Finite Group Databasebeta

← C₅₃×D₂₂₁ #37552835

Subgroups — C₅₃×D₂₂₁ #37552835

Order 23426.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×C₁₇×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₁₇×C₅₃	—	13	1	no	maximal, Hall
3	D₁₃×C₅₃	—	17	1	no	maximal, Hall
4	C₁₇×C₅₃	D₁₃	1	1	yes	characteristic, complemented, Hall
5	C₁₃×C₅₃	D₁₇	1	1	yes	characteristic, complemented, Hall
6	D₂₂₁	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₁₃×C₁₇	C₂×C₅₃	1	1	yes	characteristic, complemented, Hall
8	C₂×C₅₃	—	221	1	no	Hall
9	C₅₃	D₂₂₁	1	1	yes	characteristic, direct product factor, Sylow
10	D₁₇	—	13	1	no	Hall
11	D₁₃	—	17	1	no	Hall
12	C₁₇	D₁₃×C₅₃	1	1	yes	characteristic, complemented, Sylow
13	C₁₃	D₁₇×C₅₃	1	1	yes	characteristic, complemented, Sylow
14	C₂	—	221	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅₃	D₂₂₁
commutator, cosocle, hypocenter	C₁₃×C₁₇	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₁₃×C₁₇×C₅₃	C₂
LCS	G ▷ C₁₃×C₁₇
UCS	C₁ ◁ C₅₃

Complements

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

N	H ≅ G/N	type
1. C₁₃×C₁₇×C₅₃	C₂	semidirect
4. C₁₇×C₅₃	D₁₃	semidirect
5. C₁₃×C₅₃	D₁₇	semidirect
6. D₂₂₁	C₅₃	direct (N × H)
7. C₁₃×C₁₇	C₂×C₅₃	semidirect
9. C₅₃	D₂₂₁	direct (N × H)
12. C₁₇	D₁₃×C₅₃	semidirect
13. C₁₃	D₁₇×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×C₁₇×C₅₃	2
D₁₇×C₅₃	13
D₁₃×C₅₃	17
D₂₂₁	53