Finite Group Databasebeta

← C₂×C₁₇×C₂₃₉⋊C₇ #64205776

Subgroups — C₂×C₁₇×C₂₃₉⋊C₇ #64205776

Order 56882.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×C₂₃₉⋊C₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₁₇×C₂₃₉	C₇	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₇×C₂₃₉	C₂×C₇	1	1	yes	characteristic, complemented, Hall
4	C₂×C₂₃₉⋊C₇	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂₃₉⋊C₇	C₂×C₁₇	1	1	yes	characteristic, direct product factor, Hall
6	C₂×C₂₃₉	C₇×C₁₇	1	1	yes	characteristic, complemented, Hall
7	C₂₃₉	C₂×C₇×C₁₇	1	1	yes	characteristic, complemented, Sylow
8	C₂×C₇×C₁₇	—	239	1	no	maximal, Hall
9	C₇×C₁₇	—	239	1	no	Hall
10	C₂×C₁₇	C₂₃₉⋊C₇	1	1	yes	characteristic, direct product factor, Hall
11	C₁₇	C₂×C₂₃₉⋊C₇	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₇	—	239	1	no	Hall
13	C₇	—	239	1	no	Sylow
14	C₂	C₁₇×C₂₃₉⋊C₇	1	1	yes	characteristic, direct product factor, Sylow

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

Complements

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

N	H ≅ G/N	type
1. C₁₇×C₂₃₉⋊C₇	C₂	direct (N × H)
2. C₂×C₁₇×C₂₃₉	C₇	semidirect
3. C₁₇×C₂₃₉	C₂×C₇	semidirect
4. C₂×C₂₃₉⋊C₇	C₁₇	direct (N × H)
5. C₂₃₉⋊C₇	C₂×C₁₇	direct (N × H)
6. C₂×C₂₃₉	C₇×C₁₇	semidirect
7. C₂₃₉	C₂×C₇×C₁₇	semidirect
10. C₂×C₁₇	C₂₃₉⋊C₇	direct (N × H)
11. C₁₇	C₂×C₂₃₉⋊C₇	direct (N × H)
14. C₂	C₁₇×C₂₃₉⋊C₇	direct (N × H)

Maximal subgroups

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

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