Finite Group Databasebeta

← C₃×Q₈×C₁₉ #90721

Subgroups — C₃×Q₈×C₁₉ #90721

Order 456.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₁₉	C₂	1	3	yes	maximal
2	Q₈×C₁₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂×C₃×C₁₉	C₂²	1	1	yes	characteristic
4	C₄×C₁₉	C₂×C₃	1	3	yes	—
5	C₃×C₁₉	Q₈	1	1	yes	characteristic, direct product factor, Hall
6	C₂×C₁₉	C₂²×C₃	1	1	yes	characteristic
7	C₃×Q₈	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₁₉	C₃×Q₈	1	1	yes	characteristic, direct product factor, Sylow
9	C₃×C₄	C₂×C₁₉	1	3	yes	—
10	Q₈	C₃×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
11	C₂×C₃	C₂²×C₁₉	1	1	yes	characteristic
12	C₄	C₂×C₃×C₁₉	1	3	yes	—
13	C₃	Q₈×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	C₂²×C₃×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₃×C₁₉	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₃×C₁₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×C₁₉	C₃	direct (N × H)
5. C₃×C₁₉	Q₈	direct (N × H)
7. C₃×Q₈	C₁₉	direct (N × H)
8. C₁₉	C₃×Q₈	direct (N × H)
10. Q₈	C₃×C₁₉	direct (N × H)
13. C₃	Q₈×C₁₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₁₉	2
Q₈×C₁₉	3
C₃×Q₈	19