Finite Group Databasebeta

← C₂²×C₅×C₉ #4367

Subgroups — C₂²×C₅×C₉ #4367

Order 180.

30 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₉	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅	C₃	1	1	yes	maximal, characteristic
3	C₅×C₉	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×C₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅	C₂×C₃	1	3	yes	—
6	C₂²×C₅	C₉	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₉	C₂×C₅	1	3	yes	direct product factor
8	C₃×C₅	C₂²×C₃	1	1	yes	characteristic
9	C₂²×C₃	C₃×C₅	1	1	yes	characteristic
10	C₂×C₅	C₂×C₉	1	3	yes	direct product factor
11	C₉	C₂²×C₅	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₃	C₂×C₃×C₅	1	3	yes	—
13	C₅	C₂²×C₉	1	1	yes	characteristic, direct product factor, Sylow
14	C₂²	C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
15	C₃	C₂²×C₃×C₅	1	1	yes	characteristic
16	C₂	C₂×C₅×C₉	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂²×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₉	C₂	direct (N × H)
3. C₅×C₉	C₂²	direct (N × H)
4. C₂²×C₉	C₅	direct (N × H)
6. C₂²×C₅	C₉	direct (N × H)
7. C₂×C₉	C₂×C₅	direct (N × H)
10. C₂×C₅	C₂×C₉	direct (N × H)
11. C₉	C₂²×C₅	direct (N × H)
13. C₅	C₂²×C₉	direct (N × H)
14. C₂²	C₅×C₉	direct (N × H)
16. C₂	C₂×C₅×C₉	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×C₅×C₉	2
C₂²×C₃×C₅	3
C₂²×C₉	5