Subgroups — C₂×C₅×3₋¹⁺² #63204

Order 270.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃²×C₅	C₃	1	1	yes	maximal, characteristic
3	C₂×C₅×C₉	C₃	1	3	yes	maximal, complemented
4	C₂×3₋¹⁺²	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃²×C₅	C₂×C₃	1	1	yes	characteristic
6	C₅×C₉	C₂×C₃	1	3	yes	complemented
7	C₂×C₃×C₅	—	3	1	no	—
8	C₂×C₃×C₅	C₃²	1	1	yes	characteristic
9	3₋¹⁺²	C₂×C₅	1	1	yes	characteristic, direct product factor, Sylow
10	C₂×C₃²	C₃×C₅	1	1	yes	characteristic
11	C₂×C₉	C₃×C₅	1	3	yes	complemented
12	C₃×C₅	—	3	1	no	—
13	C₃×C₅	C₂×C₃²	1	1	yes	characteristic
14	C₂×C₅	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Hall
15	C₃²	C₂×C₃×C₅	1	1	yes	characteristic
16	C₉	C₂×C₃×C₅	1	3	yes	complemented
17	C₂×C₃	—	3	1	no	—
18	C₂×C₃	C₃²×C₅	1	1	yes	characteristic
19	C₅	C₂×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
20	C₃	—	3	1	no	—
21	C₃	C₂×C₃²×C₅	1	1	yes	characteristic
22	C₂	C₅×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, socle	C₂×C₃×C₅	C₃²
commutator, Frattini, cosocle	C₃	C₂×C₃²×C₅
prime radical	C₂×C₃²×C₅	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ G

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

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