Subgroups — C₂²×C₃×C₁₉⋊(C₅²⋊C₃) #30685962

Order 17100.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₁₉⋊(C₅²⋊C₃)	C₂	direct (N × H)
2. C₂²×C₁₉⋊(C₅²⋊C₃)	C₃	direct (N × H)
3. C₂²×C₃×C₅²×C₁₉	C₃	semidirect
4. C₃×C₁₉⋊(C₅²⋊C₃)	C₂²	direct (N × H)
5. C₂×C₁₉⋊(C₅²⋊C₃)	C₂×C₃	direct (N × H)
6. C₂×C₃×C₅²×C₁₉	C₂×C₃	semidirect
7. C₂²×C₅²×C₁₉	C₃²	semidirect
8. C₃×C₅²×C₁₉	C₂²×C₃	semidirect
9. C₁₉⋊(C₅²⋊C₃)	C₂²×C₃	direct (N × H)
11. C₂×C₅²×C₁₉	C₂×C₃²	semidirect
15. C₅²×C₁₉	C₂²×C₃²	semidirect
19. C₂²×C₃×C₅²	C₁₉⋊C₃	semidirect
22. C₂²×C₃×C₁₉	C₅²⋊C₃	semidirect
27. C₂×C₃×C₅²	C₂×C₁₉⋊C₃	semidirect
29. C₂×C₃×C₁₉	C₂×C₅²⋊C₃	semidirect
31. C₂²×C₅²	C₃×C₁₉⋊C₃	semidirect
33. C₂²×C₁₉	C₃×C₅²⋊C₃	semidirect
34. C₃×C₅²	C₂²×C₁₉⋊C₃	semidirect
37. C₃×C₁₉	C₂²×C₅²⋊C₃	semidirect
39. C₂×C₅²	C₂×C₃×C₁₉⋊C₃	semidirect
40. C₂×C₁₉	C₂×C₃×C₅²⋊C₃	semidirect
43. C₅²	C₂²×C₃×C₁₉⋊C₃	semidirect
45. C₁₉	C₂²×C₃×C₅²⋊C₃	semidirect
49. C₂²×C₃	C₁₉⋊(C₅²⋊C₃)	direct (N × H)
53. C₂×C₃	C₂×C₁₉⋊(C₅²⋊C₃)	direct (N × H)
55. C₂²	C₃×C₁₉⋊(C₅²⋊C₃)	direct (N × H)
57. C₃	C₂²×C₁₉⋊(C₅²⋊C₃)	direct (N × H)
58. C₂	C₂×C₃×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
C₂×C₃×C₁₉⋊(C₅²⋊C₃)	2
C₂²×C₁₉⋊(C₅²⋊C₃)	3
C₂²×C₃×C₅²×C₁₉	3
C₂²×C₃×C₅²⋊C₃	19
C₂²×C₃×C₁₉⋊C₃	25

Subgroups — C22×C3×C19⋊(C52⋊C3) #30685962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₁₉⋊(C₅²⋊C₃) #30685962