Subgroups — C₂×C₃×C₁₃×C₁₉×ΓL(1,𝔽₈) #44528014

Order 31122.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₇×C₁₃×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	direct product factor
5	C₃×C₇×C₁₃×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
6	C₂×C₃²×C₁₃×C₁₉	—	7	1	no	maximal, Hall
7	C₂×C₇×C₁₃×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₂×C₃×C₁₉×ΓL(1,𝔽₈)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃²×C₁₃×C₁₉	—	7	1	no	Hall
10	C₇×C₁₃×C₁₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
11	C₂×C₃×C₁₃×ΓL(1,𝔽₈)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₃×C₁₃×C₁₉	—	7	3	no	—
13	C₂×C₃×C₁₃×C₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
14	C₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₁₃	1	1	yes	characteristic, direct product factor, Hall
15	C₃×C₁₃×ΓL(1,𝔽₈)	C₂×C₁₉	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₃×C₇×C₁₉	C₃×C₁₃	1	1	yes	characteristic, complemented
17	C₂×C₁₉×ΓL(1,𝔽₈)	C₃×C₁₃	1	3	yes	direct product factor
18	C₃×C₁₃×C₁₉	—	7	3	no	—
19	C₃×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	C₂×C₃×C₇×C₁₃	C₃×C₁₉	1	1	yes	characteristic, complemented
21	C₂×C₁₃×ΓL(1,𝔽₈)	C₃×C₁₉	1	3	yes	direct product factor
22	C₂×C₁₃×C₁₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₇×C₁₉	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented
24	C₁₉×ΓL(1,𝔽₈)	C₂×C₃×C₁₃	1	3	yes	direct product factor
25	C₂×C₃²×C₁₉	—	7	1	no	Hall
26	C₃×C₇×C₁₃	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
27	C₁₃×ΓL(1,𝔽₈)	C₂×C₃×C₁₉	1	3	yes	direct product factor
28	C₂×C₇×C₁₉	C₃²×C₁₃	1	1	yes	characteristic, complemented, Hall
29	C₁₃×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₃²×C₁₃	—	7	1	no	Hall
31	C₂×C₇×C₁₃	C₃²×C₁₉	1	1	yes	characteristic, complemented, Hall
32	C₃²×C₁₉	—	7	1	no	Hall
33	C₇×C₁₉	C₂×C₃²×C₁₃	1	1	yes	characteristic, complemented, Hall
34	C₂×C₃×ΓL(1,𝔽₈)	C₁₃×C₁₉	1	1	yes	characteristic, direct product factor, Hall
35	C₃²×C₁₃	—	7	1	no	Hall
36	C₂×C₃×C₁₉	—	7	3	no	—
37	C₂×C₃×C₁₉	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
38	C₇×C₁₃	C₂×C₃²×C₁₉	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₁₃	—	7	3	no	—
40	C₂×C₃×C₁₃	C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
41	C₃×ΓL(1,𝔽₈)	C₂×C₁₃×C₁₉	1	1	yes	characteristic, direct product factor, Hall
42	C₃×C₁₉	—	7	3	no	—
43	C₃×C₁₉	C₂×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
44	C₂×C₃×C₇	C₃×C₁₃×C₁₉	1	1	yes	characteristic, complemented
45	C₂×ΓL(1,𝔽₈)	C₃×C₁₃×C₁₉	1	3	yes	direct product factor
46	C₃×C₁₃	—	7	3	no	—
47	C₃×C₁₃	C₂×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
48	C₂×C₁₉	C₃×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
49	C₂×C₁₃	C₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
50	C₃×C₇	C₂×C₃×C₁₃×C₁₉	1	1	yes	characteristic, complemented
51	ΓL(1,𝔽₈)	C₂×C₃×C₁₃×C₁₉	1	3	yes	direct product factor
52	C₁₉	C₂×C₃×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×C₃²	—	7	1	no	Hall
54	C₂×C₇	C₃²×C₁₃×C₁₉	1	1	yes	characteristic, complemented, Hall
55	C₁₃	C₂×C₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
56	C₃²	—	7	1	no	Sylow
57	C₇	C₂×C₃²×C₁₃×C₁₉	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₃	—	7	3	no	—
59	C₂×C₃	C₁₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
60	C₃	—	7	3	no	—
61	C₃	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
62	C₂	C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₃×C₁₉	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃²×C₁₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇×C₁₃×C₁₉	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₁₃×C₁₉

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₃	direct (N × H)
3. C₂×C₃×C₇×C₁₃×C₁₉	C₃	semidirect
4. C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	direct (N × H)
5. C₃×C₇×C₁₃×C₁₉	C₂×C₃	semidirect
7. C₂×C₇×C₁₃×C₁₉	C₃²	semidirect
8. C₂×C₃×C₁₉×ΓL(1,𝔽₈)	C₁₃	direct (N × H)
10. C₇×C₁₃×C₁₉	C₂×C₃²	semidirect
11. C₂×C₃×C₁₃×ΓL(1,𝔽₈)	C₁₉	direct (N × H)
13. C₂×C₃×C₁₃×C₁₉	ΓL(1,𝔽₈)	direct (N × H)
14. C₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₁₃	direct (N × H)
15. C₃×C₁₃×ΓL(1,𝔽₈)	C₂×C₁₉	direct (N × H)
16. C₂×C₃×C₇×C₁₉	C₃×C₁₃	semidirect
17. C₂×C₁₉×ΓL(1,𝔽₈)	C₃×C₁₃	direct (N × H)
19. C₃×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	direct (N × H)
20. C₂×C₃×C₇×C₁₃	C₃×C₁₉	semidirect
21. C₂×C₁₃×ΓL(1,𝔽₈)	C₃×C₁₉	direct (N × H)
22. C₂×C₁₃×C₁₉	C₃×ΓL(1,𝔽₈)	direct (N × H)
23. C₃×C₇×C₁₉	C₂×C₃×C₁₃	semidirect
24. C₁₉×ΓL(1,𝔽₈)	C₂×C₃×C₁₃	direct (N × H)
26. C₃×C₇×C₁₃	C₂×C₃×C₁₉	semidirect
27. C₁₃×ΓL(1,𝔽₈)	C₂×C₃×C₁₉	direct (N × H)
28. C₂×C₇×C₁₉	C₃²×C₁₃	semidirect
29. C₁₃×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	direct (N × H)
31. C₂×C₇×C₁₃	C₃²×C₁₉	semidirect
33. C₇×C₁₉	C₂×C₃²×C₁₃	semidirect
34. C₂×C₃×ΓL(1,𝔽₈)	C₁₃×C₁₉	direct (N × H)
37. C₂×C₃×C₁₉	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
38. C₇×C₁₃	C₂×C₃²×C₁₉	semidirect
40. C₂×C₃×C₁₃	C₁₉×ΓL(1,𝔽₈)	direct (N × H)
41. C₃×ΓL(1,𝔽₈)	C₂×C₁₃×C₁₉	direct (N × H)
43. C₃×C₁₉	C₂×C₁₃×ΓL(1,𝔽₈)	direct (N × H)
44. C₂×C₃×C₇	C₃×C₁₃×C₁₉	semidirect
45. C₂×ΓL(1,𝔽₈)	C₃×C₁₃×C₁₉	direct (N × H)
47. C₃×C₁₃	C₂×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
48. C₂×C₁₉	C₃×C₁₃×ΓL(1,𝔽₈)	direct (N × H)
49. C₂×C₁₃	C₃×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
50. C₃×C₇	C₂×C₃×C₁₃×C₁₉	semidirect
51. ΓL(1,𝔽₈)	C₂×C₃×C₁₃×C₁₉	direct (N × H)
52. C₁₉	C₂×C₃×C₁₃×ΓL(1,𝔽₈)	direct (N × H)
54. C₂×C₇	C₃²×C₁₃×C₁₉	semidirect
55. C₁₃	C₂×C₃×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
57. C₇	C₂×C₃²×C₁₃×C₁₉	semidirect
59. C₂×C₃	C₁₃×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
61. C₃	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
62. C₂	C₃×C₁₃×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
C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	2
C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	3
C₂×C₃×C₇×C₁₃×C₁₉	3
C₂×C₃²×C₁₃×C₁₉	7
C₂×C₃×C₁₉×ΓL(1,𝔽₈)	13
C₂×C₃×C₁₃×ΓL(1,𝔽₈)	19

Subgroups — C2×C3×C13×C19×ΓL(1,𝔽8) #44528014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₁₃×C₁₉×ΓL(1,𝔽₈) #44528014