Subgroups — C₅×C₉×Q₁₆ #10623619

Order 720.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×C₉	C₂	1	2	yes	maximal
2	C₅×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×Q₁₆	C₃	1	1	yes	maximal, characteristic
4	C₄×C₅×C₉	—	2	2	no	—
5	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
6	C₉×Q₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×Q₈	C₂×C₃	1	2	yes	—
8	C₃×C₅×C₈	C₂×C₃	1	1	yes	characteristic
9	C₂×C₅×C₉	D₄	1	1	yes	characteristic
10	C₅×Q₁₆	C₉	1	1	yes	characteristic, direct product factor, Hall
11	Q₈×C₉	C₂×C₅	1	2	yes	—
12	C₈×C₉	C₂×C₅	1	1	yes	characteristic
13	C₃×C₄×C₅	—	2	2	no	—
14	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
15	C₃×Q₁₆	C₃×C₅	1	1	yes	characteristic
16	C₅×C₉	Q₁₆	1	1	yes	characteristic, direct product factor, Hall
17	C₅×Q₈	C₂×C₉	1	2	yes	—
18	C₅×C₈	C₂×C₉	1	1	yes	characteristic
19	C₄×C₉	—	2	2	no	—
20	C₄×C₉	C₂²×C₅	1	1	yes	characteristic
21	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
22	C₃×Q₈	C₂×C₃×C₅	1	2	yes	—
23	C₃×C₈	C₂×C₃×C₅	1	1	yes	characteristic
24	C₄×C₅	—	2	2	no	—
25	C₄×C₅	C₂²×C₉	1	1	yes	characteristic
26	C₂×C₉	C₅×D₄	1	1	yes	characteristic
27	Q₁₆	C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
28	C₃×C₅	C₃×Q₁₆	1	1	yes	characteristic
29	C₃×C₄	—	2	2	no	—
30	C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
31	C₂×C₅	D₄×C₉	1	1	yes	characteristic
32	C₉	C₅×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
33	Q₈	C₂×C₅×C₉	1	2	yes	—
34	C₈	C₂×C₅×C₉	1	1	yes	characteristic
35	C₂×C₃	C₃×C₅×D₄	1	1	yes	characteristic
36	C₅	C₉×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
37	C₄	—	2	2	no	—
38	C₄	C₂²×C₅×C₉	1	1	yes	characteristic
39	C₃	C₃×C₅×Q₁₆	1	1	yes	characteristic
40	C₂	C₅×D₄×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₉	D₄
commutator	C₄	C₂²×C₅×C₉
Frattini, cosocle	C₃×C₄	C₂²×C₃×C₅
socle, prime radical	C₂×C₃×C₅	C₃×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₉ ◁ C₄×C₅×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₉×Q₁₆	C₅	direct (N × H)
10. C₅×Q₁₆	C₉	direct (N × H)
16. C₅×C₉	Q₁₆	direct (N × H)
27. Q₁₆	C₅×C₉	direct (N × H)
32. C₉	C₅×Q₁₆	direct (N × H)
36. C₅	C₉×Q₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×Q₈×C₉	2
C₅×C₈×C₉	2
C₃×C₅×Q₁₆	3
C₉×Q₁₆	5

Subgroups — C5×C9×Q16 #10623619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₉×Q₁₆ #10623619