Subgroups — C₂²×C₅₃×#3862 #44528482

Order 31164.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#3862	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₇²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅₃×#3862	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	2	no	maximal
6	C₇²×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
7	C₂×ΓL(1,𝔽₈)×C₅₃	—	7	6	no	—
8	C₂²×C₇×C₅₃	—	3	2	no	—
9	C₂²×C₇×C₅₃	ΓL(1,𝔽₈)	1	2	yes	complemented
10	ΓL(1,𝔽₈)×C₅₃	—	7	2	no	—
11	C₂×C₇×C₅₃	—	3	6	no	—
12	C₂×C₇×C₅₃	C₂×ΓL(1,𝔽₈)	1	6	yes	complemented
13	C₂²×C₃×C₅₃	—	49	1	no	Hall
14	C₂²×#3862	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₇×C₅₃	—	3	2	no	—
16	C₇×C₅₃	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
17	C₂×C₃×C₅₃	—	49	3	no	—
18	C₂×#3862	C₂×C₅₃	1	3	yes	direct product factor
19	C₂²×C₅₃	#3862	1	1	yes	characteristic, direct product factor, Hall
20	C₂²×C₇²	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
21	C₃×C₅₃	—	49	1	no	Hall
22	#3862	C₂²×C₅₃	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₅₃	C₂×#3862	1	3	yes	direct product factor
24	C₂×C₇²	C₂×C₃×C₅₃	1	3	yes	complemented
25	C₂²×ΓL(1,𝔽₈)	—	7	2	no	—
26	C₅₃	C₂²×#3862	1	1	yes	characteristic, direct product factor, Sylow
27	C₇²	C₂²×C₃×C₅₃	1	1	yes	characteristic, complemented, Sylow
28	C₂×ΓL(1,𝔽₈)	—	7	6	no	—
29	C₂²×C₇	—	3	2	no	—
30	C₂²×C₇	ΓL(1,𝔽₈)×C₅₃	1	2	yes	complemented
31	ΓL(1,𝔽₈)	—	7	2	no	—
32	C₂×C₇	—	3	6	no	—
33	C₂×C₇	C₂×ΓL(1,𝔽₈)×C₅₃	1	6	yes	complemented
34	C₂²×C₃	—	49	1	no	Hall
35	C₇	—	3	2	no	—
36	C₇	C₂²×ΓL(1,𝔽₈)×C₅₃	1	2	yes	complemented
37	C₂×C₃	—	49	3	no	—
38	C₂²	C₅₃×#3862	1	1	yes	characteristic, direct product factor, Sylow
39	C₃	—	49	1	no	Sylow
40	C₂	C₂×C₅₃×#3862	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅₃	#3862
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×#3862	C₂	direct (N × H)
2. C₂²×C₇²×C₅₃	C₃	semidirect
3. C₅₃×#3862	C₂²	direct (N × H)
4. C₂×C₇²×C₅₃	C₂×C₃	semidirect
6. C₇²×C₅₃	C₂²×C₃	semidirect
9. C₂²×C₇×C₅₃	ΓL(1,𝔽₈)	semidirect
12. C₂×C₇×C₅₃	C₂×ΓL(1,𝔽₈)	semidirect
14. C₂²×#3862	C₅₃	direct (N × H)
16. C₇×C₅₃	C₂²×ΓL(1,𝔽₈)	semidirect
18. C₂×#3862	C₂×C₅₃	direct (N × H)
19. C₂²×C₅₃	#3862	direct (N × H)
20. C₂²×C₇²	C₃×C₅₃	semidirect
22. #3862	C₂²×C₅₃	direct (N × H)
23. C₂×C₅₃	C₂×#3862	direct (N × H)
24. C₂×C₇²	C₂×C₃×C₅₃	semidirect
26. C₅₃	C₂²×#3862	direct (N × H)
27. C₇²	C₂²×C₃×C₅₃	semidirect
30. C₂²×C₇	ΓL(1,𝔽₈)×C₅₃	semidirect
33. C₂×C₇	C₂×ΓL(1,𝔽₈)×C₅₃	semidirect
36. C₇	C₂²×ΓL(1,𝔽₈)×C₅₃	semidirect
38. C₂²	C₅₃×#3862	direct (N × H)
40. C₂	C₂×C₅₃×#3862	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×#3862	2
C₂²×C₇²×C₅₃	3
C₂²×ΓL(1,𝔽₈)×C₅₃	7
C₂²×#3862	53

Subgroups — C22×C53×#3862 #44528482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#3862 #44528482