Subgroups — C₂³×C₅×3₋¹⁺² #11762016

Order 1080.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×3₋¹⁺²	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₃²×C₅	C₃	1	1	yes	maximal, characteristic
3	C₂³×C₅×C₉	C₃	1	3	yes	maximal, complemented
4	C₂×C₅×3₋¹⁺²	C₂²	1	7	yes	direct product factor
5	C₂³×3₋¹⁺²	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃²×C₅	C₂×C₃	1	7	yes	—
7	C₂²×C₅×C₉	C₂×C₃	1	21	yes	complemented
8	C₅×3₋¹⁺²	C₂³	1	1	yes	characteristic, direct product factor, Hall
9	C₂³×C₃×C₅	—	3	1	no	—
10	C₂³×C₃×C₅	C₃²	1	1	yes	characteristic
11	C₂²×3₋¹⁺²	C₂×C₅	1	7	yes	direct product factor
12	C₂×C₃²×C₅	C₂²×C₃	1	7	yes	—
13	C₂×C₅×C₉	C₂²×C₃	1	21	yes	complemented
14	C₂³×C₃²	C₃×C₅	1	1	yes	characteristic
15	C₂³×C₉	C₃×C₅	1	3	yes	complemented
16	C₂²×C₃×C₅	—	3	7	no	—
17	C₂²×C₃×C₅	C₂×C₃²	1	7	yes	—
18	C₂×3₋¹⁺²	C₂²×C₅	1	7	yes	direct product factor
19	C₃²×C₅	C₂³×C₃	1	1	yes	characteristic
20	C₅×C₉	C₂³×C₃	1	3	yes	complemented
21	C₂³×C₅	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Hall
22	C₂²×C₃²	C₂×C₃×C₅	1	7	yes	—
23	C₂²×C₉	C₂×C₃×C₅	1	21	yes	complemented
24	C₂×C₃×C₅	—	3	7	no	—
25	C₂×C₃×C₅	C₂²×C₃²	1	7	yes	—
26	3₋¹⁺²	C₂³×C₅	1	1	yes	characteristic, direct product factor, Sylow
27	C₂³×C₃	—	3	1	no	—
28	C₂³×C₃	C₃²×C₅	1	1	yes	characteristic
29	C₂²×C₅	C₂×3₋¹⁺²	1	7	yes	direct product factor
30	C₂×C₃²	C₂²×C₃×C₅	1	7	yes	—
31	C₂×C₉	C₂²×C₃×C₅	1	21	yes	complemented
32	C₃×C₅	—	3	1	no	—
33	C₃×C₅	C₂³×C₃²	1	1	yes	characteristic
34	C₂²×C₃	—	3	7	no	—
35	C₂²×C₃	C₂×C₃²×C₅	1	7	yes	—
36	C₂×C₅	C₂²×3₋¹⁺²	1	7	yes	direct product factor
37	C₃²	C₂³×C₃×C₅	1	1	yes	characteristic
38	C₉	C₂³×C₃×C₅	1	3	yes	complemented
39	C₂³	C₅×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₃	—	3	7	no	—
41	C₂×C₃	C₂²×C₃²×C₅	1	7	yes	—
42	C₅	C₂³×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²	C₂×C₅×3₋¹⁺²	1	7	yes	direct product factor
44	C₃	—	3	1	no	—
45	C₃	C₂³×C₃²×C₅	1	1	yes	characteristic
46	C₂	C₂²×C₅×3₋¹⁺²	1	7	yes	direct product factor

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₃²
commutator, Frattini, cosocle	C₃	C₂³×C₃²×C₅
prime radical	C₂³×C₃²×C₅	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×3₋¹⁺²	C₂	direct (N × H)
3. C₂³×C₅×C₉	C₃	semidirect
4. C₂×C₅×3₋¹⁺²	C₂²	direct (N × H)
5. C₂³×3₋¹⁺²	C₅	direct (N × H)
7. C₂²×C₅×C₉	C₂×C₃	semidirect
8. C₅×3₋¹⁺²	C₂³	direct (N × H)
11. C₂²×3₋¹⁺²	C₂×C₅	direct (N × H)
13. C₂×C₅×C₉	C₂²×C₃	semidirect
15. C₂³×C₉	C₃×C₅	semidirect
18. C₂×3₋¹⁺²	C₂²×C₅	direct (N × H)
20. C₅×C₉	C₂³×C₃	semidirect
21. C₂³×C₅	3₋¹⁺²	direct (N × H)
23. C₂²×C₉	C₂×C₃×C₅	semidirect
26. 3₋¹⁺²	C₂³×C₅	direct (N × H)
29. C₂²×C₅	C₂×3₋¹⁺²	direct (N × H)
31. C₂×C₉	C₂²×C₃×C₅	semidirect
36. C₂×C₅	C₂²×3₋¹⁺²	direct (N × H)
38. C₉	C₂³×C₃×C₅	semidirect
39. C₂³	C₅×3₋¹⁺²	direct (N × H)
42. C₅	C₂³×3₋¹⁺²	direct (N × H)
43. C₂²	C₂×C₅×3₋¹⁺²	direct (N × H)
46. C₂	C₂²×C₅×3₋¹⁺²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×3₋¹⁺²	2
C₂³×C₃²×C₅	3
C₂³×C₅×C₉	3
C₂³×3₋¹⁺²	5

Subgroups — C23×C5×3−1+2 #11762016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×3₋¹⁺² #11762016