Subgroups — C₇×ΓL(1,𝔽₁₆)×C₇₃ #44501418

Order 30660.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×D₅×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₇×AGL(1,𝔽₅)×C₇₃	—	3	1	no	maximal, Hall
3	C₃×C₅×C₇×C₇₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×Dic₃×C₇₃	—	5	1	no	maximal, Hall
5	C₇×D₅×C₇₃	D₃	1	1	yes	characteristic
6	ΓL(1,𝔽₁₆)×C₇₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₇×C₇₃	—	5	1	no	—
8	C₅×C₇×C₇₃	Dic₃	1	1	yes	characteristic, complemented, Hall
9	C₃×D₅×C₇₃	C₂×C₇	1	1	yes	characteristic
10	C₄×C₇×C₇₃	—	15	1	no	Hall
11	C₃×C₇×C₇₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
12	AGL(1,𝔽₅)×C₇₃	—	3	1	no	Hall
13	C₃×C₅×C₇₃	C₄×C₇	1	1	yes	characteristic, complemented, Hall
14	C₂×C₇×C₇₃	—	5	1	no	—
15	Dic₃×C₇₃	—	5	1	no	Hall
16	D₅×C₇₃	D₃×C₇	1	1	yes	characteristic
17	C₇×C₇₃	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₇₃	—	5	1	no	—
19	C₇×ΓL(1,𝔽₁₆)	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₅×C₇₃	C₇×Dic₃	1	1	yes	characteristic, complemented, Hall
21	C₄×C₇₃	—	15	1	no	Hall
22	C₃×C₇₃	C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
23	C₃×C₇×D₅	C₂×C₇₃	1	1	yes	characteristic
24	C₂×C₇₃	—	5	1	no	—
25	C₇×AGL(1,𝔽₅)	—	3	1	no	Hall
26	C₃×C₅×C₇	C₄×C₇₃	1	1	yes	characteristic, complemented, Hall
27	C₇×Dic₃	—	5	1	no	Hall
28	C₇₃	C₇×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
29	C₇×D₅	D₃×C₇₃	1	1	yes	characteristic
30	ΓL(1,𝔽₁₆)	C₇×C₇₃	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₇	—	5	1	no	—
32	C₅×C₇	Dic₃×C₇₃	1	1	yes	characteristic, complemented, Hall
33	C₃×D₅	C₂×C₇×C₇₃	1	1	yes	characteristic
34	C₄×C₇	—	15	1	no	Hall
35	C₃×C₇	AGL(1,𝔽₅)×C₇₃	1	1	yes	characteristic, complemented, Hall
36	AGL(1,𝔽₅)	—	3	1	no	Hall
37	C₃×C₅	C₄×C₇×C₇₃	1	1	yes	characteristic, complemented, Hall
38	C₂×C₇	—	5	1	no	—
39	Dic₃	—	5	1	no	Hall
40	D₅	D₃×C₇×C₇₃	1	1	yes	characteristic
41	C₇	ΓL(1,𝔽₁₆)×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₃	—	5	1	no	—
43	C₅	C₇×Dic₃×C₇₃	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	15	1	no	Sylow
45	C₃	C₇×AGL(1,𝔽₅)×C₇₃	1	1	yes	characteristic, complemented, Sylow
46	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₇×C₇₃	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₇×C₇₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₇₃	C₄
prime radical	C₃×C₇×D₅×C₇₃	C₂
cosocle	C₃×D₅	C₂×C₇×C₇₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₇×C₇₃

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₇×C₇₃	C₄	semidirect
6. ΓL(1,𝔽₁₆)×C₇₃	C₇	direct (N × H)
8. C₅×C₇×C₇₃	Dic₃	semidirect
11. C₃×C₇×C₇₃	AGL(1,𝔽₅)	semidirect
13. C₃×C₅×C₇₃	C₄×C₇	semidirect
17. C₇×C₇₃	ΓL(1,𝔽₁₆)	direct (N × H)
19. C₇×ΓL(1,𝔽₁₆)	C₇₃	direct (N × H)
20. C₅×C₇₃	C₇×Dic₃	semidirect
22. C₃×C₇₃	C₇×AGL(1,𝔽₅)	semidirect
26. C₃×C₅×C₇	C₄×C₇₃	semidirect
28. C₇₃	C₇×ΓL(1,𝔽₁₆)	direct (N × H)
30. ΓL(1,𝔽₁₆)	C₇×C₇₃	direct (N × H)
32. C₅×C₇	Dic₃×C₇₃	semidirect
35. C₃×C₇	AGL(1,𝔽₅)×C₇₃	semidirect
37. C₃×C₅	C₄×C₇×C₇₃	semidirect
41. C₇	ΓL(1,𝔽₁₆)×C₇₃	direct (N × H)
43. C₅	C₇×Dic₃×C₇₃	semidirect
45. C₃	C₇×AGL(1,𝔽₅)×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×D₅×C₇₃	2
C₇×AGL(1,𝔽₅)×C₇₃	3
C₇×Dic₃×C₇₃	5
ΓL(1,𝔽₁₆)×C₇₃	7
C₇×ΓL(1,𝔽₁₆)	73

Subgroups — C7×ΓL(1,𝔽16)×C73 #44501418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×ΓL(1,𝔽₁₆)×C₇₃ #44501418