Finite Group Databasebeta

← Q₁₆×C₂₂₂₁ #48001076

Subgroups — Q₁₆×C₂₂₂₁ #48001076

Order 35536.

18 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₂₂₂₁	C₂	1	2	yes	maximal
2	C₈×C₂₂₂₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₂₂₁	—	2	2	no	—
4	C₄×C₂₂₂₁	C₂²	1	1	yes	characteristic
5	C₂×C₂₂₂₁	D₄	1	1	yes	characteristic
6	C₂₂₂₁	Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
7	Q₁₆	C₂₂₂₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	Q₈	C₂×C₂₂₂₁	1	2	yes	—
9	C₈	C₂×C₂₂₂₁	1	1	yes	characteristic
10	C₄	—	2	2	no	—
11	C₄	C₂²×C₂₂₂₁	1	1	yes	characteristic
12	C₂	D₄×C₂₂₂₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₂₂₂₁	D₄
commutator, Frattini, cosocle	C₄	C₂²×C₂₂₂₁
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₂₂₁ ◁ C₄×C₂₂₂₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂₂₂₁	Q₁₆	direct (N × H)
7. Q₁₆	C₂₂₂₁	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×C₂₂₂₁	2
C₈×C₂₂₂₁	2
Q₁₆	2221