Subgroups — C₃×C₄²×C₁₉×C₂₃ #72198548

Order 20976.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₁₉×C₂₃	C₂	1	3	yes	maximal
2	C₄²×C₁₉×C₂₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂×C₄×C₁₉×C₂₃	C₂×C₃	1	3	yes	—
4	C₂²×C₃×C₁₉×C₂₃	C₂²	1	1	yes	characteristic
5	C₃×C₄×C₁₉×C₂₃	C₄	1	6	yes	direct product factor
6	C₂×C₃×C₁₉×C₂₃	C₂×C₄	1	3	yes	—
7	C₂²×C₁₉×C₂₃	C₂²×C₃	1	1	yes	characteristic
8	C₄×C₁₉×C₂₃	C₃×C₄	1	6	yes	direct product factor
9	C₃×C₁₉×C₂₃	C₄²	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₄²×C₂₃	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₄²×C₁₉	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₁₉×C₂₃	C₂×C₃×C₄	1	3	yes	—
13	C₂×C₃×C₄×C₂₃	C₂×C₁₉	1	3	yes	—
14	C₂×C₃×C₄×C₁₉	C₂×C₂₃	1	3	yes	—
15	C₁₉×C₂₃	C₃×C₄²	1	1	yes	characteristic, direct product factor, Hall
16	C₄²×C₂₃	C₃×C₁₉	1	1	yes	characteristic, direct product factor, Hall
17	C₄²×C₁₉	C₃×C₂₃	1	1	yes	characteristic, direct product factor, Hall
18	C₂²×C₃×C₂₃	C₂²×C₁₉	1	1	yes	characteristic
19	C₃×C₄×C₂₃	C₄×C₁₉	1	6	yes	direct product factor
20	C₂²×C₃×C₁₉	C₂²×C₂₃	1	1	yes	characteristic
21	C₃×C₄×C₁₉	C₄×C₂₃	1	6	yes	direct product factor
22	C₂×C₄×C₂₃	C₂×C₃×C₁₉	1	3	yes	—
23	C₂×C₄×C₁₉	C₂×C₃×C₂₃	1	3	yes	—
24	C₂×C₃×C₂₃	C₂×C₄×C₁₉	1	3	yes	—
25	C₂×C₃×C₁₉	C₂×C₄×C₂₃	1	3	yes	—
26	C₂²×C₂₃	C₂²×C₃×C₁₉	1	1	yes	characteristic
27	C₄×C₂₃	C₃×C₄×C₁₉	1	6	yes	direct product factor
28	C₂²×C₁₉	C₂²×C₃×C₂₃	1	1	yes	characteristic
29	C₄×C₁₉	C₃×C₄×C₂₃	1	6	yes	direct product factor
30	C₃×C₂₃	C₄²×C₁₉	1	1	yes	characteristic, direct product factor, Hall
31	C₃×C₁₉	C₄²×C₂₃	1	1	yes	characteristic, direct product factor, Hall
32	C₃×C₄²	C₁₉×C₂₃	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₂₃	C₂×C₃×C₄×C₁₉	1	3	yes	—
34	C₂×C₁₉	C₂×C₃×C₄×C₂₃	1	3	yes	—
35	C₂×C₃×C₄	C₂×C₁₉×C₂₃	1	3	yes	—
36	C₂₃	C₃×C₄²×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
37	C₁₉	C₃×C₄²×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
38	C₄²	C₃×C₁₉×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₃	C₂²×C₁₉×C₂₃	1	1	yes	characteristic
40	C₃×C₄	C₄×C₁₉×C₂₃	1	6	yes	direct product factor
41	C₂×C₄	C₂×C₃×C₁₉×C₂₃	1	3	yes	—
42	C₂×C₃	C₂×C₄×C₁₉×C₂₃	1	3	yes	—
43	C₂²	C₂²×C₃×C₁₉×C₂₃	1	1	yes	characteristic
44	C₄	C₃×C₄×C₁₉×C₂₃	1	6	yes	direct product factor
45	C₃	C₄²×C₁₉×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
46	C₂	C₂×C₃×C₄×C₁₉×C₂₃	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂²×C₃×C₁₉×C₂₃
socle, prime radical	C₂²×C₃×C₁₉×C₂₃	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×C₁₉×C₂₃	C₃	direct (N × H)
5. C₃×C₄×C₁₉×C₂₃	C₄	direct (N × H)
8. C₄×C₁₉×C₂₃	C₃×C₄	direct (N × H)
9. C₃×C₁₉×C₂₃	C₄²	direct (N × H)
10. C₃×C₄²×C₂₃	C₁₉	direct (N × H)
11. C₃×C₄²×C₁₉	C₂₃	direct (N × H)
15. C₁₉×C₂₃	C₃×C₄²	direct (N × H)
16. C₄²×C₂₃	C₃×C₁₉	direct (N × H)
17. C₄²×C₁₉	C₃×C₂₃	direct (N × H)
19. C₃×C₄×C₂₃	C₄×C₁₉	direct (N × H)
21. C₃×C₄×C₁₉	C₄×C₂₃	direct (N × H)
27. C₄×C₂₃	C₃×C₄×C₁₉	direct (N × H)
29. C₄×C₁₉	C₃×C₄×C₂₃	direct (N × H)
30. C₃×C₂₃	C₄²×C₁₉	direct (N × H)
31. C₃×C₁₉	C₄²×C₂₃	direct (N × H)
32. C₃×C₄²	C₁₉×C₂₃	direct (N × H)
36. C₂₃	C₃×C₄²×C₁₉	direct (N × H)
37. C₁₉	C₃×C₄²×C₂₃	direct (N × H)
38. C₄²	C₃×C₁₉×C₂₃	direct (N × H)
40. C₃×C₄	C₄×C₁₉×C₂₃	direct (N × H)
44. C₄	C₃×C₄×C₁₉×C₂₃	direct (N × H)
45. C₃	C₄²×C₁₉×C₂₃	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×C₄×C₁₉×C₂₃	2
C₄²×C₁₉×C₂₃	3
C₃×C₄²×C₂₃	19
C₃×C₄²×C₁₉	23

Subgroups — C3×C42×C19×C23 #72198548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×C₁₉×C₂₃ #72198548