Subgroups — C₂×C₅²×C₁₉×ΓL(1,𝔽₈) #34093101

Order 19950.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₅²×C₇×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×C₅×C₁₉×ΓL(1,𝔽₈)	C₅	1	6	yes	maximal, direct product factor
4	C₅²×C₇×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₅²×C₁₉	—	7	1	no	maximal, Hall
6	C₅×C₁₉×ΓL(1,𝔽₈)	C₂×C₅	1	6	yes	direct product factor
7	C₃×C₅²×C₁₉	—	7	1	no	Hall
8	C₂×C₅×C₇×C₁₉	C₃×C₅	1	6	yes	complemented
9	C₂×C₅²×ΓL(1,𝔽₈)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₅²×C₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₁₉×ΓL(1,𝔽₈)	C₅²	1	1	yes	characteristic, direct product factor, Hall
12	C₅×C₇×C₁₉	C₂×C₃×C₅	1	6	yes	complemented
13	C₂×C₃×C₅×C₁₉	—	7	6	no	—
14	C₅²×ΓL(1,𝔽₈)	C₂×C₁₉	1	1	yes	characteristic, direct product factor, Hall
15	C₅²×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
16	C₁₉×ΓL(1,𝔽₈)	C₂×C₅²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₅²×C₇	C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
18	C₃×C₅×C₁₉	—	7	6	no	—
19	C₂×C₇×C₁₉	C₃×C₅²	1	1	yes	characteristic, complemented, Hall
20	C₂×C₅×ΓL(1,𝔽₈)	C₅×C₁₉	1	6	yes	direct product factor
21	C₂×C₅×C₁₉	C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
22	C₅²×C₇	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
23	C₂×C₃×C₅²	—	7	1	no	Hall
24	C₇×C₁₉	C₂×C₃×C₅²	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₁₉	—	7	1	no	Hall
26	C₅×ΓL(1,𝔽₈)	C₂×C₅×C₁₉	1	6	yes	direct product factor
27	C₅×C₁₉	C₂×C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
28	C₃×C₅²	—	7	1	no	Hall
29	C₂×C₅×C₇	C₃×C₅×C₁₉	1	6	yes	complemented
30	C₃×C₁₉	—	7	1	no	Hall
31	C₂×C₅²	C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
32	C₂×ΓL(1,𝔽₈)	C₅²×C₁₉	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₁₉	C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
34	C₅×C₇	C₂×C₃×C₅×C₁₉	1	6	yes	complemented
35	C₂×C₃×C₅	—	7	6	no	—
36	C₅²	C₂×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
37	ΓL(1,𝔽₈)	C₂×C₅²×C₁₉	1	1	yes	characteristic, direct product factor, Hall
38	C₁₉	C₂×C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
39	C₃×C₅	—	7	6	no	—
40	C₂×C₇	C₃×C₅²×C₁₉	1	1	yes	characteristic, complemented, Hall
41	C₂×C₅	C₅×C₁₉×ΓL(1,𝔽₈)	1	6	yes	direct product factor
42	C₇	C₂×C₃×C₅²×C₁₉	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	7	1	no	Hall
44	C₅	C₂×C₅×C₁₉×ΓL(1,𝔽₈)	1	6	yes	direct product factor
45	C₃	—	7	1	no	Sylow
46	C₂	C₅²×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₅²×C₁₉	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₅²×C₁₉
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇×C₁₉	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₅²×C₁₉

Complements

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

N	H ≅ G/N	type
1. C₅²×C₁₉×ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂×C₅²×C₇×C₁₉	C₃	semidirect
3. C₂×C₅×C₁₉×ΓL(1,𝔽₈)	C₅	direct (N × H)
4. C₅²×C₇×C₁₉	C₂×C₃	semidirect
6. C₅×C₁₉×ΓL(1,𝔽₈)	C₂×C₅	direct (N × H)
8. C₂×C₅×C₇×C₁₉	C₃×C₅	semidirect
9. C₂×C₅²×ΓL(1,𝔽₈)	C₁₉	direct (N × H)
10. C₂×C₅²×C₁₉	ΓL(1,𝔽₈)	direct (N × H)
11. C₂×C₁₉×ΓL(1,𝔽₈)	C₅²	direct (N × H)
12. C₅×C₇×C₁₉	C₂×C₃×C₅	semidirect
14. C₅²×ΓL(1,𝔽₈)	C₂×C₁₉	direct (N × H)
15. C₅²×C₁₉	C₂×ΓL(1,𝔽₈)	direct (N × H)
16. C₁₉×ΓL(1,𝔽₈)	C₂×C₅²	direct (N × H)
17. C₂×C₅²×C₇	C₃×C₁₉	semidirect
19. C₂×C₇×C₁₉	C₃×C₅²	semidirect
20. C₂×C₅×ΓL(1,𝔽₈)	C₅×C₁₉	direct (N × H)
21. C₂×C₅×C₁₉	C₅×ΓL(1,𝔽₈)	direct (N × H)
22. C₅²×C₇	C₂×C₃×C₁₉	semidirect
24. C₇×C₁₉	C₂×C₃×C₅²	semidirect
26. C₅×ΓL(1,𝔽₈)	C₂×C₅×C₁₉	direct (N × H)
27. C₅×C₁₉	C₂×C₅×ΓL(1,𝔽₈)	direct (N × H)
29. C₂×C₅×C₇	C₃×C₅×C₁₉	semidirect
31. C₂×C₅²	C₁₉×ΓL(1,𝔽₈)	direct (N × H)
32. C₂×ΓL(1,𝔽₈)	C₅²×C₁₉	direct (N × H)
33. C₂×C₁₉	C₅²×ΓL(1,𝔽₈)	direct (N × H)
34. C₅×C₇	C₂×C₃×C₅×C₁₉	semidirect
36. C₅²	C₂×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
37. ΓL(1,𝔽₈)	C₂×C₅²×C₁₉	direct (N × H)
38. C₁₉	C₂×C₅²×ΓL(1,𝔽₈)	direct (N × H)
40. C₂×C₇	C₃×C₅²×C₁₉	semidirect
41. C₂×C₅	C₅×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
42. C₇	C₂×C₃×C₅²×C₁₉	semidirect
44. C₅	C₂×C₅×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
46. C₂	C₅²×C₁₉×ΓL(1,𝔽₈)	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×C₁₉×ΓL(1,𝔽₈)	2
C₂×C₅²×C₇×C₁₉	3
C₂×C₅×C₁₉×ΓL(1,𝔽₈)	5
C₂×C₃×C₅²×C₁₉	7
C₂×C₅²×ΓL(1,𝔽₈)	19

Subgroups — C2×C52×C19×ΓL(1,𝔽8) #34093101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×C₁₉×ΓL(1,𝔽₈) #34093101