Finite Group Databasebeta

← C₃×C₁₆×C₂₂₃ #72189253

Subgroups — C₃×C₁₆×C₂₂₃ #72189253

Order 10704.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₂₂₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₄×C₂₂₃	C₄	1	1	yes	characteristic
4	C₈×C₂₂₃	C₂×C₃	1	1	yes	characteristic
5	C₂×C₃×C₂₂₃	C₈	1	1	yes	characteristic
6	C₄×C₂₂₃	C₃×C₄	1	1	yes	characteristic
7	C₃×C₂₂₃	C₁₆	1	1	yes	characteristic, direct product factor, Hall
8	C₂×C₂₂₃	C₃×C₈	1	1	yes	characteristic
9	C₂₂₃	C₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
10	C₃×C₁₆	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₈	C₂×C₂₂₃	1	1	yes	characteristic
12	C₁₆	C₃×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
13	C₃×C₄	C₄×C₂₂₃	1	1	yes	characteristic
14	C₈	C₂×C₃×C₂₂₃	1	1	yes	characteristic
15	C₂×C₃	C₈×C₂₂₃	1	1	yes	characteristic
16	C₄	C₃×C₄×C₂₂₃	1	1	yes	characteristic
17	C₃	C₁₆×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
18	C₂	C₃×C₈×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₃×C₂₂₃
socle, prime radical	C₂×C₃×C₂₂₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₆×C₂₂₃	C₃	direct (N × H)
7. C₃×C₂₂₃	C₁₆	direct (N × H)
9. C₂₂₃	C₃×C₁₆	direct (N × H)
10. C₃×C₁₆	C₂₂₃	direct (N × H)
12. C₁₆	C₃×C₂₂₃	direct (N × H)
17. C₃	C₁₆×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
C₃×C₈×C₂₂₃	2
C₁₆×C₂₂₃	3
C₃×C₁₆	223