Subgroups — C₇×C₅₃×C₂²⋊ΓL(1,𝔽₈) #44528477

Order 31164.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×ΓL(1,𝔽₈)×C₅₃	—	4	1	no	maximal, Hall
3	C₂×C₇²×C₅₃	—	3	1	no	—
4	C₅₃×C₂²⋊ΓL(1,𝔽₈)	C₇	1	1	yes	maximal, characteristic, direct product factor
5	C₇×A₄×C₅₃	—	7	1	no	maximal
6	C₇²×C₅₃	A₄	1	1	yes	characteristic, complemented, Hall
7	C₂²×C₇×C₅₃	—	3	2	no	—
8	C₂²×C₇×C₅₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
9	C₂²×C₇×C₅₃	C₃×C₇	1	1	yes	characteristic, complemented
10	C₃×C₇×C₅₃	—	28	1	no	—
11	ΓL(1,𝔽₈)×C₅₃	—	4	1	no	—
12	C₂×C₇×C₅₃	—	3	6	no	—
13	C₂×C₇×C₅₃	—	3	1	no	—
14	C₂×C₇×C₅₃	—	3	1	no	—
15	A₄×C₅₃	—	7	1	no	Hall
16	C₇×C₂²⋊ΓL(1,𝔽₈)	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₇×C₅₃	—	3	2	no	—
18	C₇×C₅₃	C₇×A₄	1	1	yes	characteristic, complemented
19	C₇×C₅₃	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	C₂²×C₅₃	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
21	C₂²×C₇²	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
22	C₃×C₅₃	—	28	1	no	Hall
23	C₇×ΓL(1,𝔽₈)	—	4	1	no	Hall
24	C₂×C₅₃	—	3	1	no	—
25	C₂×C₇²	—	3	1	no	—
26	C₂²⋊ΓL(1,𝔽₈)	C₇×C₅₃	1	1	yes	characteristic, direct product factor
27	C₇×A₄	—	7	1	no	—
28	C₅₃	C₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
29	C₇²	A₄×C₅₃	1	1	yes	characteristic, complemented, Sylow
30	C₂²×C₇	—	3	2	no	—
31	C₂²×C₇	ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic, complemented
32	C₂²×C₇	C₃×C₇×C₅₃	1	1	yes	characteristic, complemented
33	C₃×C₇	—	28	1	no	—
34	ΓL(1,𝔽₈)	—	4	1	no	—
35	C₂×C₇	—	3	6	no	—
36	C₂×C₇	—	3	1	no	—
37	C₂×C₇	—	3	1	no	—
38	A₄	—	7	1	no	Hall
39	C₇	—	3	2	no	—
40	C₇	C₇×A₄×C₅₃	1	1	yes	characteristic, complemented
41	C₇	C₅₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
42	C₂²	C₇×ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic, complemented, Sylow
43	C₃	—	28	1	no	Sylow
44	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₇×C₅₃	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₇×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇²×C₅₃	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₇×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇²×C₅₃	C₃	semidirect
4. C₅₃×C₂²⋊ΓL(1,𝔽₈)	C₇	direct (N × H)
6. C₇²×C₅₃	A₄	semidirect
8. C₂²×C₇×C₅₃	ΓL(1,𝔽₈)	semidirect
9. C₂²×C₇×C₅₃	C₃×C₇	semidirect
16. C₇×C₂²⋊ΓL(1,𝔽₈)	C₅₃	direct (N × H)
18. C₇×C₅₃	C₇×A₄	semidirect
19. C₇×C₅₃	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
20. C₂²×C₅₃	C₇×ΓL(1,𝔽₈)	semidirect
21. C₂²×C₇²	C₃×C₅₃	semidirect
26. C₂²⋊ΓL(1,𝔽₈)	C₇×C₅₃	direct (N × H)
28. C₅₃	C₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
29. C₇²	A₄×C₅₃	semidirect
31. C₂²×C₇	ΓL(1,𝔽₈)×C₅₃	semidirect
32. C₂²×C₇	C₃×C₇×C₅₃	semidirect
40. C₇	C₇×A₄×C₅₃	semidirect
41. C₇	C₅₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
42. C₂²	C₇×ΓL(1,𝔽₈)×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇²×C₅₃	3
C₇×ΓL(1,𝔽₈)×C₅₃	4
C₅₃×C₂²⋊ΓL(1,𝔽₈)	7
C₇×A₄×C₅₃	7
C₇×C₂²⋊ΓL(1,𝔽₈)	53

Subgroups — C7×C53×C22⋊ΓL(1,𝔽8) #44528477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₅₃×C₂²⋊ΓL(1,𝔽₈) #44528477