Finite Group Databasebeta

← C₃×C₁₉⋊C₃×C₁₃₇ #37552844

Subgroups — C₃×C₁₉⋊C₃×C₁₃₇ #37552844

Order 23427.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₉×C₁₃₇	C₃	1	1	yes	maximal, characteristic, complemented
2	C₁₉⋊C₃×C₁₃₇	C₃	1	3	yes	maximal, direct product factor
3	C₁₉×C₁₃₇	C₃²	1	1	yes	characteristic, complemented, Hall
4	C₃²×C₁₃₇	—	19	1	no	maximal, Hall
5	C₃×C₁₃₇	—	19	3	no	—
6	C₃×C₁₃₇	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
7	C₃×C₁₉⋊C₃	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₁₃₇	C₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
9	C₃×C₁₉	C₃×C₁₃₇	1	1	yes	characteristic, complemented
10	C₁₉⋊C₃	C₃×C₁₃₇	1	3	yes	direct product factor
11	C₁₉	C₃²×C₁₃₇	1	1	yes	characteristic, complemented, Sylow
12	C₃²	—	19	1	no	Sylow
13	C₃	—	19	3	no	—
14	C₃	C₁₉⋊C₃×C₁₃₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₃×C₁₃₇	C₁₉⋊C₃
commutator, cosocle, hypocenter	C₁₉	C₃²×C₁₃₇
Frattini	C₁	G
Fitting, socle	C₃×C₁₉×C₁₃₇	C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₃×C₁₃₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₉×C₁₃₇	C₃	semidirect
2. C₁₉⋊C₃×C₁₃₇	C₃	direct (N × H)
3. C₁₉×C₁₃₇	C₃²	semidirect
6. C₃×C₁₃₇	C₁₉⋊C₃	direct (N × H)
7. C₃×C₁₉⋊C₃	C₁₃₇	direct (N × H)
8. C₁₃₇	C₃×C₁₉⋊C₃	direct (N × H)
9. C₃×C₁₉	C₃×C₁₃₇	semidirect
10. C₁₉⋊C₃	C₃×C₁₃₇	direct (N × H)
11. C₁₉	C₃²×C₁₃₇	semidirect
14. C₃	C₁₉⋊C₃×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₁₉×C₁₃₇	3
C₁₉⋊C₃×C₁₃₇	3
C₃²×C₁₃₇	19
C₃×C₁₉⋊C₃	137