Subgroups — C₄×C₅₃×C₄₉⋊C₃ #44528310

Order 31164.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×C₄₉⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄₉×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅₃×C₄₉⋊C₃	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₄₉×C₅₃	C₂×C₃	1	1	yes	characteristic
5	C₄×ΓL(1,𝔽₈)×C₅₃	—	7	1	no	maximal
6	C₄₉×C₅₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₂×ΓL(1,𝔽₈)×C₅₃	—	7	1	no	—
8	C₄×C₇×C₅₃	ΓL(1,𝔽₈)	1	1	yes	characteristic
9	ΓL(1,𝔽₈)×C₅₃	—	7	1	no	—
10	C₂×C₇×C₅₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
11	C₃×C₄×C₅₃	—	49	1	no	Hall
12	C₄×C₄₉⋊C₃	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₇×C₅₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
14	C₂×C₃×C₅₃	—	49	1	no	—
15	C₂×C₄₉⋊C₃	C₂×C₅₃	1	1	yes	characteristic
16	C₄×C₅₃	C₄₉⋊C₃	1	1	yes	characteristic, direct product factor, Hall
17	C₄×C₄₉	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
18	C₃×C₅₃	—	49	1	no	Hall
19	C₄₉⋊C₃	C₄×C₅₃	1	1	yes	characteristic, direct product factor, Hall
20	C₂×C₅₃	C₂×C₄₉⋊C₃	1	1	yes	characteristic
21	C₂×C₄₉	C₂×C₃×C₅₃	1	1	yes	characteristic
22	C₄×ΓL(1,𝔽₈)	—	7	1	no	—
23	C₅₃	C₄×C₄₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
24	C₄₉	C₃×C₄×C₅₃	1	1	yes	characteristic, complemented, Sylow
25	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
26	C₄×C₇	ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic
27	ΓL(1,𝔽₈)	—	7	1	no	—
28	C₂×C₇	C₂×ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic
29	C₃×C₄	—	49	1	no	Hall
30	C₇	C₄×ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic
31	C₂×C₃	—	49	1	no	—
32	C₄	C₅₃×C₄₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	—	49	1	no	Sylow
34	C₂	C₂×C₅₃×C₄₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄×C₅₃	C₄₉⋊C₃
commutator, hypocenter	C₄₉	C₃×C₄×C₅₃
Frattini	C₂×C₇	C₂×ΓL(1,𝔽₈)×C₅₃
Fitting	C₄×C₄₉×C₅₃	C₃
socle	C₂×C₇×C₅₃	C₂×ΓL(1,𝔽₈)
prime radical	C₂×C₅₃×C₄₉⋊C₃	C₂
cosocle	C₂×C₄₉	C₂×C₃×C₅₃
LCS	G ▷ C₄₉
UCS	C₁ ◁ C₄×C₅₃

Complements

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

N	H ≅ G/N	type
2. C₄×C₄₉×C₅₃	C₃	semidirect
3. C₅₃×C₄₉⋊C₃	C₄	direct (N × H)
6. C₄₉×C₅₃	C₃×C₄	semidirect
12. C₄×C₄₉⋊C₃	C₅₃	direct (N × H)
16. C₄×C₅₃	C₄₉⋊C₃	direct (N × H)
17. C₄×C₄₉	C₃×C₅₃	semidirect
19. C₄₉⋊C₃	C₄×C₅₃	direct (N × H)
23. C₅₃	C₄×C₄₉⋊C₃	direct (N × H)
24. C₄₉	C₃×C₄×C₅₃	semidirect
32. C₄	C₅₃×C₄₉⋊C₃	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×C₄₉⋊C₃	2
C₄×C₄₉×C₅₃	3
C₄×ΓL(1,𝔽₈)×C₅₃	7
C₄×C₄₉⋊C₃	53

Subgroups — C4×C53×C49⋊C3 #44528310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×C₄₉⋊C₃ #44528310