Subgroups — #91112

Order 480.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₂₀	C₂	1	1	yes	maximal, characteristic
3	C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
4	#3960	—	3	1	no	maximal, Hall
5	C₃×C₅×Q₈	—	2	1	no	—
6	D₆₀	—	2	1	no	—
7	C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	#828	—	5	1	no	maximal, Hall
9	C₅×Q₁₆	D₃	1	1	yes	characteristic, complemented
10	D₄₀	—	3	1	no	—
11	#672	—	3	1	no	—
12	C₂×D₁₅	—	4	1	no	—
13	C₃×C₄×C₅	—	4	1	no	—
14	C₃×C₄×C₅	D₄	1	1	yes	characteristic
15	C₃×Q₁₆	D₅	1	1	yes	characteristic, complemented
16	D₂₄	—	5	1	no	—
17	C₃⋊C₁₆	—	5	1	no	—
18	C₅×Q₈	—	2	1	no	—
19	D₂₀	—	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	C₃×Q₈	—	2	1	no	—
25	D₁₂	—	10	1	no	—
26	C₃×C₈	C₂×D₅	1	1	yes	characteristic
27	C₂×D₅	—	12	1	no	—
28	C₄×C₅	—	4	1	no	—
29	C₄×C₅	#67	1	1	yes	characteristic
30	Q₁₆	D₁₅	1	1	yes	characteristic, complemented
31	D₈	—	15	1	no	—
32	C₁₆	—	15	1	no	—
33	C₃×C₅	SD₃₂	1	1	yes	characteristic, complemented, Hall
34	C₂×D₃	—	20	1	no	—
35	C₃×C₄	—	4	1	no	—
36	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
37	C₂×C₅	#213	1	1	yes	characteristic
38	D₅	—	24	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	30	1	no	—
41	C₈	C₂×D₁₅	1	1	yes	characteristic
42	C₂×C₃	#686	1	1	yes	characteristic
43	D₃	—	40	1	no	—
44	C₅	#828	1	1	yes	characteristic, complemented, Sylow
45	C₂²	—	60	1	no	—
46	C₄	—	4	1	no	—
47	C₄	#1220	1	1	yes	characteristic
48	C₃	#3960	1	1	yes	characteristic, complemented, Sylow
49	C₂	—	120	1	no	—
50	C₂	#6715	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6715
commutator, cosocle	C₃×C₅×C₈	C₂²
Frattini	C₈	C₂×D₁₅
Fitting	C₃×C₅×Q₁₆	C₂
socle	C₂×C₃×C₅	D₈
prime radical	D₁₂₀	C₂
hypercenter	Q₁₆	D₁₅
hypocenter	C₃×C₅	SD₃₂
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×Q₁₆	C₂	semidirect
3. C₃⋊#672	C₂	semidirect
9. C₅×Q₁₆	D₃	semidirect
15. C₃×Q₁₆	D₅	semidirect
30. Q₁₆	D₁₅	semidirect
33. C₃×C₅	SD₃₂	semidirect
44. C₅	#828	semidirect
48. C₃	#3960	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×Q₁₆	2
D₁₂₀	2
C₃⋊#672	2
#3960	3
#828	5