Finite Group Databasebeta

← C₂×D₂₃×C₂₃₉ #36347124

Subgroups — C₂×D₂₃×C₂₃₉ #36347124

Order 21988.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₂₃×C₂₃₉	C₂	1	2	yes	maximal, direct product factor
3	C₂₃×C₂₃₉	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂²×C₂₃₉	—	23	1	no	maximal, Hall
5	C₂×C₂₃₉	—	23	2	no	—
6	C₂×C₂₃₉	D₂₃	1	1	yes	characteristic, direct product factor
7	C₂₃₉	C₂×D₂₃	1	1	yes	characteristic, direct product factor, Sylow
8	C₂×D₂₃	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₂₃	C₂×C₂₃₉	1	1	yes	characteristic, complemented
10	D₂₃	C₂×C₂₃₉	1	2	yes	direct product factor
11	C₂₃	C₂²×C₂₃₉	1	1	yes	characteristic, complemented, Sylow
12	C₂²	—	23	1	no	Sylow
13	C₂	—	23	2	no	—
14	C₂	D₂₃×C₂₃₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₂₃₉	D₂₃
commutator, cosocle, hypocenter	C₂₃	C₂²×C₂₃₉
Frattini	C₁	G
Fitting, socle	C₂×C₂₃×C₂₃₉	C₂
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₂×C₂₃₉

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×C₂₃₉	C₂	semidirect
2. D₂₃×C₂₃₉	C₂	direct (N × H)
3. C₂₃×C₂₃₉	C₂²	semidirect
6. C₂×C₂₃₉	D₂₃	direct (N × H)
7. C₂₃₉	C₂×D₂₃	direct (N × H)
8. C₂×D₂₃	C₂₃₉	direct (N × H)
9. C₂×C₂₃	C₂×C₂₃₉	semidirect
10. D₂₃	C₂×C₂₃₉	direct (N × H)
11. C₂₃	C₂²×C₂₃₉	semidirect
14. C₂	D₂₃×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
C₂×C₂₃×C₂₃₉	2
D₂₃×C₂₃₉	2
C₂²×C₂₃₉	23
C₂×D₂₃	239