Subgroups — #91120

Order 480.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	#6710	C₂	1	2	yes	maximal
3	#3968	—	3	1	no	maximal, Hall
4	C₃×C₅×Q₈	—	2	2	no	—
5	C₂×C₃×C₄×C₅	C₄	1	2	yes	—
6	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₃⋊#168	—	2	2	no	—
8	#836	—	5	1	no	maximal, Hall
9	C₂×C₅×Q₈	D₃	1	1	yes	characteristic
10	#681	—	3	2	no	—
11	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
12	C₃×C₄×C₅	—	2	2	no	—
13	C₃×C₄×C₅	D₄	1	2	yes	—
14	C₂×C₃×Q₈	D₅	1	1	yes	characteristic
15	#208	—	5	2	no	—
16	C₅×Q₈	—	2	2	no	—
17	C₂×C₄×C₅	Dic₃	1	2	yes	—
18	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
19	#168	—	6	2	no	—
20	#101	—	15	1	no	Sylow
21	C₂×C₃×C₅	—	2	1	no	—
22	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
23	C₃×Q₈	—	2	2	no	—
24	C₂×C₃×C₄	Dic₅	1	2	yes	—
25	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
26	C₃⋊C₈	—	10	2	no	—
27	C₂²×C₅	C₂×Dic₃	1	1	yes	characteristic
28	C₄×C₅	—	2	2	no	—
29	C₄×C₅	#67	1	2	yes	—
30	C₂×Q₈	D₁₅	1	1	yes	characteristic
31	M₁₆	—	15	2	no	—
32	C₃×C₅	#101	1	1	yes	characteristic, complemented, Hall
33	C₂²×C₃	C₂×Dic₅	1	1	yes	characteristic
34	C₃×C₄	—	2	2	no	—
35	C₃×C₄	C₂²⋊D₅	1	2	yes	—
36	C₂×C₅	—	2	1	no	—
37	C₂×C₅	#217	1	1	yes	characteristic
38	Q₈	—	2	2	no	—
39	C₂×C₄	Dic₁₅	1	2	yes	—
40	C₂×C₄	C₂×D₁₅	1	1	yes	characteristic
41	C₈	—	30	2	no	—
42	C₂×C₃	—	2	1	no	—
43	C₂×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
44	C₅	#836	1	1	yes	characteristic, complemented, Sylow
45	C₂²	C₂×Dic₁₅	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	#1220	1	2	yes	—
48	C₃	#3968	1	1	yes	characteristic, complemented, Sylow
49	C₂	—	2	1	no	—
50	C₂	#6719	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6719
commutator, prime radical	C₂²×C₃×C₅	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₅
Fitting	C₂×C₃×C₅×Q₈	C₂
socle	C₂×C₃×C₅	C₂²⋊C₄
cosocle	C₂×C₃×C₄×C₅	C₂²
hypercenter	C₂×Q₈	D₁₅
hypocenter	C₃×C₅	#101
LCS	G ▷ C₂²×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
32. C₃×C₅	#101	semidirect
44. C₅	#836	semidirect
48. C₃	#3968	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×Q₈	2
#6710	2
#3968	3
#836	5