Subgroups — C₇×C₁₉×C₃⋊ΓL(1,𝔽₁₆) #37561594

Order 23940.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇×D₅×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₇×C₁₉×ΓL(1,𝔽₁₆)	—	3	4	no	maximal
3	C₃²×C₅×C₇×C₁₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×C₁₉×C₃⋊Dic₃	—	5	1	no	maximal, Hall
5	C₃×C₇×D₅×C₁₉	D₃	1	4	yes	—
6	C₁₉×C₃⋊ΓL(1,𝔽₁₆)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₇×C₁₉×AGL(1,𝔽₅)	—	9	1	no	Hall
8	C₂×C₃²×C₇×C₁₉	—	5	1	no	—
9	C₃×C₅×C₇×C₁₉	Dic₃	1	4	yes	complemented
10	C₃²×D₅×C₁₉	C₂×C₇	1	1	yes	characteristic
11	C₇×Dic₃×C₁₉	—	15	4	no	—
12	C₇×D₅×C₁₉	C₃⋊D₃	1	1	yes	characteristic
13	C₇×C₃⋊ΓL(1,𝔽₁₆)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₃²×C₇×C₁₉	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
15	C₁₉×ΓL(1,𝔽₁₆)	—	3	4	no	—
16	C₃²×C₅×C₁₉	C₄×C₇	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₇×C₁₉	—	5	4	no	—
18	C₁₉×C₃⋊Dic₃	—	5	1	no	Hall
19	C₅×C₇×C₁₉	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
20	C₃²×C₇×D₅	C₂×C₁₉	1	1	yes	characteristic
21	C₃×D₅×C₁₉	D₃×C₇	1	4	yes	—
22	C₄×C₇×C₁₉	—	45	1	no	Hall
23	C₇×ΓL(1,𝔽₁₆)	—	3	4	no	—
24	C₃×C₇×C₁₉	ΓL(1,𝔽₁₆)	1	4	yes	complemented
25	C₁₉×AGL(1,𝔽₅)	—	9	1	no	Hall
26	C₂×C₃²×C₁₉	—	5	1	no	—
27	C₃²×C₅×C₇	C₄×C₁₉	1	1	yes	characteristic, complemented, Hall
28	C₃×C₅×C₁₉	C₇×Dic₃	1	4	yes	complemented
29	C₂×C₇×C₁₉	—	5	1	no	—
30	C₇×C₃⋊Dic₃	—	5	1	no	Hall
31	Dic₃×C₁₉	—	15	4	no	—
32	C₃×C₇×D₅	D₃×C₁₉	1	4	yes	—
33	D₅×C₁₉	C₇×C₃⋊D₃	1	1	yes	characteristic
34	C₃⋊ΓL(1,𝔽₁₆)	C₇×C₁₉	1	1	yes	characteristic, direct product factor, Hall
35	C₃²×C₁₉	C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
36	C₇×AGL(1,𝔽₅)	—	9	1	no	Hall
37	C₇×C₁₉	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Hall
38	C₂×C₃²×C₇	—	5	1	no	—
39	C₂×C₃×C₁₉	—	5	4	no	—
40	C₃×C₅×C₇	Dic₃×C₁₉	1	4	yes	complemented
41	C₅×C₁₉	C₇×C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
42	C₃²×D₅	C₂×C₇×C₁₉	1	1	yes	characteristic
43	C₇×Dic₃	—	15	4	no	—
44	C₄×C₁₉	—	45	1	no	Hall
45	C₇×D₅	C₃⋊D₃×C₁₉	1	1	yes	characteristic
46	C₃²×C₇	C₁₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
47	ΓL(1,𝔽₁₆)	—	3	4	no	—
48	C₃×C₁₉	C₇×ΓL(1,𝔽₁₆)	1	4	yes	complemented
49	C₃²×C₅	C₄×C₇×C₁₉	1	1	yes	characteristic, complemented, Hall
50	C₂×C₃×C₇	—	5	4	no	—
51	C₂×C₁₉	—	5	1	no	—
52	C₃⋊Dic₃	—	5	1	no	Hall
53	C₅×C₇	C₁₉×C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
54	C₃×D₅	D₃×C₇×C₁₉	1	4	yes	—
55	C₄×C₇	—	45	1	no	Hall
56	C₃×C₇	C₁₉×ΓL(1,𝔽₁₆)	1	4	yes	complemented
57	AGL(1,𝔽₅)	—	9	1	no	Hall
58	C₁₉	C₇×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₃²	—	5	1	no	—
60	C₃×C₅	C₇×Dic₃×C₁₉	1	4	yes	complemented
61	C₂×C₇	—	5	1	no	—
62	Dic₃	—	15	4	no	—
63	D₅	C₇×C₃⋊D₃×C₁₉	1	1	yes	characteristic
64	C₃²	C₇×C₁₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
65	C₇	C₁₉×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃	—	5	4	no	—
67	C₅	C₇×C₁₉×C₃⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	45	1	no	Sylow
69	C₃	C₇×C₁₉×ΓL(1,𝔽₁₆)	1	4	yes	complemented
70	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₇×C₁₉	C₃⋊ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃²×C₅	C₄×C₇×C₁₉
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇×C₁₉	C₄
prime radical	C₃²×C₇×D₅×C₁₉	C₂
cosocle	C₃²×D₅	C₂×C₇×C₁₉
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₇×C₁₉

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅×C₇×C₁₉	C₄	semidirect
6. C₁₉×C₃⋊ΓL(1,𝔽₁₆)	C₇	direct (N × H)
9. C₃×C₅×C₇×C₁₉	Dic₃	semidirect
13. C₇×C₃⋊ΓL(1,𝔽₁₆)	C₁₉	direct (N × H)
14. C₃²×C₇×C₁₉	AGL(1,𝔽₅)	semidirect
16. C₃²×C₅×C₁₉	C₄×C₇	semidirect
19. C₅×C₇×C₁₉	C₃⋊Dic₃	semidirect
24. C₃×C₇×C₁₉	ΓL(1,𝔽₁₆)	semidirect
27. C₃²×C₅×C₇	C₄×C₁₉	semidirect
28. C₃×C₅×C₁₉	C₇×Dic₃	semidirect
34. C₃⋊ΓL(1,𝔽₁₆)	C₇×C₁₉	direct (N × H)
35. C₃²×C₁₉	C₇×AGL(1,𝔽₅)	semidirect
37. C₇×C₁₉	C₃⋊ΓL(1,𝔽₁₆)	direct (N × H)
40. C₃×C₅×C₇	Dic₃×C₁₉	semidirect
41. C₅×C₁₉	C₇×C₃⋊Dic₃	semidirect
46. C₃²×C₇	C₁₉×AGL(1,𝔽₅)	semidirect
48. C₃×C₁₉	C₇×ΓL(1,𝔽₁₆)	semidirect
49. C₃²×C₅	C₄×C₇×C₁₉	semidirect
53. C₅×C₇	C₁₉×C₃⋊Dic₃	semidirect
56. C₃×C₇	C₁₉×ΓL(1,𝔽₁₆)	semidirect
58. C₁₉	C₇×C₃⋊ΓL(1,𝔽₁₆)	direct (N × H)
60. C₃×C₅	C₇×Dic₃×C₁₉	semidirect
64. C₃²	C₇×C₁₉×AGL(1,𝔽₅)	semidirect
65. C₇	C₁₉×C₃⋊ΓL(1,𝔽₁₆)	direct (N × H)
67. C₅	C₇×C₁₉×C₃⋊Dic₃	semidirect
69. C₃	C₇×C₁₉×ΓL(1,𝔽₁₆)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×C₇×D₅×C₁₉	2
C₇×C₁₉×ΓL(1,𝔽₁₆)	3
C₇×C₁₉×C₃⋊Dic₃	5
C₁₉×C₃⋊ΓL(1,𝔽₁₆)	7
C₇×C₃⋊ΓL(1,𝔽₁₆)	19

Subgroups — C7×C19×C3⋊ΓL(1,𝔽16) #37561594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₁₉×C₃⋊ΓL(1,𝔽₁₆) #37561594