Finite Group Databasebeta

← C₂×C₇×C₂₃₉⋊C₇ #37552819

Subgroups — C₂×C₇×C₂₃₉⋊C₇ #37552819

Order 23422.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂₃₉⋊C₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₇×C₂₃₉	C₇	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃₉⋊C₇	C₇	1	7	yes	maximal, direct product factor
4	C₇×C₂₃₉	C₂×C₇	1	1	yes	characteristic, complemented
5	C₂₃₉⋊C₇	C₂×C₇	1	7	yes	direct product factor
6	C₂×C₂₃₉	C₇²	1	1	yes	characteristic, complemented, Hall
7	C₂₃₉	C₂×C₇²	1	1	yes	characteristic, complemented, Sylow
8	C₂×C₇²	—	239	1	no	maximal, Hall
9	C₇²	—	239	1	no	Sylow
10	C₂×C₇	—	239	7	no	—
11	C₂×C₇	C₂₃₉⋊C₇	1	1	yes	characteristic, direct product factor
12	C₇	—	239	7	no	—
13	C₇	C₂×C₂₃₉⋊C₇	1	1	yes	characteristic, direct product factor
14	C₂	C₇×C₂₃₉⋊C₇	1	1	yes	characteristic, direct product factor, Sylow

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

Complements

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

N	H ≅ G/N	type
1. C₇×C₂₃₉⋊C₇	C₂	direct (N × H)
2. C₂×C₇×C₂₃₉	C₇	semidirect
3. C₂×C₂₃₉⋊C₇	C₇	direct (N × H)
4. C₇×C₂₃₉	C₂×C₇	semidirect
5. C₂₃₉⋊C₇	C₂×C₇	direct (N × H)
6. C₂×C₂₃₉	C₇²	semidirect
7. C₂₃₉	C₂×C₇²	semidirect
11. C₂×C₇	C₂₃₉⋊C₇	direct (N × H)
13. C₇	C₂×C₂₃₉⋊C₇	direct (N × H)
14. C₂	C₇×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
C₇×C₂₃₉⋊C₇	2
C₂×C₇×C₂₃₉	7
C₂×C₂₃₉⋊C₇	7
C₂×C₇²	239