Subgroups — C₇₃×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅) #44501303

Order 30660.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×ΓL(1,𝔽₈)×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₇₃×C₇⋊AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×ΓL(1,𝔽₈)×C₇₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇₃×ΓL(1,𝔽₈)⋊C₄	—	5	1	no	maximal, Hall
5	C₇×D₅×C₇₃	C₂×C₃	1	1	yes	characteristic
6	C₃×AGL(1,𝔽₅)×C₇₃	—	7	1	no	maximal, Hall
7	C₂×ΓL(1,𝔽₈)×C₇₃	—	5	1	no	—
8	C₅×C₇×C₇₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₃×D₅×C₇₃	—	7	1	no	—
10	Dic₇×C₇₃	—	5	1	no	Hall
11	ΓL(1,𝔽₈)×C₇₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
12	AGL(1,𝔽₅)×C₇₃	—	7	1	no	Hall
13	C₃×C₅×C₇₃	—	7	1	no	Hall
14	C₂×C₇×C₇₃	—	5	1	no	—
15	C₃×C₄×C₇₃	—	35	1	no	Hall
16	D₅×C₇₃	AGL(1,𝔽₇)	1	1	yes	characteristic
17	C₇×C₇₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₇₃	—	35	1	no	—
19	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₅×C₇₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Hall
21	C₄×C₇₃	—	35	1	no	Hall
22	C₃×C₇₃	—	7	1	no	Hall
23	D₅×ΓL(1,𝔽₈)	C₂×C₇₃	1	1	yes	characteristic
24	C₂×C₇₃	—	5	1	no	—
25	C₇⋊AGL(1,𝔽₅)	C₃×C₇₃	1	1	yes	characteristic, complemented, Hall
26	C₅×ΓL(1,𝔽₈)	C₄×C₇₃	1	1	yes	characteristic, complemented, Hall
27	ΓL(1,𝔽₈)⋊C₄	—	5	1	no	Hall
28	C₇₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
29	C₇×D₅	C₂×C₃×C₇₃	1	1	yes	characteristic
30	C₃×AGL(1,𝔽₅)	—	7	1	no	Hall
31	C₂×ΓL(1,𝔽₈)	—	5	1	no	—
32	C₅×C₇	C₃×C₄×C₇₃	1	1	yes	characteristic, complemented, Hall
33	C₃×D₅	—	7	1	no	—
34	Dic₇	—	5	1	no	Hall
35	ΓL(1,𝔽₈)	AGL(1,𝔽₅)×C₇₃	1	1	yes	characteristic, complemented, Hall
36	AGL(1,𝔽₅)	—	7	1	no	Hall
37	C₃×C₅	—	7	1	no	Hall
38	C₂×C₇	—	5	1	no	—
39	C₃×C₄	—	35	1	no	Hall
40	D₅	AGL(1,𝔽₇)×C₇₃	1	1	yes	characteristic
41	C₇	C₃×AGL(1,𝔽₅)×C₇₃	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₃	—	35	1	no	—
43	C₅	C₇₃×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	35	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₇₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₇	C₃×C₄×C₇₃
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₇₃	C₃×C₄
prime radical	D₅×ΓL(1,𝔽₈)×C₇₃	C₂
cosocle	C₇×D₅	C₂×C₃×C₇₃
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₇₃

Complements

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

N	H ≅ G/N	type
2. C₇₃×C₇⋊AGL(1,𝔽₅)	C₃	semidirect
3. C₅×ΓL(1,𝔽₈)×C₇₃	C₄	semidirect
8. C₅×C₇×C₇₃	C₃×C₄	semidirect
11. ΓL(1,𝔽₈)×C₇₃	AGL(1,𝔽₅)	semidirect
17. C₇×C₇₃	C₃×AGL(1,𝔽₅)	semidirect
19. ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	C₇₃	direct (N × H)
20. C₅×C₇₃	ΓL(1,𝔽₈)⋊C₄	semidirect
25. C₇⋊AGL(1,𝔽₅)	C₃×C₇₃	semidirect
26. C₅×ΓL(1,𝔽₈)	C₄×C₇₃	semidirect
28. C₇₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	direct (N × H)
32. C₅×C₇	C₃×C₄×C₇₃	semidirect
35. ΓL(1,𝔽₈)	AGL(1,𝔽₅)×C₇₃	semidirect
41. C₇	C₃×AGL(1,𝔽₅)×C₇₃	semidirect
43. C₅	C₇₃×ΓL(1,𝔽₈)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
D₅×ΓL(1,𝔽₈)×C₇₃	2
C₇₃×C₇⋊AGL(1,𝔽₅)	3
C₇₃×ΓL(1,𝔽₈)⋊C₄	5
C₃×AGL(1,𝔽₅)×C₇₃	7
ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	73

Subgroups — C73×ΓL(1,𝔽8)⋊AGL(1,𝔽5) #44501303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅) #44501303