Subgroups — C₃×C₇×ΓL(1,𝔽₈)×C₇₃ #44567161

Order 32193.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×ΓL(1,𝔽₈)×C₇₃	C₃	1	3	yes	maximal, direct product factor
2	C₃×C₇²×C₇₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×ΓL(1,𝔽₈)×C₇₃	C₇	1	1	yes	maximal, characteristic, direct product factor
4	C₃²×C₇×C₇₃	—	7	1	no	maximal
5	C₇²×C₇₃	C₃²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₇×C₇₃	—	3	2	no	—
7	C₃×C₇×C₇₃	—	7	3	no	—
8	C₃×C₇×C₇₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
9	C₃×C₇×C₇₃	C₃×C₇	1	1	yes	characteristic, complemented
10	ΓL(1,𝔽₈)×C₇₃	C₃×C₇	1	3	yes	direct product factor
11	C₃²×C₇₃	—	7	1	no	Hall
12	C₇×C₇₃	—	3	2	no	—
13	C₇×C₇₃	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
14	C₇×C₇₃	C₃²×C₇	1	1	yes	characteristic, complemented
15	C₃×C₇×ΓL(1,𝔽₈)	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₃×C₇₃	—	7	3	no	—
17	C₃×C₇₃	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
18	C₃×C₇²	C₃×C₇₃	1	1	yes	characteristic, complemented
19	C₇×ΓL(1,𝔽₈)	C₃×C₇₃	1	3	yes	direct product factor
20	C₇₃	C₃×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
21	C₃²×C₇	—	7	1	no	—
22	C₃×ΓL(1,𝔽₈)	C₇×C₇₃	1	1	yes	characteristic, direct product factor
23	C₇²	C₃²×C₇₃	1	1	yes	characteristic, complemented, Sylow
24	C₃×C₇	—	3	2	no	—
25	C₃×C₇	—	7	3	no	—
26	C₃×C₇	ΓL(1,𝔽₈)×C₇₃	1	1	yes	characteristic, direct product factor
27	C₃×C₇	C₃×C₇×C₇₃	1	1	yes	characteristic, complemented
28	ΓL(1,𝔽₈)	C₃×C₇×C₇₃	1	3	yes	direct product factor
29	C₃²	—	7	1	no	Sylow
30	C₇	—	3	2	no	—
31	C₇	C₃²×C₇×C₇₃	1	1	yes	characteristic, complemented
32	C₇	C₃×ΓL(1,𝔽₈)×C₇₃	1	1	yes	characteristic, direct product factor
33	C₃	—	7	3	no	—
34	C₃	C₇×ΓL(1,𝔽₈)×C₇₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₃×C₇×C₇₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃²×C₇×C₇₃
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₇₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₇×C₇₃

Complements

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

N	H ≅ G/N	type
1. C₇×ΓL(1,𝔽₈)×C₇₃	C₃	direct (N × H)
2. C₃×C₇²×C₇₃	C₃	semidirect
3. C₃×ΓL(1,𝔽₈)×C₇₃	C₇	direct (N × H)
5. C₇²×C₇₃	C₃²	semidirect
8. C₃×C₇×C₇₃	ΓL(1,𝔽₈)	direct (N × H)
9. C₃×C₇×C₇₃	C₃×C₇	semidirect
10. ΓL(1,𝔽₈)×C₇₃	C₃×C₇	direct (N × H)
13. C₇×C₇₃	C₃×ΓL(1,𝔽₈)	direct (N × H)
14. C₇×C₇₃	C₃²×C₇	semidirect
15. C₃×C₇×ΓL(1,𝔽₈)	C₇₃	direct (N × H)
17. C₃×C₇₃	C₇×ΓL(1,𝔽₈)	direct (N × H)
18. C₃×C₇²	C₃×C₇₃	semidirect
19. C₇×ΓL(1,𝔽₈)	C₃×C₇₃	direct (N × H)
20. C₇₃	C₃×C₇×ΓL(1,𝔽₈)	direct (N × H)
22. C₃×ΓL(1,𝔽₈)	C₇×C₇₃	direct (N × H)
23. C₇²	C₃²×C₇₃	semidirect
26. C₃×C₇	ΓL(1,𝔽₈)×C₇₃	direct (N × H)
27. C₃×C₇	C₃×C₇×C₇₃	semidirect
28. ΓL(1,𝔽₈)	C₃×C₇×C₇₃	direct (N × H)
31. C₇	C₃²×C₇×C₇₃	semidirect
32. C₇	C₃×ΓL(1,𝔽₈)×C₇₃	direct (N × H)
34. C₃	C₇×ΓL(1,𝔽₈)×C₇₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×ΓL(1,𝔽₈)×C₇₃	3
C₃×C₇²×C₇₃	3
C₃×ΓL(1,𝔽₈)×C₇₃	7
C₃²×C₇×C₇₃	7
C₃×C₇×ΓL(1,𝔽₈)	73

Subgroups — C3×C7×ΓL(1,𝔽8)×C73 #44567161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×ΓL(1,𝔽₈)×C₇₃ #44567161