Finite Group Databasebeta

← C₇×D₁₇×C₂₃₉ #64205779

Subgroups — C₇×D₁₇×C₂₃₉ #64205779

Order 56882.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₇×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₁₇×C₂₃₉	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₁₇×C₂₃₉	C₂×C₇	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₂₃₉	—	17	1	no	maximal, Hall
5	C₇×C₂₃₉	D₁₇	1	1	yes	characteristic, direct product factor, Hall
6	C₂×C₂₃₉	—	17	1	no	Hall
7	C₂₃₉	C₇×D₁₇	1	1	yes	characteristic, direct product factor, Sylow
8	C₇×D₁₇	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₇×C₁₇	C₂×C₂₃₉	1	1	yes	characteristic, complemented, Hall
10	D₁₇	C₇×C₂₃₉	1	1	yes	characteristic, direct product factor, Hall
11	C₁₇	C₂×C₇×C₂₃₉	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₇	—	17	1	no	Hall
13	C₇	D₁₇×C₂₃₉	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	17	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇×C₂₃₉	D₁₇
commutator, cosocle, hypocenter	C₁₇	C₂×C₇×C₂₃₉
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₂₃₉	C₂
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₇×C₂₃₉

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₇×C₂₃₉	C₂	semidirect
2. D₁₇×C₂₃₉	C₇	direct (N × H)
3. C₁₇×C₂₃₉	C₂×C₇	semidirect
5. C₇×C₂₃₉	D₁₇	direct (N × H)
7. C₂₃₉	C₇×D₁₇	direct (N × H)
8. C₇×D₁₇	C₂₃₉	direct (N × H)
9. C₇×C₁₇	C₂×C₂₃₉	semidirect
10. D₁₇	C₇×C₂₃₉	direct (N × H)
11. C₁₇	C₂×C₇×C₂₃₉	semidirect
13. C₇	D₁₇×C₂₃₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×C₁₇×C₂₃₉	2
D₁₇×C₂₃₉	7
C₂×C₇×C₂₃₉	17
C₇×D₁₇	239