²×C₃×C₅	2
C₂³×C₅	3
C₂³×C₃	5