Subgroups — C₅×AGL(1,ℤ/9) #63193

Order 270.

26 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, complemented, Hall
2	C₃×C₅×D₃	—	3	1	no	maximal
3	C₅×D₉	C₃	1	1	yes	maximal, characteristic, complemented
4	AGL(1,ℤ/9)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃²×C₅	D₃	1	1	yes	characteristic
6	C₅×C₉	—	2	1	no	—
7	C₅×C₉	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₃×C₅	—	9	1	no	—
9	C₅×D₃	—	3	1	no	—
10	3₋¹⁺²	C₂×C₅	1	1	yes	characteristic, complemented, Sylow
11	C₃×D₃	—	3	1	no	—
12	D₉	C₃×C₅	1	1	yes	characteristic, complemented
13	C₃×C₅	—	3	1	no	—
14	C₃×C₅	C₃×D₃	1	1	yes	characteristic
15	C₂×C₅	—	9	1	no	Hall
16	C₃²	C₅×D₃	1	1	yes	characteristic
17	C₉	—	2	1	no	—
18	C₉	C₂×C₃×C₅	1	1	yes	characteristic, complemented
19	C₂×C₃	—	9	1	no	—
20	D₃	—	3	1	no	—
21	C₅	AGL(1,ℤ/9)	1	1	yes	characteristic, direct product factor, Sylow
22	C₃	—	3	1	no	—
23	C₃	C₃×C₅×D₃	1	1	yes	characteristic
24	C₂	—	9	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅	AGL(1,ℤ/9)
commutator, cosocle, hypocenter	C₉	C₂×C₃×C₅
Frattini	C₃	C₃×C₅×D₃
Fitting	C₅×3₋¹⁺²	C₂
socle	C₃×C₅	C₃×D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₅

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.