Subgroups — C₃×C₄×C₅×C₂₂₃ #27211413

Order 13380.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₂₂₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₂₂₃	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₃×C₄×C₂₂₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₅×C₂₂₃	C₂×C₃	1	1	yes	characteristic
6	C₂×C₃×C₂₂₃	C₂×C₅	1	1	yes	characteristic
7	C₅×C₂₂₃	C₃×C₄	1	1	yes	characteristic, direct product factor, Hall
8	C₄×C₂₂₃	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
9	C₃×C₂₂₃	C₄×C₅	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₂₂₃	C₂×C₃×C₅	1	1	yes	characteristic
11	C₂₂₃	C₃×C₄×C₅	1	1	yes	characteristic, direct product factor, Sylow
12	C₃×C₄×C₅	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅	C₂×C₂₂₃	1	1	yes	characteristic
14	C₄×C₅	C₃×C₂₂₃	1	1	yes	characteristic, direct product factor, Hall
15	C₃×C₅	C₄×C₂₂₃	1	1	yes	characteristic, direct product factor, Hall
16	C₃×C₄	C₅×C₂₂₃	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₅	C₂×C₃×C₂₂₃	1	1	yes	characteristic
18	C₂×C₃	C₂×C₅×C₂₂₃	1	1	yes	characteristic
19	C₅	C₃×C₄×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
20	C₄	C₃×C₅×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
21	C₃	C₄×C₅×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
22	C₂	C₂×C₃×C₅×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃×C₅×C₂₂₃
socle, prime radical	C₂×C₃×C₅×C₂₂₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×C₂₂₃	C₃	direct (N × H)
3. C₃×C₅×C₂₂₃	C₄	direct (N × H)
4. C₃×C₄×C₂₂₃	C₅	direct (N × H)
7. C₅×C₂₂₃	C₃×C₄	direct (N × H)
8. C₄×C₂₂₃	C₃×C₅	direct (N × H)
9. C₃×C₂₂₃	C₄×C₅	direct (N × H)
11. C₂₂₃	C₃×C₄×C₅	direct (N × H)
12. C₃×C₄×C₅	C₂₂₃	direct (N × H)
14. C₄×C₅	C₃×C₂₂₃	direct (N × H)
15. C₃×C₅	C₄×C₂₂₃	direct (N × H)
16. C₃×C₄	C₅×C₂₂₃	direct (N × H)
19. C₅	C₃×C₄×C₂₂₃	direct (N × H)
20. C₄	C₃×C₅×C₂₂₃	direct (N × H)
21. C₃	C₄×C₅×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₂₂₃	2
C₄×C₅×C₂₂₃	3
C₃×C₄×C₂₂₃	5
C₃×C₄×C₅	223

Subgroups — C3×C4×C5×C223 #27211413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₅×C₂₂₃ #27211413