Finite Group Databasebeta

← C₃×C₁₇×D₂₃₃ #37557725

Subgroups — C₃×C₁₇×D₂₃₃ #37557725

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	C₁₇×D₂₃₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₁₇×C₂₃₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₃×D₂₃₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₂₃₃	C₂×C₁₇	1	1	yes	characteristic, complemented, Hall
6	D₂₃₃	C₃×C₁₇	1	1	yes	characteristic, direct product factor, Hall
7	C₂₃₃	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
8	C₂×C₃×C₁₇	—	233	1	no	maximal, Hall
9	C₃×C₁₇	D₂₃₃	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₁₇	—	233	1	no	Hall
11	C₁₇	C₃×D₂₃₃	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₃	—	233	1	no	Hall
13	C₃	C₁₇×D₂₃₃	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	233	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₁₇	D₂₃₃
commutator, cosocle, hypocenter	C₂₃₃	C₂×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₃×C₁₇×C₂₃₃	C₂
LCS	G ▷ C₂₃₃
UCS	C₁ ◁ C₃×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₇×C₂₃₃	C₂	semidirect
2. C₁₇×D₂₃₃	C₃	direct (N × H)
3. C₁₇×C₂₃₃	C₂×C₃	semidirect
4. C₃×D₂₃₃	C₁₇	direct (N × H)
5. C₃×C₂₃₃	C₂×C₁₇	semidirect
6. D₂₃₃	C₃×C₁₇	direct (N × H)
7. C₂₃₃	C₂×C₃×C₁₇	semidirect
9. C₃×C₁₇	D₂₃₃	direct (N × H)
11. C₁₇	C₃×D₂₃₃	direct (N × H)
13. C₃	C₁₇×D₂₃₃	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×C₁₇×C₂₃₃	2
C₁₇×D₂₃₃	3
C₃×D₂₃₃	17
C₂×C₃×C₁₇	233