Finite Group Databasebeta

← C₆₁×C₇⋊(C₅²⋊C₃) #44545063

Subgroups — C₆₁×C₇⋊(C₅²⋊C₃) #44545063

Order 32025.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₇×C₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₆₁×C₅²⋊C₃	—	7	1	no	maximal, Hall
3	C₅×C₇×C₆₁	—	3	2	no	—
4	C₅²×C₆₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
5	ΓL(1,𝔽₈)×C₆₁	—	25	1	no	maximal, Hall
6	C₇⋊(C₅²⋊C₃)	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₇×C₆₁	C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
8	C₅×C₆₁	—	3	2	no	—
9	C₃×C₆₁	—	175	1	no	Hall
10	C₅²×C₇	C₃×C₆₁	1	1	yes	characteristic, complemented, Hall
11	C₅²⋊C₃	—	7	1	no	Hall
12	C₆₁	C₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
13	C₅×C₇	—	3	2	no	—
14	C₅²	ΓL(1,𝔽₈)×C₆₁	1	1	yes	characteristic, complemented, Sylow
15	ΓL(1,𝔽₈)	—	25	1	no	Hall
16	C₇	C₆₁×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
17	C₅	—	3	2	no	—
18	C₃	—	175	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₆₁	C₇⋊(C₅²⋊C₃)
commutator, cosocle, hypocenter	C₅²×C₇	C₃×C₆₁
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₆₁	C₃
LCS	G ▷ C₅²×C₇
UCS	C₁ ◁ C₆₁

Complements

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

N	H ≅ G/N	type
1. C₅²×C₇×C₆₁	C₃	semidirect
4. C₅²×C₆₁	ΓL(1,𝔽₈)	semidirect
6. C₇⋊(C₅²⋊C₃)	C₆₁	direct (N × H)
7. C₇×C₆₁	C₅²⋊C₃	semidirect
10. C₅²×C₇	C₃×C₆₁	semidirect
12. C₆₁	C₇⋊(C₅²⋊C₃)	direct (N × H)
14. C₅²	ΓL(1,𝔽₈)×C₆₁	semidirect
16. C₇	C₆₁×C₅²⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₇×C₆₁	3
C₆₁×C₅²⋊C₃	7
ΓL(1,𝔽₈)×C₆₁	25
C₇⋊(C₅²⋊C₃)	61