Subgroups — #72271787

Order 10240.

At least 24645 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has at least 24645 conjugacy classes of subgroups — too many to draw. Only its 15 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	#72268210	C₂	1	1	yes	maximal, characteristic
2	2₋¹⁺⁴²	D₅	1	1	yes	characteristic
3	C₂×2₋¹⁺⁴	(C₂⁴⋊C₅)⋊C₂	1	1	yes	characteristic
4	C₂×2₋¹⁺⁴	(C₂⁴⋊C₅)⋊C₂	1	1	yes	characteristic
5	C₂×2₋¹⁺⁴	(C₂⁴⋊C₅)⋊C₂	1	1	yes	characteristic
6	#564	(C₂⁴⋊C₅)⋊C₂	1	2	yes	—
7	2₋¹⁺⁴	#66243	1	1	yes	characteristic, complemented
8	2₋¹⁺⁴	#66244	1	1	yes	characteristic
9	C₂²	#72209701	1	1	yes	characteristic
10	C₂	#72211305	1	1	yes	characteristic
11	C₂	#72212713	1	1	yes	characteristic
12	C₂	#72227762	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂²	#72209701
commutator, prime radical, cosocle, hypocenter	#72268210	C₂
Fitting	2₋¹⁺⁴²	D₅
LCS	G ▷ #72268210
UCS	C₁ ◁ C₂²

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

N	H ≅ G/N	type
7. 2₋¹⁺⁴	#66243	semidirect

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