Subgroups — C₂×C₅²×SL(2,𝔽₃) #11925670

Order 1200.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, direct product factor
2	C₂×C₅²×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×C₃×C₅²	—	4	1	no	maximal
4	C₂×C₅×SL(2,𝔽₃)	C₅	1	6	yes	maximal, direct product factor
5	C₅²×Q₈	—	3	1	no	—
6	C₅²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₄×C₅²	—	3	1	no	—
8	C₂×C₃×C₅²	—	4	2	no	—
9	C₂×C₃×C₅²	—	4	1	no	—
10	C₅×SL(2,𝔽₃)	C₂×C₅	1	6	yes	direct product factor
11	C₂²×C₅²	A₄	1	1	yes	characteristic
12	C₄×C₅²	—	3	1	no	—
13	C₄×C₅²	—	3	1	no	—
14	C₂×C₅×Q₈	C₃×C₅	1	6	yes	complemented
15	C₃×C₅²	—	4	1	no	Hall
16	C₂²×C₃×C₅	—	4	6	no	—
17	C₂×C₅²	SL(2,𝔽₃)	1	2	yes	direct product factor
18	C₂×C₅²	C₂×A₄	1	1	yes	characteristic
19	C₂×SL(2,𝔽₃)	C₅²	1	1	yes	characteristic, direct product factor, Hall
20	C₅×Q₈	—	3	6	no	—
21	C₅×Q₈	C₂×C₃×C₅	1	6	yes	complemented
22	C₂×C₄×C₅	—	3	6	no	—
23	C₂×C₃×C₅	—	4	12	no	—
24	C₂×C₃×C₅	—	4	6	no	—
25	C₅²	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
26	SL(2,𝔽₃)	C₂×C₅²	1	1	yes	characteristic, direct product factor
27	C₂²×C₅	C₅×A₄	1	6	yes	—
28	C₄×C₅	—	3	6	no	—
29	C₄×C₅	—	3	6	no	—
30	C₂×Q₈	C₃×C₅²	1	1	yes	characteristic, complemented, Sylow
31	C₃×C₅	—	4	6	no	—
32	C₂²×C₃	—	4	1	no	—
33	C₂×C₅	C₅×SL(2,𝔽₃)	1	12	yes	direct product factor
34	C₂×C₅	C₂×C₅×A₄	1	6	yes	—
35	Q₈	—	3	1	no	—
36	Q₈	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
37	C₂×C₄	—	3	1	no	—
38	C₂×C₃	—	4	2	no	—
39	C₂×C₃	—	4	1	no	—
40	C₅	C₂×C₅×SL(2,𝔽₃)	1	6	yes	direct product factor
41	C₂²	C₅²×A₄	1	1	yes	characteristic
42	C₄	—	3	1	no	—
43	C₄	—	3	1	no	—
44	C₃	—	4	1	no	Sylow
45	C₂	C₅²×SL(2,𝔽₃)	1	2	yes	direct product factor
46	C₂	C₂×C₅²×A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂²×C₅²	A₄
commutator, cosocle, hypocenter	Q₈	C₂×C₃×C₅²
Frattini	C₂	C₂×C₅²×A₄
Fitting	C₂×C₅²×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂²×C₅²

Complements

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

N	H ≅ G/N	type
1. C₅²×SL(2,𝔽₃)	C₂	direct (N × H)
2. C₂×C₅²×Q₈	C₃	semidirect
4. C₂×C₅×SL(2,𝔽₃)	C₅	direct (N × H)
6. C₅²×Q₈	C₂×C₃	semidirect
10. C₅×SL(2,𝔽₃)	C₂×C₅	direct (N × H)
14. C₂×C₅×Q₈	C₃×C₅	semidirect
17. C₂×C₅²	SL(2,𝔽₃)	direct (N × H)
19. C₂×SL(2,𝔽₃)	C₅²	direct (N × H)
21. C₅×Q₈	C₂×C₃×C₅	semidirect
25. C₅²	C₂×SL(2,𝔽₃)	direct (N × H)
26. SL(2,𝔽₃)	C₂×C₅²	direct (N × H)
30. C₂×Q₈	C₃×C₅²	semidirect
33. C₂×C₅	C₅×SL(2,𝔽₃)	direct (N × H)
36. Q₈	C₂×C₃×C₅²	semidirect
40. C₅	C₂×C₅×SL(2,𝔽₃)	direct (N × H)
45. C₂	C₅²×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
C₅²×SL(2,𝔽₃)	2
C₂×C₅²×Q₈	3
C₂²×C₃×C₅²	4
C₂×C₅×SL(2,𝔽₃)	5

Subgroups — C2×C52×SL(2,𝔽3) #11925670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×SL(2,𝔽₃) #11925670