Subgroups — #91184

Order 480.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#3967	—	3	1	no	maximal, Hall
3	C₂×Dic₁₅	—	4	1	no	maximal
4	C₅×SL(2,𝔽₃)	C₄	1	1	yes	characteristic, complemented
5	#859	—	5	1	no	maximal, Hall
6	C₂×C₅×Q₈	D₃	1	1	yes	characteristic
7	C₄⋊Dic₅	—	3	1	no	—
8	C₂×#168	—	3	1	no	—
9	C₂²×C₃×C₅	—	4	1	no	—
10	Dic₁₅	—	4	2	no	—
11	C₂×SL(2,𝔽₃)	D₅	1	1	yes	characteristic
12	C₅×Q₈	—	3	1	no	—
13	C₅×Q₈	Dic₃	1	1	yes	characteristic, complemented
14	C₂×C₄×C₅	—	3	1	no	—
15	C₂×Dic₅	—	6	1	no	—
16	#168	—	6	1	no	—
17	#103	—	15	1	no	Sylow
18	C₂×C₃×C₅	—	4	1	no	—
19	C₂×C₃×C₅	—	4	1	no	—
20	C₂×C₃×C₅	—	4	1	no	—
21	C₂×Dic₃	—	20	1	no	—
22	SL(2,𝔽₃)	Dic₅	1	1	yes	characteristic, complemented
23	C₂²×C₅	S₄	1	1	yes	characteristic
24	C₄×C₅	—	3	1	no	—
25	C₄×C₅	—	3	1	no	—
26	Dic₅	—	12	1	no	—
27	C₂×Q₈	D₁₅	1	1	yes	characteristic
28	C₂×C₈	—	15	1	no	—
29	C₄⋊C₄	—	15	1	no	—
30	C₃×C₅	—	4	1	no	Hall
31	C₂²×C₃	—	4	1	no	—
32	Dic₃	—	20	2	no	—
33	C₂×C₅	#226	1	1	yes	characteristic
34	C₂×C₅	GL(2,𝔽₃)	1	1	yes	characteristic
35	C₂×C₅	SL(2,ℤ/4)	1	1	yes	characteristic
36	Q₈	—	3	1	no	—
37	Q₈	Dic₁₅	1	1	yes	characteristic, complemented
38	C₂×C₄	—	3	1	no	—
39	C₂×C₄	—	30	1	no	—
40	C₈	—	30	1	no	—
41	C₂×C₃	—	4	1	no	—
42	C₂×C₃	—	4	1	no	—
43	C₂×C₃	—	4	1	no	—
44	C₅	#859	1	1	yes	characteristic, complemented, Sylow
45	C₂²	C₅⋊S₄	1	1	yes	characteristic
46	C₄	—	3	1	no	—
47	C₄	—	3	1	no	—
48	C₄	—	60	1	no	—
49	C₃	—	4	1	no	Sylow
50	C₂	C₅⋊#226	1	1	yes	characteristic
51	C₂	C₅⋊GL(2,𝔽₃)	1	1	yes	characteristic
52	C₂	A₄⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	C₅⋊S₄
commutator, hypocenter	C₅×SL(2,𝔽₃)	C₄
Fitting	C₂×C₅×Q₈	D₃
socle	C₂²×C₅	S₄
prime radical, cosocle	C₂×C₅×SL(2,𝔽₃)	C₂
LCS	G ▷ C₅×SL(2,𝔽₃)
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
4. C₅×SL(2,𝔽₃)	C₄	semidirect
13. C₅×Q₈	Dic₃	semidirect
22. SL(2,𝔽₃)	Dic₅	semidirect
37. Q₈	Dic₁₅	semidirect
44. C₅	#859	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×SL(2,𝔽₃)	2
#3967	3
C₂×Dic₁₅	4
#859	5