Subgroups — C₃²×C₄×C₉×C₇₃ #72201003

Order 23652.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₉×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄×C₇₃	C₃	1	1	yes	maximal, characteristic
3	C₃×C₄×C₉×C₇₃	C₃	1	12	yes	maximal, direct product factor
4	C₂×C₃³×C₇₃	C₂×C₃	1	1	yes	characteristic
5	C₂×C₃×C₉×C₇₃	C₂×C₃	1	12	yes	—
6	C₃²×C₉×C₇₃	C₄	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×C₄×C₇₃	C₉	1	9	yes	direct product factor
8	C₃²×C₄×C₇₃	C₃²	1	4	yes	—
9	C₄×C₉×C₇₃	C₃²	1	9	yes	direct product factor
10	C₃³×C₇₃	C₃×C₄	1	1	yes	characteristic
11	C₃×C₉×C₇₃	C₃×C₄	1	12	yes	direct product factor
12	C₂×C₃²×C₇₃	C₂×C₉	1	9	yes	—
13	C₂×C₃²×C₇₃	C₂×C₃²	1	4	yes	—
14	C₂×C₉×C₇₃	C₂×C₃²	1	9	yes	—
15	C₃×C₄×C₇₃	C₃×C₉	1	12	yes	direct product factor
16	C₃×C₄×C₇₃	C₃³	1	1	yes	characteristic
17	C₃²×C₇₃	C₄×C₉	1	9	yes	direct product factor
18	C₃²×C₇₃	C₃²×C₄	1	4	yes	—
19	C₉×C₇₃	C₃²×C₄	1	9	yes	direct product factor
20	C₂×C₃×C₇₃	C₂×C₃×C₉	1	12	yes	—
21	C₂×C₃×C₇₃	C₂×C₃³	1	1	yes	characteristic
22	C₃²×C₄×C₉	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₄×C₇₃	C₃²×C₉	1	1	yes	characteristic, direct product factor, Hall
24	C₃×C₇₃	C₃×C₄×C₉	1	12	yes	direct product factor
25	C₃×C₇₃	C₃³×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₉	C₂×C₇₃	1	1	yes	characteristic
27	C₂×C₇₃	C₂×C₃²×C₉	1	1	yes	characteristic
28	C₃³×C₄	C₃×C₇₃	1	1	yes	characteristic
29	C₃×C₄×C₉	C₃×C₇₃	1	12	yes	direct product factor
30	C₃²×C₉	C₄×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
31	C₇₃	C₃²×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₃³	C₂×C₃×C₇₃	1	1	yes	characteristic
33	C₂×C₃×C₉	C₂×C₃×C₇₃	1	12	yes	—
34	C₃²×C₄	C₉×C₇₃	1	9	yes	direct product factor
35	C₃²×C₄	C₃²×C₇₃	1	4	yes	—
36	C₄×C₉	C₃²×C₇₃	1	9	yes	direct product factor
37	C₃³	C₃×C₄×C₇₃	1	1	yes	characteristic
38	C₃×C₉	C₃×C₄×C₇₃	1	12	yes	direct product factor
39	C₂×C₃²	C₂×C₉×C₇₃	1	9	yes	—
40	C₂×C₃²	C₂×C₃²×C₇₃	1	4	yes	—
41	C₂×C₉	C₂×C₃²×C₇₃	1	9	yes	—
42	C₃×C₄	C₃×C₉×C₇₃	1	12	yes	direct product factor
43	C₃×C₄	C₃³×C₇₃	1	1	yes	characteristic
44	C₃²	C₄×C₉×C₇₃	1	9	yes	direct product factor
45	C₃²	C₃²×C₄×C₇₃	1	4	yes	—
46	C₉	C₃²×C₄×C₇₃	1	9	yes	direct product factor
47	C₂×C₃	C₂×C₃×C₉×C₇₃	1	12	yes	—
48	C₂×C₃	C₂×C₃³×C₇₃	1	1	yes	characteristic
49	C₄	C₃²×C₉×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
50	C₃	C₃×C₄×C₉×C₇₃	1	12	yes	direct product factor
51	C₃	C₃³×C₄×C₇₃	1	1	yes	characteristic
52	C₂	C₂×C₃²×C₉×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂×C₃³×C₇₃
socle, prime radical	C₂×C₃³×C₇₃	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₉×C₇₃	C₃	direct (N × H)
6. C₃²×C₉×C₇₃	C₄	direct (N × H)
7. C₃²×C₄×C₇₃	C₉	direct (N × H)
9. C₄×C₉×C₇₃	C₃²	direct (N × H)
11. C₃×C₉×C₇₃	C₃×C₄	direct (N × H)
15. C₃×C₄×C₇₃	C₃×C₉	direct (N × H)
17. C₃²×C₇₃	C₄×C₉	direct (N × H)
19. C₉×C₇₃	C₃²×C₄	direct (N × H)
22. C₃²×C₄×C₉	C₇₃	direct (N × H)
23. C₄×C₇₃	C₃²×C₉	direct (N × H)
24. C₃×C₇₃	C₃×C₄×C₉	direct (N × H)
29. C₃×C₄×C₉	C₃×C₇₃	direct (N × H)
30. C₃²×C₉	C₄×C₇₃	direct (N × H)
31. C₇₃	C₃²×C₄×C₉	direct (N × H)
34. C₃²×C₄	C₉×C₇₃	direct (N × H)
36. C₄×C₉	C₃²×C₇₃	direct (N × H)
38. C₃×C₉	C₃×C₄×C₇₃	direct (N × H)
42. C₃×C₄	C₃×C₉×C₇₃	direct (N × H)
44. C₃²	C₄×C₉×C₇₃	direct (N × H)
46. C₉	C₃²×C₄×C₇₃	direct (N × H)
49. C₄	C₃²×C₉×C₇₃	direct (N × H)
50. C₃	C₃×C₄×C₉×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×C₃²×C₉×C₇₃	2
C₃³×C₄×C₇₃	3
C₃×C₄×C₉×C₇₃	3
C₃²×C₄×C₉	73

Subgroups — C32×C4×C9×C73 #72201003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₉×C₇₃ #72201003