Finite Group Databasebeta

← A₄×C₃₁×C₅₃ #34089331

Subgroups — A₄×C₃₁×C₅₃ #34089331

Order 19716.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₃₁×C₅₃	—	4	1	no	maximal, Hall
3	C₂×C₃₁×C₅₃	—	3	1	no	—
4	C₃₁×C₅₃	A₄	1	1	yes	characteristic, direct product factor, Hall
5	A₄×C₅₃	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
6	A₄×C₃₁	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₅₃	C₃×C₃₁	1	1	yes	characteristic, complemented, Hall
8	C₃×C₅₃	—	4	1	no	Hall
9	C₂²×C₃₁	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₅₃	—	3	1	no	—
11	C₃×C₃₁	—	4	1	no	Hall
12	C₂×C₃₁	—	3	1	no	—
13	C₅₃	A₄×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
14	C₃₁	A₄×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
15	A₄	C₃₁×C₅₃	1	1	yes	characteristic, direct product factor, Hall
16	C₂²	C₃×C₃₁×C₅₃	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	4	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃₁×C₅₃	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₃₁×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃₁×C₅₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃₁×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×C₅₃	C₃	semidirect
4. C₃₁×C₅₃	A₄	direct (N × H)
5. A₄×C₅₃	C₃₁	direct (N × H)
6. A₄×C₃₁	C₅₃	direct (N × H)
7. C₂²×C₅₃	C₃×C₃₁	semidirect
9. C₂²×C₃₁	C₃×C₅₃	semidirect
13. C₅₃	A₄×C₃₁	direct (N × H)
14. C₃₁	A₄×C₅₃	direct (N × H)
15. A₄	C₃₁×C₅₃	direct (N × H)
16. C₂²	C₃×C₃₁×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₅₃	3
C₃×C₃₁×C₅₃	4
A₄×C₅₃	31
A₄×C₃₁	53