Subgroups — #90939

Order 480.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₂₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#672	C₂	1	1	yes	maximal, characteristic, complemented
4	#3958	—	3	1	no	maximal, Hall
5	D₆₀	—	2	1	no	—
6	C₅×D₁₂	—	2	1	no	—
7	C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	D₄₈	—	5	1	no	maximal, Hall
9	C₅×D₈	—	3	1	no	—
10	D₄₀	—	3	1	no	—
11	#672	D₃	1	1	yes	characteristic, complemented
12	C₂×D₁₅	—	4	1	no	—
13	C₂×C₅×D₃	—	4	1	no	—
14	C₃×C₄×C₅	D₄	1	1	yes	characteristic
15	D₂₄	—	5	1	no	—
16	D₂₄	D₅	1	1	yes	characteristic, complemented
17	C₃×C₁₆	—	5	1	no	—
18	C₅×D₄	—	6	1	no	—
19	D₂₀	—	6	1	no	—
20	C₅×C₈	C₂×D₃	1	1	yes	characteristic
21	D₁₆	—	15	1	no	Sylow
22	C₂×C₃×C₅	D₈	1	1	yes	characteristic
23	D₁₅	—	8	1	no	—
24	C₅×D₃	—	8	1	no	—
25	D₁₂	—	2	1	no	—
26	D₁₂	—	10	1	no	—
27	C₃×C₈	C₂×D₅	1	1	yes	characteristic
28	C₂²×C₅	—	12	1	no	—
29	C₂×D₅	—	12	1	no	—
30	C₄×C₅	D₁₂	1	1	yes	characteristic
31	D₈	—	3	1	no	—
32	D₈	—	15	1	no	—
33	C₁₆	—	5	1	no	—
34	C₃×C₅	D₁₆	1	1	yes	characteristic, complemented, Hall
35	C₂×D₃	—	4	1	no	—
36	C₂×D₃	—	20	1	no	—
37	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
38	C₂×C₅	—	24	1	no	—
39	C₂×C₅	D₂₄	1	1	yes	characteristic
40	D₅	—	24	1	no	—
41	D₄	—	6	1	no	—
42	D₄	—	30	1	no	—
43	C₈	D₃×D₅	1	1	yes	characteristic
44	C₂×C₃	#686	1	1	yes	characteristic
45	D₃	—	8	1	no	—
46	D₃	—	40	1	no	—
47	C₅	D₄₈	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	12	1	no	—
49	C₂²	—	60	1	no	—
50	C₄	#1203	1	1	yes	characteristic
51	C₃	#3958	1	1	yes	characteristic, complemented, Sylow
52	C₂	—	24	1	no	—
53	C₂	—	120	1	no	—
54	C₂	#6654	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6654
commutator, Fitting, cosocle	C₃×C₅×C₈	C₂²
Frattini, hypercenter	C₈	D₃×D₅
socle	C₂×C₃×C₅	D₈
hypocenter	C₃×C₅	D₁₆
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. D₁₂₀	C₂	semidirect
2. C₅×D₂₄	C₂	semidirect
3. C₃×#672	C₂	semidirect
11. #672	D₃	semidirect
16. D₂₄	D₅	semidirect
34. C₃×C₅	D₁₆	semidirect
47. C₅	D₄₈	semidirect
51. C₃	#3958	semidirect

Maximal subgroups

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

subgroup	index
D₁₂₀	2
C₅×D₂₄	2
C₃×#672	2
#3958	3
D₄₈	5