Subgroups — C₁₉×C₂²⋊#63270 #35209179

Order 20748.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₁₃×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₉×#63270	—	4	1	no	maximal, Hall
3	C₂×C₇×C₁₃×C₁₉	—	3	1	no	—
4	C₁₉×C₂²⋊(C₁₃⋊C₃)	—	7	1	no	maximal, Hall
5	C₇×C₁₃×C₁₉	A₄	1	1	yes	characteristic, complemented, Hall
6	C₁₉×C₂²⋊ΓL(1,𝔽₈)	—	13	1	no	maximal, Hall
7	C₂²⋊#63270	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₁₃×C₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
9	C₁₉×C₁₃⋊C₃	—	28	1	no	Hall
10	C₂²×C₇×C₁₉	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₁₃×C₁₉	—	3	1	no	—
12	C₁₉×ΓL(1,𝔽₈)	—	52	1	no	Hall
13	C₂²×C₇×C₁₃	C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
14	#63270	—	4	1	no	Hall
15	C₂×C₇×C₁₉	—	3	1	no	—
16	C₁₃×C₁₉	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
17	A₄×C₁₉	—	91	1	no	Hall
18	C₂×C₇×C₁₃	—	3	1	no	—
19	C₂²⋊(C₁₃⋊C₃)	—	7	1	no	Hall
20	C₇×C₁₉	C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic, complemented, Hall
21	C₇×C₁₃	A₄×C₁₉	1	1	yes	characteristic, complemented, Hall
22	C₂²⋊ΓL(1,𝔽₈)	—	13	1	no	Hall
23	C₂²×C₁₉	#63270	1	1	yes	characteristic, complemented, Hall
24	C₃×C₁₉	—	364	1	no	Hall
25	C₂²×C₁₃	C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
26	C₁₃⋊C₃	—	28	1	no	Hall
27	C₂×C₁₉	—	3	1	no	—
28	C₂²×C₇	C₁₉×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₁₃	—	3	1	no	—
30	ΓL(1,𝔽₈)	—	52	1	no	Hall
31	C₁₉	C₂²⋊#63270	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₇	—	3	1	no	—
33	C₁₃	C₁₉×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
34	A₄	—	91	1	no	Hall
35	C₇	C₁₉×C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic, complemented, Sylow
36	C₂²	C₁₉×#63270	1	1	yes	characteristic, complemented, Sylow
37	C₃	—	364	1	no	Sylow
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₉	C₂²⋊#63270
commutator, cosocle, hypocenter	C₂²×C₇×C₁₃	C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₁₃×C₁₉	C₃
LCS	G ▷ C₂²×C₇×C₁₃
UCS	C₁ ◁ C₁₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₁₃×C₁₉	C₃	semidirect
5. C₇×C₁₃×C₁₉	A₄	semidirect
7. C₂²⋊#63270	C₁₉	direct (N × H)
8. C₂²×C₁₃×C₁₉	ΓL(1,𝔽₈)	semidirect
10. C₂²×C₇×C₁₉	C₁₃⋊C₃	semidirect
13. C₂²×C₇×C₁₃	C₃×C₁₉	semidirect
16. C₁₃×C₁₉	C₂²⋊ΓL(1,𝔽₈)	semidirect
20. C₇×C₁₉	C₂²⋊(C₁₃⋊C₃)	semidirect
21. C₇×C₁₃	A₄×C₁₉	semidirect
23. C₂²×C₁₉	#63270	semidirect
25. C₂²×C₁₃	C₁₉×ΓL(1,𝔽₈)	semidirect
28. C₂²×C₇	C₁₉×C₁₃⋊C₃	semidirect
31. C₁₉	C₂²⋊#63270	direct (N × H)
33. C₁₃	C₁₉×C₂²⋊ΓL(1,𝔽₈)	semidirect
35. C₇	C₁₉×C₂²⋊(C₁₃⋊C₃)	semidirect
36. C₂²	C₁₉×#63270	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₁₃×C₁₉	3
C₁₉×#63270	4
C₁₉×C₂²⋊(C₁₃⋊C₃)	7
C₁₉×C₂²⋊ΓL(1,𝔽₈)	13
C₂²⋊#63270	19

Subgroups — C19×C22⋊#63270 #35209179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×C₂²⋊#63270 #35209179