Subgroups — C₂²×C₅₃×#3861 #44528481

Order 31164.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#3861	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₇²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅₃×#3861	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₇²×C₅₃	C₂×C₃	1	3	yes	complemented
5	C₂²×ΓL(1,𝔽₈)×C₅₃	—	7	8	no	maximal
6	C₇²×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
7	C₂×ΓL(1,𝔽₈)×C₅₃	—	7	24	no	—
8	C₂²×C₇×C₅₃	ΓL(1,𝔽₈)	1	8	yes	complemented
9	ΓL(1,𝔽₈)×C₅₃	—	7	8	no	—
10	C₂×C₇×C₅₃	C₂×ΓL(1,𝔽₈)	1	24	yes	complemented
11	C₂²×C₃×C₅₃	—	49	1	no	Hall
12	C₂²×#3861	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₇×C₅₃	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
14	C₂×C₃×C₅₃	—	49	3	no	—
15	C₂×#3861	C₂×C₅₃	1	3	yes	direct product factor
16	C₂²×C₅₃	#3861	1	1	yes	characteristic, direct product factor, Hall
17	C₂²×C₇²	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
18	C₃×C₅₃	—	49	1	no	Hall
19	#3861	C₂²×C₅₃	1	1	yes	characteristic, direct product factor, Hall
20	C₂×C₅₃	C₂×#3861	1	3	yes	direct product factor
21	C₂×C₇²	C₂×C₃×C₅₃	1	3	yes	complemented
22	C₂²×ΓL(1,𝔽₈)	—	7	8	no	—
23	C₅₃	C₂²×#3861	1	1	yes	characteristic, direct product factor, Sylow
24	C₇²	C₂²×C₃×C₅₃	1	1	yes	characteristic, complemented, Sylow
25	C₂×ΓL(1,𝔽₈)	—	7	24	no	—
26	C₂²×C₇	ΓL(1,𝔽₈)×C₅₃	1	8	yes	complemented
27	ΓL(1,𝔽₈)	—	7	8	no	—
28	C₂×C₇	C₂×ΓL(1,𝔽₈)×C₅₃	1	24	yes	complemented
29	C₂²×C₃	—	49	1	no	Hall
30	C₇	C₂²×ΓL(1,𝔽₈)×C₅₃	1	8	yes	complemented
31	C₂×C₃	—	49	3	no	—
32	C₂²	C₅₃×#3861	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	—	49	1	no	Sylow
34	C₂	C₂×C₅₃×#3861	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅₃	#3861
commutator, cosocle, hypocenter	C₇²	C₂²×C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇²×C₅₃	C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂²×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#3861	C₂	direct (N × H)
2. C₂²×C₇²×C₅₃	C₃	semidirect
3. C₅₃×#3861	C₂²	direct (N × H)
4. C₂×C₇²×C₅₃	C₂×C₃	semidirect
6. C₇²×C₅₃	C₂²×C₃	semidirect
8. C₂²×C₇×C₅₃	ΓL(1,𝔽₈)	semidirect
10. C₂×C₇×C₅₃	C₂×ΓL(1,𝔽₈)	semidirect
12. C₂²×#3861	C₅₃	direct (N × H)
13. C₇×C₅₃	C₂²×ΓL(1,𝔽₈)	semidirect
15. C₂×#3861	C₂×C₅₃	direct (N × H)
16. C₂²×C₅₃	#3861	direct (N × H)
17. C₂²×C₇²	C₃×C₅₃	semidirect
19. #3861	C₂²×C₅₃	direct (N × H)
20. C₂×C₅₃	C₂×#3861	direct (N × H)
21. C₂×C₇²	C₂×C₃×C₅₃	semidirect
23. C₅₃	C₂²×#3861	direct (N × H)
24. C₇²	C₂²×C₃×C₅₃	semidirect
26. C₂²×C₇	ΓL(1,𝔽₈)×C₅₃	semidirect
28. C₂×C₇	C₂×ΓL(1,𝔽₈)×C₅₃	semidirect
30. C₇	C₂²×ΓL(1,𝔽₈)×C₅₃	semidirect
32. C₂²	C₅₃×#3861	direct (N × H)
34. C₂	C₂×C₅₃×#3861	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#3861	2
C₂²×C₇²×C₅₃	3
C₂²×ΓL(1,𝔽₈)×C₅₃	7
C₂²×#3861	53

Subgroups — C22×C53×#3861 #44528481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#3861 #44528481