Subgroups — C₄×C₅×SL(2,𝔽₃) #91579

Order 480.

60 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	C₄×C₅×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×C₃×C₄×C₅	—	4	1	no	maximal
4	C₅×SL(2,𝔽₃)	C₄	1	1	yes	characteristic, direct product factor
5	C₄×SL(2,𝔽₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
7	C₅×C₄⋊C₄	—	3	1	no	—
8	C₄²×C₅	—	3	1	no	—
9	C₂²×C₃×C₅	—	4	1	no	—
10	C₃×C₄×C₅	—	4	2	no	—
11	C₂×SL(2,𝔽₃)	C₂×C₅	1	1	yes	characteristic
12	C₅×Q₈	—	3	1	no	—
13	C₅×Q₈	C₃×C₄	1	1	yes	characteristic, complemented
14	C₂×C₄×C₅	—	3	1	no	—
15	C₂×C₄×C₅	—	3	1	no	—
16	C₂×C₄×C₅	A₄	1	1	yes	characteristic
17	C₄×Q₈	C₃×C₅	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₃×C₅	—	4	1	no	—
19	C₂×C₃×C₅	—	4	1	no	—
20	C₂×C₃×C₅	—	4	1	no	—
21	C₂×C₃×C₄	—	4	1	no	—
22	SL(2,𝔽₃)	C₄×C₅	1	1	yes	characteristic, direct product factor
23	C₂²×C₅	C₂×A₄	1	1	yes	characteristic
24	C₄×C₅	—	3	1	no	—
25	C₄×C₅	—	3	1	no	—
26	C₄×C₅	—	6	1	no	—
27	C₄×C₅	SL(2,𝔽₃)	1	2	yes	direct product factor
28	C₂×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
29	C₄⋊C₄	—	3	1	no	—
30	C₄²	—	3	1	no	—
31	C₃×C₅	—	4	1	no	Hall
32	C₂²×C₃	—	4	1	no	—
33	C₃×C₄	—	4	2	no	—
34	C₂×C₅	C₄×A₄	1	1	yes	characteristic
35	C₂×C₅	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
36	C₂×C₅	#41⋊C₃	1	1	yes	characteristic
37	Q₈	—	3	1	no	—
38	Q₈	C₃×C₄×C₅	1	1	yes	characteristic, complemented
39	C₂×C₄	—	3	1	no	—
40	C₂×C₄	—	3	1	no	—
41	C₂×C₄	C₅×A₄	1	1	yes	characteristic
42	C₂×C₃	—	4	1	no	—
43	C₂×C₃	—	4	1	no	—
44	C₂×C₃	—	4	1	no	—
45	C₅	C₄×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²	C₂×C₅×A₄	1	1	yes	characteristic
47	C₄	—	3	1	no	—
48	C₄	—	3	1	no	—
49	C₄	—	6	1	no	—
50	C₄	C₅×SL(2,𝔽₃)	1	2	yes	direct product factor
51	C₃	—	4	1	no	Sylow
52	C₂	C₄×C₅×A₄	1	1	yes	characteristic
53	C₂	C₂×C₅×SL(2,𝔽₃)	1	1	yes	characteristic
54	C₂	C₅×#41⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	A₄
commutator, hypocenter	Q₈	C₃×C₄×C₅
Frattini	C₂²	C₂×C₅×A₄
Fitting	C₄×C₅×Q₈	C₃
socle	C₂²×C₅	C₂×A₄
prime radical	C₂×C₅×SL(2,𝔽₃)	C₂
cosocle	C₂×Q₈	C₂×C₃×C₅
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×Q₈	C₃	semidirect
4. C₅×SL(2,𝔽₃)	C₄	direct (N × H)
5. C₄×SL(2,𝔽₃)	C₅	direct (N × H)
13. C₅×Q₈	C₃×C₄	semidirect
17. C₄×Q₈	C₃×C₅	semidirect
22. SL(2,𝔽₃)	C₄×C₅	direct (N × H)
27. C₄×C₅	SL(2,𝔽₃)	direct (N × H)
38. Q₈	C₃×C₄×C₅	semidirect
45. C₅	C₄×SL(2,𝔽₃)	direct (N × H)
50. C₄	C₅×SL(2,𝔽₃)	direct (N × H)

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
C₄×C₅×Q₈	3
C₂×C₃×C₄×C₅	4
C₄×SL(2,𝔽₃)	5

Subgroups — C4×C5×SL(2,𝔽3) #91579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×SL(2,𝔽₃) #91579