Finite Group Databasebeta

← A₄×C₁₉×C₆₇ #29504167

Subgroups — A₄×C₁₉×C₆₇ #29504167

Order 15276.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×C₆₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₁₉×C₆₇	—	4	1	no	maximal, Hall
3	C₂×C₁₉×C₆₇	—	3	1	no	—
4	C₁₉×C₆₇	A₄	1	1	yes	characteristic, direct product factor, Hall
5	A₄×C₆₇	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₆₇	C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
7	A₄×C₁₉	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₆₇	—	4	1	no	Hall
9	C₂×C₆₇	—	3	1	no	—
10	C₂²×C₁₉	C₃×C₆₇	1	1	yes	characteristic, complemented, Hall
11	C₆₇	A₄×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
12	C₃×C₁₉	—	4	1	no	Hall
13	C₂×C₁₉	—	3	1	no	—
14	C₁₉	A₄×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
15	A₄	C₁₉×C₆₇	1	1	yes	characteristic, direct product factor, Hall
16	C₂²	C₃×C₁₉×C₆₇	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	4	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₉×C₆₇	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₁₉×C₆₇
Frattini	C₁	G
Fitting, socle	C₂²×C₁₉×C₆₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₁₉×C₆₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₉×C₆₇	C₃	semidirect
4. C₁₉×C₆₇	A₄	direct (N × H)
5. A₄×C₆₇	C₁₉	direct (N × H)
6. C₂²×C₆₇	C₃×C₁₉	semidirect
7. A₄×C₁₉	C₆₇	direct (N × H)
10. C₂²×C₁₉	C₃×C₆₇	semidirect
11. C₆₇	A₄×C₁₉	direct (N × H)
14. C₁₉	A₄×C₆₇	direct (N × H)
15. A₄	C₁₉×C₆₇	direct (N × H)
16. C₂²	C₃×C₁₉×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×C₆₇	3
C₃×C₁₉×C₆₇	4
A₄×C₆₇	19
A₄×C₁₉	67