Subgroups — C₅×C₉×M₁₆ #10623616

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₉	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×C₉	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×M₁₆	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₅×C₉	C₄	1	1	yes	characteristic
5	C₄×C₅×C₉	C₄	1	1	yes	characteristic
6	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
7	C₉×M₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic
9	C₃×C₅×C₈	C₂×C₃	1	2	yes	—
10	C₂×C₅×C₉	—	2	1	no	—
11	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
12	C₅×M₁₆	C₉	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₄×C₉	C₂×C₅	1	1	yes	characteristic
14	C₈×C₉	C₂×C₅	1	2	yes	complemented
15	C₂²×C₃×C₅	C₃×C₄	1	1	yes	characteristic
16	C₃×C₄×C₅	C₃×C₄	1	1	yes	characteristic
17	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
18	C₃×M₁₆	C₃×C₅	1	1	yes	characteristic
19	C₅×C₉	M₁₆	1	1	yes	characteristic, direct product factor, Hall
20	C₂×C₄×C₅	C₂×C₉	1	1	yes	characteristic
21	C₅×C₈	C₂×C₉	1	2	yes	complemented
22	C₂²×C₉	C₄×C₅	1	1	yes	characteristic
23	C₄×C₉	C₄×C₅	1	1	yes	characteristic
24	C₄×C₉	C₂²×C₅	1	1	yes	characteristic
25	C₂×C₃×C₅	—	2	1	no	—
26	C₂×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂×C₃×C₅	1	1	yes	characteristic
28	C₃×C₈	C₂×C₃×C₅	1	2	yes	—
29	C₂²×C₅	C₄×C₉	1	1	yes	characteristic
30	C₄×C₅	C₄×C₉	1	1	yes	characteristic
31	C₄×C₅	C₂²×C₉	1	1	yes	characteristic
32	C₂×C₉	—	2	1	no	—
33	C₂×C₉	C₂×C₄×C₅	1	1	yes	characteristic
34	M₁₆	C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
35	C₃×C₅	C₃×M₁₆	1	1	yes	characteristic
36	C₂²×C₃	C₃×C₄×C₅	1	1	yes	characteristic
37	C₃×C₄	C₃×C₄×C₅	1	1	yes	characteristic
38	C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
39	C₂×C₅	—	2	1	no	—
40	C₂×C₅	C₂×C₄×C₉	1	1	yes	characteristic
41	C₉	C₅×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₄	C₂×C₅×C₉	1	1	yes	characteristic
43	C₈	C₂×C₅×C₉	1	2	yes	complemented
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	C₂×C₃×C₄×C₅	1	1	yes	characteristic
46	C₅	C₉×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²	C₄×C₅×C₉	1	1	yes	characteristic
48	C₄	C₄×C₅×C₉	1	1	yes	characteristic
49	C₄	C₂²×C₅×C₉	1	1	yes	characteristic
50	C₃	C₃×C₅×M₁₆	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	C₂×C₄×C₅×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅×C₉	C₂²
commutator	C₂	C₂×C₄×C₅×C₉
Frattini, cosocle	C₃×C₄	C₂²×C₃×C₅
socle	C₂×C₃×C₅	C₂×C₃×C₄
prime radical	C₂²×C₃×C₅	C₃×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×C₈×C₉	C₂	semidirect
7. C₉×M₁₆	C₅	direct (N × H)
12. C₅×M₁₆	C₉	direct (N × H)
14. C₈×C₉	C₂×C₅	semidirect
19. C₅×C₉	M₁₆	direct (N × H)
21. C₅×C₈	C₂×C₉	semidirect
34. M₁₆	C₅×C₉	direct (N × H)
41. C₉	C₅×M₁₆	direct (N × H)
43. C₈	C₂×C₅×C₉	semidirect
46. C₅	C₉×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×C₄×C₅×C₉	2
C₅×C₈×C₉	2
C₃×C₅×M₁₆	3
C₉×M₁₆	5

Subgroups — C5×C9×M16 #10623616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₉×M₁₆ #10623616