Finite Group Databasebeta

← C₂²⋊ΓL(1,𝔽₈)×C₄₃₃ #49138459

Subgroups — C₂²⋊ΓL(1,𝔽₈)×C₄₃₃ #49138459

Order 36372.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄₃₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)×C₄₃₃	—	4	1	no	maximal, Hall
3	C₂×C₇×C₄₃₃	—	3	1	no	—
4	A₄×C₄₃₃	—	7	1	no	maximal, Hall
5	C₇×C₄₃₃	A₄	1	1	yes	characteristic, complemented, Hall
6	C₂²×C₄₃₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
7	C₃×C₄₃₃	—	28	1	no	Hall
8	C₂×C₄₃₃	—	3	1	no	—
9	C₄₃₃	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
10	C₂²⋊ΓL(1,𝔽₈)	C₄₃₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₇	C₃×C₄₃₃	1	1	yes	characteristic, complemented, Hall
12	ΓL(1,𝔽₈)	—	4	1	no	Hall
13	C₂×C₇	—	3	1	no	—
14	A₄	—	7	1	no	Hall
15	C₇	A₄×C₄₃₃	1	1	yes	characteristic, complemented, Sylow
16	C₂²	ΓL(1,𝔽₈)×C₄₃₃	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	28	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₄₃₃	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₄₃₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₄₃₃	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₄₃₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₄₃₃	C₃	semidirect
5. C₇×C₄₃₃	A₄	semidirect
6. C₂²×C₄₃₃	ΓL(1,𝔽₈)	semidirect
9. C₄₃₃	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
10. C₂²⋊ΓL(1,𝔽₈)	C₄₃₃	direct (N × H)
11. C₂²×C₇	C₃×C₄₃₃	semidirect
15. C₇	A₄×C₄₃₃	semidirect
16. C₂²	ΓL(1,𝔽₈)×C₄₃₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₄₃₃	3
ΓL(1,𝔽₈)×C₄₃₃	4
A₄×C₄₃₃	7
C₂²⋊ΓL(1,𝔽₈)	433