Subgroups — C₅×C₇₃×C₂²⋊ΓL(1,𝔽₈) #44501452

Order 30660.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₇×C₇₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×ΓL(1,𝔽₈)×C₇₃	—	4	1	no	maximal, Hall
3	C₇₃×C₂²⋊ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₅×C₇×C₇₃	—	3	1	no	—
5	C₅×A₄×C₇₃	—	7	1	no	maximal, Hall
6	C₅×C₇×C₇₃	A₄	1	1	yes	characteristic, complemented, Hall
7	C₂²×C₇×C₇₃	C₃×C₅	1	1	yes	characteristic, complemented, Hall
8	ΓL(1,𝔽₈)×C₇₃	—	4	1	no	Hall
9	C₂²×C₅×C₇₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
10	C₃×C₅×C₇₃	—	28	1	no	Hall
11	C₂×C₇×C₇₃	—	3	1	no	—
12	A₄×C₇₃	—	7	1	no	Hall
13	C₂×C₅×C₇₃	—	3	1	no	—
14	C₇×C₇₃	C₅×A₄	1	1	yes	characteristic, complemented, Hall
15	C₅×C₂²⋊ΓL(1,𝔽₈)	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₅×C₇₃	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
17	C₂²×C₇₃	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
18	C₃×C₇₃	—	28	1	no	Hall
19	C₂×C₇₃	—	3	1	no	—
20	C₂²×C₅×C₇	C₃×C₇₃	1	1	yes	characteristic, complemented, Hall
21	C₅×ΓL(1,𝔽₈)	—	4	1	no	Hall
22	C₂²⋊ΓL(1,𝔽₈)	C₅×C₇₃	1	1	yes	characteristic, direct product factor, Hall
23	C₇₃	C₅×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₅×C₇	—	3	1	no	—
25	C₅×A₄	—	7	1	no	Hall
26	C₅×C₇	A₄×C₇₃	1	1	yes	characteristic, complemented, Hall
27	C₂²×C₇	C₃×C₅×C₇₃	1	1	yes	characteristic, complemented, Hall
28	ΓL(1,𝔽₈)	—	4	1	no	Hall
29	C₂²×C₅	ΓL(1,𝔽₈)×C₇₃	1	1	yes	characteristic, complemented, Hall
30	C₃×C₅	—	28	1	no	Hall
31	C₂×C₇	—	3	1	no	—
32	A₄	—	7	1	no	Hall
33	C₂×C₅	—	3	1	no	—
34	C₇	C₅×A₄×C₇₃	1	1	yes	characteristic, complemented, Sylow
35	C₅	C₇₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²	C₅×ΓL(1,𝔽₈)×C₇₃	1	1	yes	characteristic, complemented, Sylow
37	C₃	—	28	1	no	Sylow
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅×C₇₃	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₅×C₇₃
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇×C₇₃	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₅×C₇₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₇×C₇₃	C₃	semidirect
3. C₇₃×C₂²⋊ΓL(1,𝔽₈)	C₅	direct (N × H)
6. C₅×C₇×C₇₃	A₄	semidirect
7. C₂²×C₇×C₇₃	C₃×C₅	semidirect
9. C₂²×C₅×C₇₃	ΓL(1,𝔽₈)	semidirect
14. C₇×C₇₃	C₅×A₄	semidirect
15. C₅×C₂²⋊ΓL(1,𝔽₈)	C₇₃	direct (N × H)
16. C₅×C₇₃	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
17. C₂²×C₇₃	C₅×ΓL(1,𝔽₈)	semidirect
20. C₂²×C₅×C₇	C₃×C₇₃	semidirect
22. C₂²⋊ΓL(1,𝔽₈)	C₅×C₇₃	direct (N × H)
23. C₇₃	C₅×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
26. C₅×C₇	A₄×C₇₃	semidirect
27. C₂²×C₇	C₃×C₅×C₇₃	semidirect
29. C₂²×C₅	ΓL(1,𝔽₈)×C₇₃	semidirect
34. C₇	C₅×A₄×C₇₃	semidirect
35. C₅	C₇₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
36. C₂²	C₅×ΓL(1,𝔽₈)×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₇×C₇₃	3
C₅×ΓL(1,𝔽₈)×C₇₃	4
C₇₃×C₂²⋊ΓL(1,𝔽₈)	5
C₅×A₄×C₇₃	7
C₅×C₂²⋊ΓL(1,𝔽₈)	73

Subgroups — C5×C73×C22⋊ΓL(1,𝔽8) #44501452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇₃×C₂²⋊ΓL(1,𝔽₈) #44501452