Subgroups — C₃²×C₅×2₋¹⁺⁴ #13089523

Order 1440.

876 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×#41	C₂	1	10	yes	maximal, complemented
2	C₂×C₃²×C₅×Q₈	C₂	1	5	yes	maximal, complemented
3	C₃×C₅×2₋¹⁺⁴	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₅×Q₈	C₂²	1	10	yes	—
5	C₃²×C₅×D₄	C₂²	1	10	yes	—
6	C₂×C₃²×C₄×C₅	C₂²	1	15	yes	—
7	C₃²×2₋¹⁺⁴	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₅×#41	C₂×C₃	1	40	yes	complemented
9	C₂×C₃×C₅×Q₈	C₂×C₃	1	20	yes	complemented
10	C₂²×C₃²×C₅	C₂³	1	5	yes	—
11	C₃²×C₄×C₅	C₂³	1	10	yes	—
12	C₅×2₋¹⁺⁴	C₃²	1	1	yes	characteristic, direct product factor, Hall
13	C₃²×#41	C₂×C₅	1	10	yes	complemented
14	C₂×C₃²×Q₈	C₂×C₅	1	5	yes	complemented
15	C₃×C₅×Q₈	C₂²×C₃	1	40	yes	—
16	C₃×C₅×D₄	C₂²×C₃	1	40	yes	—
17	C₂×C₃×C₄×C₅	C₂²×C₃	1	60	yes	—
18	C₃×2₋¹⁺⁴	C₃×C₅	1	4	yes	direct product factor
19	C₂×C₃²×C₅	—	2	5	no	—
20	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
21	C₅×#41	C₂×C₃²	1	10	yes	complemented
22	C₂×C₅×Q₈	C₂×C₃²	1	5	yes	complemented
23	C₃²×Q₈	C₂²×C₅	1	10	yes	—
24	C₃²×D₄	C₂²×C₅	1	10	yes	—
25	C₂×C₃²×C₄	C₂²×C₅	1	15	yes	—
26	C₂²×C₃×C₅	C₂³×C₃	1	20	yes	—
27	C₃×C₄×C₅	C₂³×C₃	1	40	yes	—
28	C₃×#41	C₂×C₃×C₅	1	40	yes	complemented
29	C₂×C₃×Q₈	C₂×C₃×C₅	1	20	yes	complemented
30	C₃²×C₅	2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Hall
31	C₅×Q₈	C₂²×C₃²	1	10	yes	—
32	C₅×D₄	C₂²×C₃²	1	10	yes	—
33	C₂×C₄×C₅	C₂²×C₃²	1	15	yes	—
34	C₂²×C₃²	C₂³×C₅	1	5	yes	—
35	C₃²×C₄	C₂³×C₅	1	10	yes	—
36	2₋¹⁺⁴	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×C₃×C₅	—	2	20	no	—
38	C₂×C₃×C₅	C₂⁴×C₃	1	4	yes	—
39	C₃×Q₈	C₂²×C₃×C₅	1	40	yes	—
40	C₃×D₄	C₂²×C₃×C₅	1	40	yes	—
41	C₂×C₃×C₄	C₂²×C₃×C₅	1	60	yes	—
42	C₂²×C₅	C₂³×C₃²	1	5	yes	—
43	C₄×C₅	C₂³×C₃²	1	10	yes	—
44	C₂×C₃²	—	2	5	no	—
45	C₂×C₃²	C₂⁴×C₅	1	1	yes	characteristic
46	#41	C₂×C₃²×C₅	1	10	yes	complemented
47	C₂×Q₈	C₂×C₃²×C₅	1	5	yes	complemented
48	C₃×C₅	C₃×2₋¹⁺⁴	1	4	yes	direct product factor
49	C₂²×C₃	C₂³×C₃×C₅	1	20	yes	—
50	C₃×C₄	C₂³×C₃×C₅	1	40	yes	—
51	C₂×C₅	—	2	5	no	—
52	C₂×C₅	C₂⁴×C₃²	1	1	yes	characteristic
53	C₃²	C₅×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
54	Q₈	C₂²×C₃²×C₅	1	10	yes	—
55	D₄	C₂²×C₃²×C₅	1	10	yes	—
56	C₂×C₄	C₂²×C₃²×C₅	1	15	yes	—
57	C₂×C₃	—	2	20	no	—
58	C₂×C₃	C₂⁴×C₃×C₅	1	4	yes	—
59	C₅	C₃²×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
60	C₂²	C₂³×C₃²×C₅	1	5	yes	—
61	C₄	C₂³×C₃²×C₅	1	10	yes	—
62	C₃	C₃×C₅×2₋¹⁺⁴	1	4	yes	direct product factor
63	C₂	—	2	5	no	—
64	C₂	C₂⁴×C₃²×C₅	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²×C₅	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁴×C₃²×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×#41	C₂	semidirect
2. C₂×C₃²×C₅×Q₈	C₂	semidirect
3. C₃×C₅×2₋¹⁺⁴	C₃	direct (N × H)
7. C₃²×2₋¹⁺⁴	C₅	direct (N × H)
8. C₃×C₅×#41	C₂×C₃	semidirect
9. C₂×C₃×C₅×Q₈	C₂×C₃	semidirect
12. C₅×2₋¹⁺⁴	C₃²	direct (N × H)
13. C₃²×#41	C₂×C₅	semidirect
14. C₂×C₃²×Q₈	C₂×C₅	semidirect
18. C₃×2₋¹⁺⁴	C₃×C₅	direct (N × H)
21. C₅×#41	C₂×C₃²	semidirect
22. C₂×C₅×Q₈	C₂×C₃²	semidirect
28. C₃×#41	C₂×C₃×C₅	semidirect
29. C₂×C₃×Q₈	C₂×C₃×C₅	semidirect
30. C₃²×C₅	2₋¹⁺⁴	direct (N × H)
36. 2₋¹⁺⁴	C₃²×C₅	direct (N × H)
46. #41	C₂×C₃²×C₅	semidirect
47. C₂×Q₈	C₂×C₃²×C₅	semidirect
48. C₃×C₅	C₃×2₋¹⁺⁴	direct (N × H)
53. C₃²	C₅×2₋¹⁺⁴	direct (N × H)
59. C₅	C₃²×2₋¹⁺⁴	direct (N × H)
62. C₃	C₃×C₅×2₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅×#41	2
C₂×C₃²×C₅×Q₈	2
C₃×C₅×2₋¹⁺⁴	3
C₃²×2₋¹⁺⁴	5

Subgroups — C32×C5×2−1+4 #13089523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×2₋¹⁺⁴ #13089523