Subgroups — C₅×Q₈ #178

Order 40.

12 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅	C₂	1	3	yes	maximal
2	C₂×C₅	C₂²	1	1	yes	characteristic
3	Q₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₅	Q₈	1	1	yes	characteristic, direct product factor, Sylow
5	C₄	C₂×C₅	1	3	yes	—
6	C₂	C₂²×C₅	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ G

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

N	H ≅ G/N	type
3. Q₈	C₅	direct (N × H)
4. C₅	Q₈	direct (N × H)

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

subgroup	index
C₄×C₅	2
Q₈	5