Finite Group Databasebeta

← C₃²×C₅×C₅₂₃ #37555261

Subgroups — C₃²×C₅×C₅₂₃ #37555261

Order 23535.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₅₂₃	C₃	1	4	yes	maximal, direct product factor
2	C₃²×C₅₂₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₅×C₅₂₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
4	C₃×C₅₂₃	C₃×C₅	1	4	yes	direct product factor
5	C₅₂₃	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
6	C₃²×C₅	C₅₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅	C₃×C₅₂₃	1	4	yes	direct product factor
8	C₃²	C₅×C₅₂₃	1	1	yes	characteristic, direct product factor, Sylow
9	C₅	C₃²×C₅₂₃	1	1	yes	characteristic, direct product factor, Sylow
10	C₃	C₃×C₅×C₅₂₃	1	4	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₅₂₃	C₃	direct (N × H)
2. C₃²×C₅₂₃	C₅	direct (N × H)
3. C₅×C₅₂₃	C₃²	direct (N × H)
4. C₃×C₅₂₃	C₃×C₅	direct (N × H)
5. C₅₂₃	C₃²×C₅	direct (N × H)
6. C₃²×C₅	C₅₂₃	direct (N × H)
7. C₃×C₅	C₃×C₅₂₃	direct (N × H)
8. C₃²	C₅×C₅₂₃	direct (N × H)
9. C₅	C₃²×C₅₂₃	direct (N × H)
10. C₃	C₃×C₅×C₅₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₅₂₃	3
C₃²×C₅₂₃	5
C₃²×C₅	523