Subgroups — C₃×C₅×M₁₆ #6724

Order 240.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₈	C₂	1	2	yes	maximal, complemented
3	C₅×M₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₅	C₄	1	1	yes	characteristic
5	C₃×C₄×C₅	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₃×M₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₄×C₅	C₂×C₃	1	1	yes	characteristic
9	C₅×C₈	C₂×C₃	1	2	yes	complemented
10	C₂×C₃×C₅	—	2	1	no	—
11	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃×C₄	C₂×C₅	1	1	yes	characteristic
13	C₃×C₈	C₂×C₅	1	2	yes	complemented
14	C₂²×C₅	C₃×C₄	1	1	yes	characteristic
15	C₄×C₅	C₃×C₄	1	1	yes	characteristic
16	C₄×C₅	C₂²×C₃	1	1	yes	characteristic
17	M₁₆	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
18	C₃×C₅	M₁₆	1	1	yes	characteristic, direct product factor, Hall
19	C₂²×C₃	C₄×C₅	1	1	yes	characteristic
20	C₃×C₄	C₄×C₅	1	1	yes	characteristic
21	C₃×C₄	C₂²×C₅	1	1	yes	characteristic
22	C₂×C₅	—	2	1	no	—
23	C₂×C₅	C₂×C₃×C₄	1	1	yes	characteristic
24	C₂×C₄	C₂×C₃×C₅	1	1	yes	characteristic
25	C₈	C₂×C₃×C₅	1	2	yes	complemented
26	C₂×C₃	—	2	1	no	—
27	C₂×C₃	C₂×C₄×C₅	1	1	yes	characteristic
28	C₅	C₃×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
29	C₂²	C₃×C₄×C₅	1	1	yes	characteristic
30	C₄	C₃×C₄×C₅	1	1	yes	characteristic
31	C₄	C₂²×C₃×C₅	1	1	yes	characteristic
32	C₃	C₅×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
33	C₂	—	2	1	no	—
34	C₂	C₂×C₃×C₄×C₅	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	C₂²
commutator	C₂	C₂×C₃×C₄×C₅
Frattini, cosocle	C₄	C₂²×C₃×C₅
socle	C₂×C₃×C₅	C₂×C₄
prime radical	C₂²×C₃×C₅	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₈	C₂	semidirect
3. C₅×M₁₆	C₃	direct (N × H)
7. C₃×M₁₆	C₅	direct (N × H)
9. C₅×C₈	C₂×C₃	semidirect
13. C₃×C₈	C₂×C₅	semidirect
17. M₁₆	C₃×C₅	direct (N × H)
18. C₃×C₅	M₁₆	direct (N × H)
25. C₈	C₂×C₃×C₅	semidirect
28. C₅	C₃×M₁₆	direct (N × H)
32. C₃	C₅×M₁₆	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×C₄×C₅	2
C₃×C₅×C₈	2
C₅×M₁₆	3
C₃×M₁₆	5

Subgroups — C3×C5×M16 #6724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×M₁₆ #6724