Subgroups — C₃×C₅×#126 #91855

Order 480.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₃×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#126	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄×C₅	C₂²	1	3	yes	—
7	C₂×C₃×C₄×C₅	C₂²	1	3	yes	—
8	C₃×#126	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
10	C₅×C₂²⋊C₄	C₂×C₃	1	3	yes	—
11	C₄²×C₅	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂²×C₃×C₅	—	2	3	no	—
13	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
14	C₃×C₄×C₅	—	2	3	no	—
15	C₃×C₄×C₅	—	2	3	no	—
16	C₃×C₄⋊C₄	C₂×C₅	1	3	yes	complemented
17	C₃×C₂²⋊C₄	C₂×C₅	1	3	yes	—
18	C₃×C₄²	C₂×C₅	1	1	yes	characteristic, complemented
19	C₂³×C₅	C₂²×C₃	1	1	yes	characteristic
20	C₂×C₄×C₅	C₂²×C₃	1	3	yes	—
21	C₂×C₄×C₅	C₂²×C₃	1	3	yes	—
22	#126	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
23	C₂×C₃×C₅	—	4	1	no	—
24	C₂×C₃×C₅	#41	1	3	yes	—
25	C₂³×C₃	C₂²×C₅	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂²×C₅	1	3	yes	—
27	C₂×C₃×C₄	C₂²×C₅	1	3	yes	—
28	C₂²×C₅	—	2	3	no	—
29	C₂²×C₅	C₂³×C₃	1	1	yes	characteristic
30	C₄×C₅	—	2	3	no	—
31	C₄×C₅	—	2	3	no	—
32	C₄⋊C₄	C₂×C₃×C₅	1	3	yes	complemented
33	C₂²⋊C₄	C₂×C₃×C₅	1	3	yes	—
34	C₄²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
35	C₃×C₅	#126	1	1	yes	characteristic, direct product factor, Hall
36	C₂²×C₃	—	2	3	no	—
37	C₂²×C₃	C₂³×C₅	1	1	yes	characteristic
38	C₃×C₄	—	2	3	no	—
39	C₃×C₄	—	2	3	no	—
40	C₂×C₅	—	4	1	no	—
41	C₂×C₅	C₃×#41	1	3	yes	—
42	C₂³	C₂²×C₃×C₅	1	1	yes	characteristic
43	C₂×C₄	C₂²×C₃×C₅	1	3	yes	—
44	C₂×C₄	C₂²×C₃×C₅	1	3	yes	—
45	C₂×C₃	—	4	1	no	—
46	C₂×C₃	C₅×#41	1	3	yes	—
47	C₅	C₃×#126	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²	—	2	3	no	—
49	C₂²	C₂³×C₃×C₅	1	1	yes	characteristic
50	C₄	—	2	3	no	—
51	C₄	—	2	3	no	—
52	C₃	C₅×#126	1	1	yes	characteristic, direct product factor, Sylow
53	C₂	—	4	1	no	—
54	C₂	C₃×C₅×#41	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₄⋊C₄	C₂	semidirect
3. C₃×C₄²×C₅	C₂	semidirect
4. C₅×#126	C₃	direct (N × H)
8. C₃×#126	C₅	direct (N × H)
9. C₅×C₄⋊C₄	C₂×C₃	semidirect
11. C₄²×C₅	C₂×C₃	semidirect
16. C₃×C₄⋊C₄	C₂×C₅	semidirect
18. C₃×C₄²	C₂×C₅	semidirect
22. #126	C₃×C₅	direct (N × H)
32. C₄⋊C₄	C₂×C₃×C₅	semidirect
34. C₄²	C₂×C₃×C₅	semidirect
35. C₃×C₅	#126	direct (N × H)
47. C₅	C₃×#126	direct (N × H)
52. C₃	C₅×#126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₄⋊C₄	2
C₃×C₅×C₂²⋊C₄	2
C₃×C₄²×C₅	2
C₅×#126	3
C₃×#126	5

Subgroups — C3×C5×#126 #91855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#126 #91855