Finite Group Databasebeta

← C₇×D₇×C₂₃₉ #37552821

Subgroups — C₇×D₇×C₂₃₉ #37552821

Order 23422.

22 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×C₇×C₂₃₉	—	7	1	no	maximal
3	D₇×C₂₃₉	C₇	1	1	yes	maximal, characteristic, direct product factor
4	C₇×C₂₃₉	—	2	3	no	—
5	C₇×C₂₃₉	D₇	1	1	yes	characteristic, direct product factor
6	C₇×C₂₃₉	C₂×C₇	1	1	yes	characteristic, complemented
7	C₂×C₂₃₉	—	7	1	no	Hall
8	C₂₃₉	C₇×D₇	1	1	yes	characteristic, direct product factor, Sylow
9	C₇×D₇	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₇²	C₂×C₂₃₉	1	1	yes	characteristic, complemented, Sylow
11	C₂×C₇	—	7	1	no	—
12	D₇	C₇×C₂₃₉	1	1	yes	characteristic, direct product factor
13	C₇	—	2	3	no	—
14	C₇	D₇×C₂₃₉	1	1	yes	characteristic, direct product factor
15	C₇	C₂×C₇×C₂₃₉	1	1	yes	characteristic, complemented
16	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇×C₂₃₉	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₇×C₂₃₉
Frattini	C₁	G
Fitting, socle	C₇²×C₂₃₉	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₇×C₂₃₉

Complements

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

N	H ≅ G/N	type
1. C₇²×C₂₃₉	C₂	semidirect
3. D₇×C₂₃₉	C₇	direct (N × H)
5. C₇×C₂₃₉	D₇	direct (N × H)
6. C₇×C₂₃₉	C₂×C₇	semidirect
8. C₂₃₉	C₇×D₇	direct (N × H)
9. C₇×D₇	C₂₃₉	direct (N × H)
10. C₇²	C₂×C₂₃₉	semidirect
12. D₇	C₇×C₂₃₉	direct (N × H)
14. C₇	D₇×C₂₃₉	direct (N × H)
15. C₇	C₂×C₇×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₇²×C₂₃₉	2
C₂×C₇×C₂₃₉	7
D₇×C₂₃₉	7
C₇×D₇	239