Finite Group Databasebeta

← C₃×C₇×C₃₄₉⋊C₃ #36347113

Subgroups — C₃×C₇×C₃₄₉⋊C₃ #36347113

Order 21987.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₃₄₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇×C₃₄₉⋊C₃	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₃₄₉⋊C₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₇×C₃₄₉	C₃²	1	1	yes	characteristic, complemented, Hall
5	C₃×C₃₄₉	C₃×C₇	1	1	yes	characteristic, complemented
6	C₃₄₉⋊C₃	C₃×C₇	1	3	yes	direct product factor
7	C₃₄₉	C₃²×C₇	1	1	yes	characteristic, complemented, Sylow
8	C₃²×C₇	—	349	1	no	maximal, Hall
9	C₃×C₇	—	349	3	no	—
10	C₃×C₇	C₃₄₉⋊C₃	1	1	yes	characteristic, direct product factor
11	C₃²	—	349	1	no	Sylow
12	C₇	C₃×C₃₄₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
13	C₃	—	349	3	no	—
14	C₃	C₇×C₃₄₉⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₃×C₇	C₃₄₉⋊C₃
commutator, cosocle, hypocenter	C₃₄₉	C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₃₄₉	C₃
LCS	G ▷ C₃₄₉
UCS	C₁ ◁ C₃×C₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₃₄₉	C₃	semidirect
2. C₇×C₃₄₉⋊C₃	C₃	direct (N × H)
3. C₃×C₃₄₉⋊C₃	C₇	direct (N × H)
4. C₇×C₃₄₉	C₃²	semidirect
5. C₃×C₃₄₉	C₃×C₇	semidirect
6. C₃₄₉⋊C₃	C₃×C₇	direct (N × H)
7. C₃₄₉	C₃²×C₇	semidirect
10. C₃×C₇	C₃₄₉⋊C₃	direct (N × H)
12. C₇	C₃×C₃₄₉⋊C₃	direct (N × H)
14. C₃	C₇×C₃₄₉⋊C₃	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×C₇×C₃₄₉	3
C₇×C₃₄₉⋊C₃	3
C₃×C₃₄₉⋊C₃	7
C₃²×C₇	349