Subgroups — #72219852

Order 15360.

At least 8260 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has at least 8260 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72218553	C₂	1	1	yes	maximal, characteristic
2	#72218551	C₄	1	1	yes	characteristic, complemented
3	#72209701	D₃	1	1	yes	characteristic
4	C₂⁴⋊(C₂⁴⋊C₅)	Dic₃	1	1	yes	characteristic, complemented
5	C₂⁴×C₂²⋊A₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
6	C₂⁸	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
7	C₂²⋊A₄	#66297	1	1	yes	characteristic, complemented
8	C₂⁴	AΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
9	C₂⁴	C₃⋊#66297	1	1	yes	characteristic, complemented

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	#72218551	C₄
Fitting, socle	C₂⁸	ΓL(1,𝔽₁₆)
prime radical, cosocle	#72218553	C₂
LCS	G ▷ #72218551
UCS	C₁

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