Finite Group Databasebeta

← D₃×C₁₇×C₂₃₃ #37557728

Subgroups — D₃×C₁₇×C₂₃₃ #37557728

Order 23766.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₇×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×C₁₇×C₂₃₃	—	3	1	no	maximal, Hall
3	C₁₇×C₂₃₃	D₃	1	1	yes	characteristic, direct product factor, Hall
4	D₃×C₂₃₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₂₃₃	C₂×C₁₇	1	1	yes	characteristic, complemented, Hall
6	C₂×C₂₃₃	—	3	1	no	Hall
7	C₂₃₃	D₃×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
8	D₃×C₁₇	C₂₃₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₁₇	C₂×C₂₃₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₁₇	—	3	1	no	Hall
11	C₁₇	D₃×C₂₃₃	1	1	yes	characteristic, direct product factor, Sylow
12	D₃	C₁₇×C₂₃₃	1	1	yes	characteristic, direct product factor, Hall
13	C₃	C₂×C₁₇×C₂₃₃	1	1	yes	characteristic, complemented, Sylow
14	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₇×C₂₃₃	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₁₇×C₂₃₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₇×C₂₃₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₁₇×C₂₃₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₇×C₂₃₃	C₂	semidirect
3. C₁₇×C₂₃₃	D₃	direct (N × H)
4. D₃×C₂₃₃	C₁₇	direct (N × H)
5. C₃×C₂₃₃	C₂×C₁₇	semidirect
7. C₂₃₃	D₃×C₁₇	direct (N × H)
8. D₃×C₁₇	C₂₃₃	direct (N × H)
9. C₃×C₁₇	C₂×C₂₃₃	semidirect
11. C₁₇	D₃×C₂₃₃	direct (N × H)
12. D₃	C₁₇×C₂₃₃	direct (N × H)
13. C₃	C₂×C₁₇×C₂₃₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₇×C₂₃₃	2
C₂×C₁₇×C₂₃₃	3
D₃×C₂₃₃	17
D₃×C₁₇	233