Subgroups — C₃×C₅₃×C₂²⋊ΓL(1,𝔽₈) #27211323

Order 13356.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×C₂²⋊ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₇×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×ΓL(1,𝔽₈)×C₅₃	—	4	1	no	maximal, Hall
4	C₂×C₃×C₇×C₅₃	—	3	1	no	—
5	C₃×A₄×C₅₃	—	7	1	no	maximal, Hall
6	C₂²×C₇×C₅₃	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₇×C₅₃	A₄	1	1	yes	characteristic, complemented
8	ΓL(1,𝔽₈)×C₅₃	—	4	3	no	—
9	C₂×C₇×C₅₃	—	3	1	no	—
10	C₂²×C₃×C₅₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
11	A₄×C₅₃	—	7	3	no	—
12	C₃²×C₅₃	—	28	1	no	Hall
13	C₇×C₅₃	C₃×A₄	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅₃	—	3	1	no	—
15	C₃×C₂²⋊ΓL(1,𝔽₈)	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₅₃	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
17	C₃×C₅₃	—	28	3	no	—
18	C₃×C₅₃	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
19	C₂×C₅₃	—	3	1	no	—
20	C₂²×C₃×C₇	C₃×C₅₃	1	1	yes	characteristic, complemented
21	C₂²⋊ΓL(1,𝔽₈)	C₃×C₅₃	1	3	yes	direct product factor
22	C₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
23	C₅₃	C₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₃×C₇	—	3	1	no	—
25	C₃×A₄	—	7	1	no	Hall
26	C₂²×C₇	C₃²×C₅₃	1	1	yes	characteristic, complemented, Hall
27	C₃×C₇	A₄×C₅₃	1	1	yes	characteristic, complemented
28	ΓL(1,𝔽₈)	—	4	3	no	—
29	C₂×C₇	—	3	1	no	—
30	C₂²×C₃	ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic, complemented
31	A₄	—	7	3	no	—
32	C₃²	—	28	1	no	Sylow
33	C₇	C₃×A₄×C₅₃	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃	—	3	1	no	—
35	C₂²	C₃×ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic, complemented, Sylow
36	C₃	—	28	3	no	—
37	C₃	C₅₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅₃	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃²×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇×C₅₃	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₃×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₅₃×C₂²⋊ΓL(1,𝔽₈)	C₃	direct (N × H)
2. C₂²×C₃×C₇×C₅₃	C₃	semidirect
6. C₂²×C₇×C₅₃	C₃²	semidirect
7. C₃×C₇×C₅₃	A₄	semidirect
10. C₂²×C₃×C₅₃	ΓL(1,𝔽₈)	semidirect
13. C₇×C₅₃	C₃×A₄	semidirect
15. C₃×C₂²⋊ΓL(1,𝔽₈)	C₅₃	direct (N × H)
16. C₂²×C₅₃	C₃×ΓL(1,𝔽₈)	semidirect
18. C₃×C₅₃	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
20. C₂²×C₃×C₇	C₃×C₅₃	semidirect
21. C₂²⋊ΓL(1,𝔽₈)	C₃×C₅₃	direct (N × H)
23. C₅₃	C₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
26. C₂²×C₇	C₃²×C₅₃	semidirect
27. C₃×C₇	A₄×C₅₃	semidirect
30. C₂²×C₃	ΓL(1,𝔽₈)×C₅₃	semidirect
33. C₇	C₃×A₄×C₅₃	semidirect
35. C₂²	C₃×ΓL(1,𝔽₈)×C₅₃	semidirect
37. C₃	C₅₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×C₂²⋊ΓL(1,𝔽₈)	3
C₂²×C₃×C₇×C₅₃	3
C₃×ΓL(1,𝔽₈)×C₅₃	4
C₃×A₄×C₅₃	7
C₃×C₂²⋊ΓL(1,𝔽₈)	53

Subgroups — C3×C53×C22⋊ΓL(1,𝔽8) #27211323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅₃×C₂²⋊ΓL(1,𝔽₈) #27211323