Subgroups — #91084

Order 480.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₆₀	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₂₀	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#3932	—	3	1	no	maximal, Hall
5	D₆₀	—	2	1	no	—
6	Dic₃₀	—	2	1	no	—
7	C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	#800	—	5	1	no	maximal, Hall
9	Dic₂₀	—	3	1	no	—
10	D₄₀	—	3	1	no	—
11	C₅×C₁₆	D₃	1	1	yes	characteristic, complemented
12	C₂×D₁₅	—	4	1	no	—
13	C₃×C₄×C₅	D₄	1	1	yes	characteristic
14	Dic₁₅	—	4	1	no	—
15	Dic₁₂	—	5	1	no	—
16	D₂₄	—	5	1	no	—
17	C₃×C₁₆	D₅	1	1	yes	characteristic, complemented
18	D₂₀	—	6	1	no	—
19	Dic₁₀	—	6	1	no	—
20	C₅×C₈	C₂×D₃	1	1	yes	characteristic
21	SD₃₂	—	15	1	no	Sylow
22	C₂×C₃×C₅	D₈	1	1	yes	characteristic
23	D₁₅	—	8	1	no	—
24	D₁₂	—	10	1	no	—
25	Dic₆	—	10	1	no	—
26	C₃×C₈	C₂×D₅	1	1	yes	characteristic
27	C₂×D₅	—	12	1	no	—
28	C₄×C₅	D₁₂	1	1	yes	characteristic
29	Dic₅	—	12	1	no	—
30	Q₁₆	—	15	1	no	—
31	D₈	—	15	1	no	—
32	C₁₆	D₁₅	1	1	yes	characteristic, complemented
33	C₃×C₅	SD₃₂	1	1	yes	characteristic, complemented, Hall
34	C₂×D₃	—	20	1	no	—
35	C₃×C₄	D₂₀	1	1	yes	characteristic
36	Dic₃	—	20	1	no	—
37	C₂×C₅	D₂₄	1	1	yes	characteristic
38	D₅	—	24	1	no	—
39	Q₈	—	30	1	no	—
40	D₄	—	30	1	no	—
41	C₈	C₂×D₁₅	1	1	yes	characteristic
42	C₂×C₃	D₄₀	1	1	yes	characteristic
43	D₃	—	40	1	no	—
44	C₅	#800	1	1	yes	characteristic, complemented, Sylow
45	C₂²	—	60	1	no	—
46	C₄	—	60	1	no	—
47	C₄	D₆₀	1	1	yes	characteristic
48	C₃	#3932	1	1	yes	characteristic, complemented, Sylow
49	C₂	—	120	1	no	—
50	C₂	D₁₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₂₀
commutator, cosocle	C₃×C₅×C₈	C₂²
Frattini	C₈	C₂×D₁₅
Fitting	C₃×C₅×C₁₆	C₂
socle	C₂×C₃×C₅	D₈
prime radical	D₁₂₀	C₂
hypercenter	C₁₆	D₁₅
hypocenter	C₃×C₅	SD₃₂
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
1. Dic₆₀	C₂	semidirect
3. C₃×C₅×C₁₆	C₂	semidirect
11. C₅×C₁₆	D₃	semidirect
17. C₃×C₁₆	D₅	semidirect
32. C₁₆	D₁₅	semidirect
33. C₃×C₅	SD₃₂	semidirect
44. C₅	#800	semidirect
48. C₃	#3932	semidirect

Maximal subgroups

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

subgroup	index
Dic₆₀	2
D₁₂₀	2
C₃×C₅×C₁₆	2
#3932	3
#800	5