Finite Group Databasebeta

← C₆₁×#63271 #30659352

Subgroups — C₆₁×#63271 #30659352

Order 16653.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₃×C₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₃⋊C₃×C₆₁	—	7	1	no	maximal, Hall
3	ΓL(1,𝔽₈)×C₆₁	—	13	1	no	maximal, Hall
4	C₁₃×C₆₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
5	C₇×C₆₁	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
6	#63271	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₆₁	—	91	1	no	Hall
8	C₇×C₁₃	C₃×C₆₁	1	1	yes	characteristic, complemented, Hall
9	C₆₁	#63271	1	1	yes	characteristic, direct product factor, Sylow
10	C₁₃⋊C₃	—	7	1	no	Hall
11	ΓL(1,𝔽₈)	—	13	1	no	Hall
12	C₁₃	ΓL(1,𝔽₈)×C₆₁	1	1	yes	characteristic, complemented, Sylow
13	C₇	C₁₃⋊C₃×C₆₁	1	1	yes	characteristic, complemented, Sylow
14	C₃	—	91	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₆₁	#63271
commutator, cosocle, hypocenter	C₇×C₁₃	C₃×C₆₁
Frattini	C₁	G
Fitting, socle	C₇×C₁₃×C₆₁	C₃
LCS	G ▷ C₇×C₁₃
UCS	C₁ ◁ C₆₁

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₃×C₆₁	C₃	semidirect
4. C₁₃×C₆₁	ΓL(1,𝔽₈)	semidirect
5. C₇×C₆₁	C₁₃⋊C₃	semidirect
6. #63271	C₆₁	direct (N × H)
8. C₇×C₁₃	C₃×C₆₁	semidirect
9. C₆₁	#63271	direct (N × H)
12. C₁₃	ΓL(1,𝔽₈)×C₆₁	semidirect
13. C₇	C₁₃⋊C₃×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₁₃×C₆₁	3
C₁₃⋊C₃×C₆₁	7
ΓL(1,𝔽₈)×C₆₁	13
#63271	61