Subgroups — C₃×C₄²×C₇×C₇₃ #72201782

Order 24528.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₇×C₇₃	C₂	1	3	yes	maximal
2	C₄²×C₇×C₇₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂×C₄×C₇×C₇₃	C₂×C₃	1	3	yes	—
4	C₃×C₄²×C₇₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₇×C₇₃	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₇×C₇₃	C₄	1	6	yes	direct product factor
7	C₂×C₃×C₇×C₇₃	C₂×C₄	1	3	yes	—
8	C₂²×C₇×C₇₃	C₂²×C₃	1	1	yes	characteristic
9	C₄×C₇×C₇₃	C₃×C₄	1	6	yes	direct product factor
10	C₂×C₃×C₄×C₇₃	C₂×C₇	1	3	yes	—
11	C₃×C₇×C₇₃	C₄²	1	1	yes	characteristic, direct product factor, Hall
12	C₄²×C₇₃	C₃×C₇	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₇×C₇₃	C₂×C₃×C₄	1	3	yes	—
14	C₂²×C₃×C₇₃	C₂²×C₇	1	1	yes	characteristic
15	C₃×C₄×C₇₃	C₄×C₇	1	6	yes	direct product factor
16	C₂×C₄×C₇₃	C₂×C₃×C₇	1	3	yes	—
17	C₇×C₇₃	C₃×C₄²	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₇₃	C₂×C₄×C₇	1	3	yes	—
19	C₃×C₄²×C₇	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂²×C₇₃	C₂²×C₃×C₇	1	1	yes	characteristic
21	C₄×C₇₃	C₃×C₄×C₇	1	6	yes	direct product factor
22	C₃×C₇₃	C₄²×C₇	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₃×C₄×C₇	C₂×C₇₃	1	3	yes	—
24	C₂×C₇₃	C₂×C₃×C₄×C₇	1	3	yes	—
25	C₄²×C₇	C₃×C₇₃	1	1	yes	characteristic, direct product factor, Hall
26	C₂²×C₃×C₇	C₂²×C₇₃	1	1	yes	characteristic
27	C₃×C₄×C₇	C₄×C₇₃	1	6	yes	direct product factor
28	C₇₃	C₃×C₄²×C₇	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×C₄×C₇	C₂×C₃×C₇₃	1	3	yes	—
30	C₃×C₄²	C₇×C₇₃	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₇	C₂×C₄×C₇₃	1	3	yes	—
32	C₂²×C₇	C₂²×C₃×C₇₃	1	1	yes	characteristic
33	C₄×C₇	C₃×C₄×C₇₃	1	6	yes	direct product factor
34	C₂×C₃×C₄	C₂×C₇×C₇₃	1	3	yes	—
35	C₃×C₇	C₄²×C₇₃	1	1	yes	characteristic, direct product factor, Hall
36	C₄²	C₃×C₇×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×C₇	C₂×C₃×C₄×C₇₃	1	3	yes	—
38	C₂²×C₃	C₂²×C₇×C₇₃	1	1	yes	characteristic
39	C₃×C₄	C₄×C₇×C₇₃	1	6	yes	direct product factor
40	C₂×C₄	C₂×C₃×C₇×C₇₃	1	3	yes	—
41	C₇	C₃×C₄²×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₃	C₂×C₄×C₇×C₇₃	1	3	yes	—
43	C₂²	C₂²×C₃×C₇×C₇₃	1	1	yes	characteristic
44	C₄	C₃×C₄×C₇×C₇₃	1	6	yes	direct product factor
45	C₃	C₄²×C₇×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
46	C₂	C₂×C₃×C₄×C₇×C₇₃	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂²×C₃×C₇×C₇₃
socle, prime radical	C₂²×C₃×C₇×C₇₃	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×C₇×C₇₃	C₃	direct (N × H)
4. C₃×C₄²×C₇₃	C₇	direct (N × H)
6. C₃×C₄×C₇×C₇₃	C₄	direct (N × H)
9. C₄×C₇×C₇₃	C₃×C₄	direct (N × H)
11. C₃×C₇×C₇₃	C₄²	direct (N × H)
12. C₄²×C₇₃	C₃×C₇	direct (N × H)
15. C₃×C₄×C₇₃	C₄×C₇	direct (N × H)
17. C₇×C₇₃	C₃×C₄²	direct (N × H)
19. C₃×C₄²×C₇	C₇₃	direct (N × H)
21. C₄×C₇₃	C₃×C₄×C₇	direct (N × H)
22. C₃×C₇₃	C₄²×C₇	direct (N × H)
25. C₄²×C₇	C₃×C₇₃	direct (N × H)
27. C₃×C₄×C₇	C₄×C₇₃	direct (N × H)
28. C₇₃	C₃×C₄²×C₇	direct (N × H)
30. C₃×C₄²	C₇×C₇₃	direct (N × H)
33. C₄×C₇	C₃×C₄×C₇₃	direct (N × H)
35. C₃×C₇	C₄²×C₇₃	direct (N × H)
36. C₄²	C₃×C₇×C₇₃	direct (N × H)
39. C₃×C₄	C₄×C₇×C₇₃	direct (N × H)
41. C₇	C₃×C₄²×C₇₃	direct (N × H)
44. C₄	C₃×C₄×C₇×C₇₃	direct (N × H)
45. C₃	C₄²×C₇×C₇₃	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×C₇×C₇₃	2
C₄²×C₇×C₇₃	3
C₃×C₄²×C₇₃	7
C₃×C₄²×C₇	73

Subgroups — C3×C42×C7×C73 #72201782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×C₇×C₇₃ #72201782