Subgroups — C₁₉×ΓL(1,𝔽₈)×D₃₉ #44527988

Order 31122.

72 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, complemented, Hall
2	C₁₉×ΓL(1,𝔽₈)×D₁₃	—	3	1	no	maximal
3	C₇×C₁₉×D₃₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
5	C₁₃×C₁₉×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
6	C₃×C₇×C₁₃×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₁₉×D₃₉	—	7	1	no	maximal, Hall
8	C₇×C₁₉×D₁₃	—	3	1	no	Hall
9	D₃×C₁₉×ΓL(1,𝔽₈)	—	13	1	no	maximal, Hall
10	C₃²×C₁₃×C₁₉	—	7	1	no	Hall
11	C₇×C₁₃×C₁₉	C₃×D₃	1	1	yes	characteristic, complemented, Hall
12	ΓL(1,𝔽₈)×D₃₉	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₁₉×D₃₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
14	C₃×C₁₉×D₁₃	—	21	1	no	—
15	C₃×C₁₉×ΓL(1,𝔽₈)	D₁₃	1	1	yes	characteristic, complemented, Hall
16	C₃×C₁₃×ΓL(1,𝔽₈)	C₂×C₁₉	1	1	yes	characteristic, complemented, Hall
17	D₃×C₇×C₁₉	—	13	1	no	—
18	C₂×C₁₉×ΓL(1,𝔽₈)	—	39	1	no	—
19	C₃×C₁₃×C₁₉	—	7	1	no	—
20	C₃×C₁₃×C₁₉	—	14	1	no	—
21	C₃×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₇×D₃₉	C₃×C₁₉	1	1	yes	characteristic, complemented
23	ΓL(1,𝔽₈)×D₁₃	—	3	1	no	—
24	C₁₉×D₁₃	—	3	1	no	Hall
25	C₃×C₇×C₁₉	C₃×D₁₃	1	1	yes	characteristic, complemented
26	C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
27	C₁₉×ΓL(1,𝔽₈)	D₃₉	1	1	yes	characteristic, direct product factor
28	C₃×D₃×C₁₉	—	91	1	no	Hall
29	C₃×C₇×C₁₃	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
30	C₁₃×ΓL(1,𝔽₈)	—	2	1	no	—
31	C₁₃×ΓL(1,𝔽₈)	D₃×C₁₉	1	1	yes	characteristic, complemented
32	C₂×C₇×C₁₉	—	39	1	no	Hall
33	C₁₃×C₁₉	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
34	C₃×D₃₉	—	7	1	no	Hall
35	C₇×D₁₃	—	3	1	no	Hall
36	C₃²×C₁₉	—	7	1	no	Hall
37	C₇×C₁₉	C₃×D₃₉	1	1	yes	characteristic, complemented, Hall
38	D₃×ΓL(1,𝔽₈)	—	13	1	no	Hall
39	C₃²×C₁₃	—	7	1	no	Hall
40	C₂×C₃×C₁₉	—	273	1	no	—
41	D₃×C₁₉	—	13	1	no	—
42	C₇×C₁₃	C₃×D₃×C₁₉	1	1	yes	characteristic, complemented, Hall
43	D₃₉	C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
44	C₃×D₁₃	—	21	1	no	—
45	C₃×ΓL(1,𝔽₈)	C₁₉×D₁₃	1	1	yes	characteristic, complemented, Hall
46	C₃×C₁₉	—	7	1	no	—
47	C₃×C₁₉	—	14	1	no	—
48	C₃×C₁₉	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
49	D₃×C₇	—	13	1	no	—
50	C₂×ΓL(1,𝔽₈)	—	39	1	no	—
51	C₃×C₁₃	—	7	1	no	—
52	C₃×C₁₃	—	14	1	no	—
53	C₃×C₁₃	C₂×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
54	C₂×C₁₉	—	39	1	no	Hall
55	D₁₃	—	3	1	no	Hall
56	C₃×C₇	C₃×C₁₉×D₁₃	1	1	yes	characteristic, complemented
57	ΓL(1,𝔽₈)	—	2	1	no	—
58	ΓL(1,𝔽₈)	C₁₉×D₃₉	1	1	yes	characteristic, direct product factor
59	C₁₉	ΓL(1,𝔽₈)×D₃₉	1	1	yes	characteristic, direct product factor, Sylow
60	C₃×D₃	—	91	1	no	Hall
61	C₂×C₇	—	39	1	no	Hall
62	C₁₃	D₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
63	C₃²	—	7	1	no	Sylow
64	C₇	C₃×C₁₉×D₃₉	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	273	1	no	—
66	D₃	—	13	1	no	—
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	C₁₉×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
70	C₂	—	39	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₉	ΓL(1,𝔽₈)×D₃₉
commutator, cosocle, hypocenter	C₃×C₇×C₁₃	C₂×C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₃×C₁₉	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₃
UCS	C₁ ◁ 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	semidirect
3. C₇×C₁₉×D₃₉	C₃	semidirect
5. C₁₃×C₁₉×ΓL(1,𝔽₈)	D₃	semidirect
6. C₃×C₇×C₁₃×C₁₉	C₂×C₃	semidirect
11. C₇×C₁₃×C₁₉	C₃×D₃	semidirect
12. ΓL(1,𝔽₈)×D₃₉	C₁₉	direct (N × H)
13. C₁₉×D₃₉	ΓL(1,𝔽₈)	direct (N × H)
15. C₃×C₁₉×ΓL(1,𝔽₈)	D₁₃	semidirect
16. C₃×C₁₃×ΓL(1,𝔽₈)	C₂×C₁₉	semidirect
21. C₃×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
22. C₇×D₃₉	C₃×C₁₉	semidirect
25. C₃×C₇×C₁₉	C₃×D₁₃	semidirect
27. C₁₉×ΓL(1,𝔽₈)	D₃₉	direct (N × H)
29. C₃×C₇×C₁₃	C₂×C₃×C₁₉	semidirect
31. C₁₃×ΓL(1,𝔽₈)	D₃×C₁₉	semidirect
33. C₁₃×C₁₉	D₃×ΓL(1,𝔽₈)	semidirect
37. C₇×C₁₉	C₃×D₃₉	semidirect
42. C₇×C₁₃	C₃×D₃×C₁₉	semidirect
43. D₃₉	C₁₉×ΓL(1,𝔽₈)	direct (N × H)
45. C₃×ΓL(1,𝔽₈)	C₁₉×D₁₃	semidirect
48. C₃×C₁₉	ΓL(1,𝔽₈)×D₁₃	semidirect
53. C₃×C₁₃	C₂×C₁₉×ΓL(1,𝔽₈)	semidirect
56. C₃×C₇	C₃×C₁₉×D₁₃	semidirect
58. ΓL(1,𝔽₈)	C₁₉×D₃₉	direct (N × H)
59. C₁₉	ΓL(1,𝔽₈)×D₃₉	direct (N × H)
62. C₁₃	D₃×C₁₉×ΓL(1,𝔽₈)	semidirect
64. C₇	C₃×C₁₉×D₃₉	semidirect
69. C₃	C₁₉×ΓL(1,𝔽₈)×D₁₃	semidirect

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
C₁₉×ΓL(1,𝔽₈)×D₁₃	3
C₇×C₁₉×D₃₉	3
C₃×C₁₉×D₃₉	7
D₃×C₁₉×ΓL(1,𝔽₈)	13
ΓL(1,𝔽₈)×D₃₉	19

Subgroups — C19×ΓL(1,𝔽8)×D39 #44527988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×ΓL(1,𝔽₈)×D₃₉ #44527988