Subgroups — #90991

Order 480.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6710	C₂	1	1	yes	maximal, characteristic
2	C₅×#208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#168	C₂	1	1	yes	maximal, characteristic
4	#3954	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₃⋊#168	—	2	1	no	—
7	C₃×#168	C₄	1	2	yes	—
8	C₅×C₃⋊C₈	—	2	1	no	—
9	#819	—	5	1	no	maximal, Hall
10	C₅×M₁₆	—	3	1	no	—
11	#681	—	3	1	no	—
12	C₂×#168	D₃	1	1	yes	characteristic
13	C₂²×C₃×C₅	Q₈	1	1	yes	characteristic
14	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄×C₅	D₄	1	1	yes	characteristic
16	C₂×C₃×C₈	—	5	1	no	—
17	#208	—	5	1	no	—
18	#208	D₅	1	1	yes	characteristic
19	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
20	C₅×C₈	—	6	1	no	—
21	#168	—	6	1	no	—
22	#168	Dic₃	1	2	yes	—
23	#108	—	15	1	no	Sylow
24	C₂×C₃×C₅	—	2	1	no	—
25	C₂×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
27	C₃×C₈	—	5	2	no	—
28	C₃⋊C₈	—	2	1	no	—
29	C₃⋊C₈	—	10	1	no	—
30	C₂²×C₅	Dic₆	1	1	yes	characteristic
31	C₄×C₅	D₁₂	1	1	yes	characteristic
32	C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
33	M₁₆	—	3	1	no	—
34	M₁₆	—	15	1	no	—
35	C₂×C₈	—	5	1	no	—
36	C₃×C₅	#108	1	1	yes	characteristic, complemented, Hall
37	C₂²×C₃	Dic₁₀	1	1	yes	characteristic
38	C₃×C₄	C₄×D₅	1	1	yes	characteristic
39	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
40	C₂×C₅	—	2	1	no	—
41	C₂×C₅	C₄⋊Dic₃	1	1	yes	characteristic
42	C₂×C₄	D₃×D₅	1	1	yes	characteristic
43	C₈	—	5	2	no	—
44	C₈	—	6	1	no	—
45	C₈	—	30	1	no	—
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	#683	1	1	yes	characteristic
48	C₅	#819	1	1	yes	characteristic, complemented, Sylow
49	C₂²	#1204	1	1	yes	characteristic
50	C₄	D₅×Dic₃	1	1	yes	characteristic
51	C₄	#1203	1	1	yes	characteristic
52	C₃	#3954	1	1	yes	characteristic, complemented, Sylow
53	C₂	—	2	1	no	—
54	C₂	#6668	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1203
commutator	C₃×C₄×C₅	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃×D₅
Fitting, cosocle	C₂×C₃×C₄×C₅	C₂²
socle	C₂×C₃×C₅	C₄⋊C₄
prime radical	C₂²×C₃×C₅	Q₈
hypocenter	C₃×C₅	#108
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
36. C₃×C₅	#108	semidirect
48. C₅	#819	semidirect
52. C₃	#3954	semidirect

Maximal subgroups

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

subgroup	index
#6710	2
C₅×#208	2
C₂×C₃×#168	2
#3954	3
#819	5