Subgroups — C₃×C₂²⋊(C₁₉⋊(C₅²⋊C₃)) #30685958

Order 17100.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₁₉⋊(C₅²⋊C₃))	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅²×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₁₉⋊(C₅²⋊C₃)	—	4	1	no	maximal, Hall
4	C₂×C₃×C₅²×C₁₉	—	3	1	no	—
5	C₂²×C₅²×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₅²×C₁₉	A₄	1	1	yes	characteristic, complemented
7	C₁₉⋊(C₅²⋊C₃)	—	4	3	no	—
8	C₂²×C₃×C₅×C₁₉	—	3	2	no	—
9	C₂×C₅²×C₁₉	—	3	1	no	—
10	C₃×C₂²⋊(C₅²⋊C₃)	—	19	1	no	maximal, Hall
11	C₃×C₂²⋊(C₁₉⋊C₃)	—	25	1	no	maximal, Hall
12	C₂×C₃×C₅×C₁₉	—	3	6	no	—
13	C₅²×C₁₉	C₃×A₄	1	1	yes	characteristic, complemented, Hall
14	C₂²×C₅×C₁₉	—	3	2	no	—
15	C₂²×C₃×C₅²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
16	C₂²⋊(C₅²⋊C₃)	—	19	3	no	—
17	C₃×C₅×C₁₉	—	3	2	no	—
18	C₂²×C₃×C₁₉	C₅²⋊C₃	1	1	yes	characteristic, complemented
19	C₂²⋊(C₁₉⋊C₃)	—	25	3	no	—
20	C₃×C₅²⋊C₃	—	76	1	no	Hall
21	C₂×C₅×C₁₉	—	3	6	no	—
22	C₃×C₁₉⋊C₃	—	100	1	no	Hall
23	C₂×C₃×C₅²	—	3	1	no	—
24	C₂×C₃×C₁₉	—	3	1	no	—
25	C₂²×C₅²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
26	C₅×C₁₉	—	3	2	no	—
27	C₂²×C₁₉	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
28	C₃×C₅²	C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, complemented
29	C₅²⋊C₃	—	76	3	no	—
30	C₂²×C₃×C₅	—	3	2	no	—
31	C₃×C₁₉	C₂²⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
32	C₁₉⋊C₃	—	100	3	no	—
33	C₂×C₅²	—	3	1	no	—
34	C₂×C₁₉	—	3	1	no	—
35	C₃×A₄	—	475	1	no	Hall
36	C₂×C₃×C₅	—	3	6	no	—
37	C₅²	C₃×C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, complemented, Sylow
38	C₂²×C₅	—	3	2	no	—
39	C₁₉	C₃×C₂²⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented, Sylow
40	C₃×C₅	—	3	2	no	—
41	C₂²×C₃	C₁₉⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
42	A₄	—	475	3	no	—
43	C₂×C₅	—	3	6	no	—
44	C₃²	—	1900	1	no	Sylow
45	C₂×C₃	—	3	1	no	—
46	C₅	—	3	2	no	—
47	C₂²	C₃×C₁₉⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented, Sylow
48	C₃	—	1900	3	no	—
49	C₃	C₂²⋊(C₁₉⋊(C₅²⋊C₃))	1	1	yes	characteristic, direct product factor
50	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃	C₂²⋊(C₁₉⋊(C₅²⋊C₃))
commutator, cosocle, hypocenter	C₂²×C₅²×C₁₉	C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₁₉	C₃
LCS	G ▷ C₂²×C₅²×C₁₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₁₉⋊(C₅²⋊C₃))	C₃	direct (N × H)
2. C₂²×C₃×C₅²×C₁₉	C₃	semidirect
5. C₂²×C₅²×C₁₉	C₃²	semidirect
6. C₃×C₅²×C₁₉	A₄	semidirect
13. C₅²×C₁₉	C₃×A₄	semidirect
15. C₂²×C₃×C₅²	C₁₉⋊C₃	semidirect
18. C₂²×C₃×C₁₉	C₅²⋊C₃	semidirect
25. C₂²×C₅²	C₃×C₁₉⋊C₃	semidirect
27. C₂²×C₁₉	C₃×C₅²⋊C₃	semidirect
28. C₃×C₅²	C₂²⋊(C₁₉⋊C₃)	semidirect
31. C₃×C₁₉	C₂²⋊(C₅²⋊C₃)	semidirect
37. C₅²	C₃×C₂²⋊(C₁₉⋊C₃)	semidirect
39. C₁₉	C₃×C₂²⋊(C₅²⋊C₃)	semidirect
41. C₂²×C₃	C₁₉⋊(C₅²⋊C₃)	semidirect
47. C₂²	C₃×C₁₉⋊(C₅²⋊C₃)	semidirect
49. C₃	C₂²⋊(C₁₉⋊(C₅²⋊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
C₂²⋊(C₁₉⋊(C₅²⋊C₃))	3
C₂²×C₃×C₅²×C₁₉	3
C₃×C₁₉⋊(C₅²⋊C₃)	4
C₃×C₂²⋊(C₅²⋊C₃)	19
C₃×C₂²⋊(C₁₉⋊C₃)	25

Subgroups — C3×C22⋊(C19⋊(C52⋊C3)) #30685958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊(C₁₉⋊(C₅²⋊C₃)) #30685958