Subgroups — C₂×C₅×Q₈×C₉ #10623718

Order 720.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×C₉	C₂	1	4	yes	maximal, direct product factor
2	C₂×C₄×C₅×C₉	C₂	1	3	yes	maximal
3	C₂×C₃×C₅×Q₈	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
5	C₄×C₅×C₉	C₂²	1	6	yes	—
6	C₂×Q₈×C₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×Q₈	C₂×C₃	1	4	yes	—
8	C₂×C₃×C₄×C₅	C₂×C₃	1	3	yes	—
9	C₂×C₅×C₉	Q₈	1	2	yes	direct product factor
10	C₂×C₅×C₉	C₂³	1	1	yes	characteristic
11	C₂×C₅×Q₈	C₉	1	1	yes	characteristic, direct product factor, Hall
12	Q₈×C₉	C₂×C₅	1	4	yes	direct product factor
13	C₂×C₄×C₉	C₂×C₅	1	3	yes	—
14	C₂²×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
15	C₃×C₄×C₅	C₂²×C₃	1	6	yes	—
16	C₂×C₃×Q₈	C₃×C₅	1	1	yes	characteristic
17	C₅×C₉	C₂×Q₈	1	1	yes	characteristic, direct product factor, Hall
18	C₅×Q₈	C₂×C₉	1	4	yes	direct product factor
19	C₂×C₄×C₅	C₂×C₉	1	3	yes	—
20	C₂²×C₉	C₂²×C₅	1	1	yes	characteristic
21	C₄×C₉	C₂²×C₅	1	6	yes	—
22	C₂×C₃×C₅	C₃×Q₈	1	2	yes	—
23	C₂×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
24	C₃×Q₈	C₂×C₃×C₅	1	4	yes	—
25	C₂×C₃×C₄	C₂×C₃×C₅	1	3	yes	—
26	C₂²×C₅	C₂²×C₉	1	1	yes	characteristic
27	C₄×C₅	C₂²×C₉	1	6	yes	—
28	C₂×C₉	C₅×Q₈	1	2	yes	direct product factor
29	C₂×C₉	C₂³×C₅	1	1	yes	characteristic
30	C₂×Q₈	C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
31	C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
32	C₂²×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
33	C₃×C₄	C₂²×C₃×C₅	1	6	yes	—
34	C₂×C₅	Q₈×C₉	1	2	yes	direct product factor
35	C₂×C₅	C₂³×C₉	1	1	yes	characteristic
36	C₉	C₂×C₅×Q₈	1	1	yes	characteristic, direct product factor, Sylow
37	Q₈	C₂×C₅×C₉	1	4	yes	direct product factor
38	C₂×C₄	C₂×C₅×C₉	1	3	yes	—
39	C₂×C₃	C₃×C₅×Q₈	1	2	yes	—
40	C₂×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
41	C₅	C₂×Q₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	C₂²×C₅×C₉	1	1	yes	characteristic
43	C₄	C₂²×C₅×C₉	1	6	yes	—
44	C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
45	C₂	C₅×Q₈×C₉	1	2	yes	direct product factor
46	C₂	C₂³×C₅×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅×C₉	C₂²
commutator	C₂	C₂³×C₅×C₉
Frattini, cosocle	C₂×C₃	C₂³×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₂²×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×C₉	C₂	direct (N × H)
6. C₂×Q₈×C₉	C₅	direct (N × H)
9. C₂×C₅×C₉	Q₈	direct (N × H)
11. C₂×C₅×Q₈	C₉	direct (N × H)
12. Q₈×C₉	C₂×C₅	direct (N × H)
17. C₅×C₉	C₂×Q₈	direct (N × H)
18. C₅×Q₈	C₂×C₉	direct (N × H)
28. C₂×C₉	C₅×Q₈	direct (N × H)
30. C₂×Q₈	C₅×C₉	direct (N × H)
34. C₂×C₅	Q₈×C₉	direct (N × H)
36. C₉	C₂×C₅×Q₈	direct (N × H)
37. Q₈	C₂×C₅×C₉	direct (N × H)
41. C₅	C₂×Q₈×C₉	direct (N × H)
45. C₂	C₅×Q₈×C₉	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
C₂×C₄×C₅×C₉	2
C₂×C₃×C₅×Q₈	3
C₂×Q₈×C₉	5

Subgroups — C2×C5×Q8×C9 #10623718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×Q₈×C₉ #10623718