Subgroups — AGL(1,𝔽₉) #642

Order 72.

10 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#157	C₂	1	1	yes	maximal, characteristic
2	C₃⋊D₃	C₄	1	1	yes	characteristic
3	C₃²	C₈	1	1	yes	characteristic, complemented, Sylow
4	C₈	—	9	1	no	maximal, Sylow
5	D₃	—	12	1	no	—
6	C₄	—	9	1	no	—
7	C₃	—	4	1	no	—
8	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²	C₈
prime radical	C₃⋊D₃	C₄
cosocle	#157	C₂
LCS	G ▷ C₃²
UCS	C₁

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

N	H ≅ G/N	type
3. C₃²	C₈	semidirect

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

subgroup	index
#157	2
C₈	9