Finite Group Databasebeta

← C₃×C₁₃×C₅₃⋊C₁₃ #41001802

Subgroups — C₃×C₁₃×C₅₃⋊C₁₃ #41001802

Order 26871.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×C₅₃⋊C₁₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₃×C₁₃×C₅₃	C₁₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅₃⋊C₁₃	C₁₃	1	13	yes	maximal, direct product factor
4	C₁₃×C₅₃	C₃×C₁₃	1	1	yes	characteristic, complemented
5	C₅₃⋊C₁₃	C₃×C₁₃	1	13	yes	direct product factor
6	C₃×C₁₃²	—	53	1	no	maximal, Hall
7	C₁₃²	—	53	1	no	Sylow
8	C₃×C₅₃	C₁₃²	1	1	yes	characteristic, complemented, Hall
9	C₅₃	C₃×C₁₃²	1	1	yes	characteristic, complemented, Sylow
10	C₃×C₁₃	—	53	13	no	—
11	C₃×C₁₃	C₅₃⋊C₁₃	1	1	yes	characteristic, direct product factor
12	C₁₃	—	53	13	no	—
13	C₁₃	C₃×C₅₃⋊C₁₃	1	1	yes	characteristic, direct product factor
14	C₃	C₁₃×C₅₃⋊C₁₃	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₃×C₁₃	C₅₃⋊C₁₃
commutator, cosocle, hypocenter	C₅₃	C₃×C₁₃²
Frattini	C₁	G
Fitting, socle	C₃×C₁₃×C₅₃	C₁₃
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₃×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₁₃×C₅₃⋊C₁₃	C₃	direct (N × H)
2. C₃×C₁₃×C₅₃	C₁₃	semidirect
3. C₃×C₅₃⋊C₁₃	C₁₃	direct (N × H)
4. C₁₃×C₅₃	C₃×C₁₃	semidirect
5. C₅₃⋊C₁₃	C₃×C₁₃	direct (N × H)
8. C₃×C₅₃	C₁₃²	semidirect
9. C₅₃	C₃×C₁₃²	semidirect
11. C₃×C₁₃	C₅₃⋊C₁₃	direct (N × H)
13. C₁₃	C₃×C₅₃⋊C₁₃	direct (N × H)
14. C₃	C₁₃×C₅₃⋊C₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×C₅₃⋊C₁₃	3
C₃×C₁₃×C₅₃	13
C₃×C₅₃⋊C₁₃	13
C₃×C₁₃²	53