Finite Group Databasebeta

← C₁₇×C₂²⋊(C₁₃₉⋊C₃) #43248596

Subgroups — C₁₇×C₂²⋊(C₁₃₉⋊C₃) #43248596

Order 28356.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₇×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₇×C₁₃₉⋊C₃	—	4	1	no	maximal, Hall
3	C₂×C₁₇×C₁₃₉	—	3	1	no	—
4	C₁₇×C₁₃₉	A₄	1	1	yes	characteristic, complemented, Hall
5	C₂²⋊(C₁₃₉⋊C₃)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₁₃₉	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
7	C₁₃₉⋊C₃	—	4	1	no	Hall
8	C₂×C₁₃₉	—	3	1	no	—
9	A₄×C₁₇	—	139	1	no	maximal, Hall
10	C₁₃₉	A₄×C₁₇	1	1	yes	characteristic, complemented, Sylow
11	C₂²×C₁₇	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented, Hall
12	C₃×C₁₇	—	556	1	no	Hall
13	C₂×C₁₇	—	3	1	no	—
14	C₁₇	C₂²⋊(C₁₃₉⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
15	A₄	—	139	1	no	Hall
16	C₂²	C₁₇×C₁₃₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	556	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₇	C₂²⋊(C₁₃₉⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₁₃₉	C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₂²×C₁₇×C₁₃₉	C₃
LCS	G ▷ C₂²×C₁₃₉
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₇×C₁₃₉	C₃	semidirect
4. C₁₇×C₁₃₉	A₄	semidirect
5. C₂²⋊(C₁₃₉⋊C₃)	C₁₇	direct (N × H)
6. C₂²×C₁₃₉	C₃×C₁₇	semidirect
10. C₁₃₉	A₄×C₁₇	semidirect
11. C₂²×C₁₇	C₁₃₉⋊C₃	semidirect
14. C₁₇	C₂²⋊(C₁₃₉⋊C₃)	direct (N × H)
16. C₂²	C₁₇×C₁₃₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₇×C₁₃₉	3
C₁₇×C₁₃₉⋊C₃	4
C₂²⋊(C₁₃₉⋊C₃)	17
A₄×C₁₇	139