Subgroups — C₂²×C₅×SL(2,𝔽₃) #92052

Order 480.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×SL(2,𝔽₃)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂³×C₃×C₅	—	4	1	no	maximal
4	C₅×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, direct product factor
5	C₂²×SL(2,𝔽₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₅×Q₈	—	3	3	no	—
7	C₂×C₅×Q₈	C₂×C₃	1	3	yes	complemented
8	C₂²×C₄×C₅	—	3	1	no	—
9	C₂²×C₃×C₅	—	4	4	no	—
10	C₂²×C₃×C₅	—	4	3	no	—
11	C₂×SL(2,𝔽₃)	C₂×C₅	1	3	yes	direct product factor
12	C₂³×C₅	A₄	1	1	yes	characteristic
13	C₅×Q₈	—	3	3	no	—
14	C₅×Q₈	—	3	2	no	—
15	C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
16	C₂×C₄×C₅	—	3	3	no	—
17	C₂×C₄×C₅	—	3	3	no	—
18	C₂²×Q₈	C₃×C₅	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₃×C₅	—	4	6	no	—
20	C₂×C₃×C₅	—	4	1	no	—
21	C₂³×C₃	—	4	1	no	—
22	SL(2,𝔽₃)	C₂²×C₅	1	1	yes	characteristic, direct product factor
23	C₂²×C₅	SL(2,𝔽₃)	1	4	yes	direct product factor
24	C₂²×C₅	C₂×A₄	1	3	yes	—
25	C₄×C₅	—	3	3	no	—
26	C₄×C₅	—	3	1	no	—
27	C₂×Q₈	—	3	3	no	—
28	C₂×Q₈	C₂×C₃×C₅	1	3	yes	complemented
29	C₂²×C₄	—	3	1	no	—
30	C₃×C₅	—	4	1	no	Hall
31	C₂²×C₃	—	4	4	no	—
32	C₂²×C₃	—	4	3	no	—
33	C₂×C₅	C₂×SL(2,𝔽₃)	1	6	yes	direct product factor
34	C₂×C₅	C₂²×A₄	1	1	yes	characteristic
35	C₂³	C₅×A₄	1	1	yes	characteristic
36	Q₈	—	3	3	no	—
37	Q₈	—	3	2	no	—
38	Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
39	C₂×C₄	—	3	3	no	—
40	C₂×C₄	—	3	3	no	—
41	C₂×C₃	—	4	6	no	—
42	C₂×C₃	—	4	1	no	—
43	C₅	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²	C₅×SL(2,𝔽₃)	1	4	yes	direct product factor
45	C₂²	C₂×C₅×A₄	1	3	yes	—
46	C₄	—	3	1	no	—
47	C₄	—	3	3	no	—
48	C₃	—	4	1	no	Sylow
49	C₂	C₂×C₅×SL(2,𝔽₃)	1	6	yes	direct product factor
50	C₂	C₂²×C₅×A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂³×C₅	A₄
commutator, cosocle, hypocenter	Q₈	C₂²×C₃×C₅
Frattini	C₂	C₂²×C₅×A₄
Fitting	C₂²×C₅×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂³×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×SL(2,𝔽₃)	C₂	direct (N × H)
2. C₂²×C₅×Q₈	C₃	semidirect
4. C₅×SL(2,𝔽₃)	C₂²	direct (N × H)
5. C₂²×SL(2,𝔽₃)	C₅	direct (N × H)
7. C₂×C₅×Q₈	C₂×C₃	semidirect
11. C₂×SL(2,𝔽₃)	C₂×C₅	direct (N × H)
15. C₅×Q₈	C₂²×C₃	semidirect
18. C₂²×Q₈	C₃×C₅	semidirect
22. SL(2,𝔽₃)	C₂²×C₅	direct (N × H)
23. C₂²×C₅	SL(2,𝔽₃)	direct (N × H)
28. C₂×Q₈	C₂×C₃×C₅	semidirect
33. C₂×C₅	C₂×SL(2,𝔽₃)	direct (N × H)
38. Q₈	C₂²×C₃×C₅	semidirect
43. C₅	C₂²×SL(2,𝔽₃)	direct (N × H)
44. C₂²	C₅×SL(2,𝔽₃)	direct (N × H)
49. C₂	C₂×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
C₂²×C₅×Q₈	3
C₂³×C₃×C₅	4
C₂²×SL(2,𝔽₃)	5

Subgroups — C22×C5×SL(2,𝔽3) #92052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×SL(2,𝔽₃) #92052