Subgroups — C₂²×C₅²×C₉×C₃₁ #42124469

Order 27900.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₉×C₃₁	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅²×C₃₁	C₃	1	1	yes	maximal, characteristic
3	C₅²×C₉×C₃₁	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×C₅×C₉×C₃₁	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₃×C₅²×C₃₁	C₂×C₃	1	3	yes	—
6	C₂²×C₅²×C₃₁	C₉	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₅×C₉×C₃₁	C₂×C₅	1	18	yes	direct product factor
8	C₃×C₅²×C₃₁	C₂²×C₃	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₃₁	C₃×C₅	1	6	yes	—
10	C₂×C₅²×C₃₁	C₂×C₉	1	3	yes	direct product factor
11	C₅×C₉×C₃₁	C₂²×C₅	1	6	yes	direct product factor
12	C₂²×C₉×C₃₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₃×C₅×C₃₁	C₂×C₃×C₅	1	18	yes	—
14	C₂²×C₅²×C₉	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅²×C₃₁	C₂²×C₉	1	1	yes	characteristic, direct product factor, Hall
16	C₂²×C₅×C₃₁	C₅×C₉	1	6	yes	direct product factor
17	C₂×C₉×C₃₁	C₂×C₅²	1	3	yes	direct product factor
18	C₃×C₅×C₃₁	C₂²×C₃×C₅	1	6	yes	—
19	C₂×C₅²×C₉	C₂×C₃₁	1	3	yes	direct product factor
20	C₂²×C₃×C₃₁	C₃×C₅²	1	1	yes	characteristic
21	C₂×C₅×C₃₁	C₂×C₅×C₉	1	18	yes	direct product factor
22	C₂²×C₃×C₅²	C₃×C₃₁	1	1	yes	characteristic
23	C₉×C₃₁	C₂²×C₅²	1	1	yes	characteristic, direct product factor, Hall
24	C₅²×C₉	C₂²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₃×C₃₁	C₂×C₃×C₅²	1	3	yes	—
26	C₂²×C₅×C₉	C₅×C₃₁	1	6	yes	direct product factor
27	C₅×C₃₁	C₂²×C₅×C₉	1	6	yes	direct product factor
28	C₂×C₃×C₅²	C₂×C₃×C₃₁	1	3	yes	—
29	C₂²×C₃₁	C₅²×C₉	1	1	yes	characteristic, direct product factor, Hall
30	C₂²×C₅²	C₉×C₃₁	1	1	yes	characteristic, direct product factor, Hall
31	C₃×C₃₁	C₂²×C₃×C₅²	1	1	yes	characteristic
32	C₂×C₅×C₉	C₂×C₅×C₃₁	1	18	yes	direct product factor
33	C₃×C₅²	C₂²×C₃×C₃₁	1	1	yes	characteristic
34	C₂×C₃₁	C₂×C₅²×C₉	1	3	yes	direct product factor
35	C₂²×C₃×C₅	C₃×C₅×C₃₁	1	6	yes	—
36	C₂×C₅²	C₂×C₉×C₃₁	1	3	yes	direct product factor
37	C₅×C₉	C₂²×C₅×C₃₁	1	6	yes	direct product factor
38	C₂²×C₉	C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
39	C₃₁	C₂²×C₅²×C₉	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₃×C₅	C₂×C₃×C₅×C₃₁	1	18	yes	—
41	C₅²	C₂²×C₉×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×C₅	C₅×C₉×C₃₁	1	6	yes	direct product factor
43	C₂×C₉	C₂×C₅²×C₃₁	1	3	yes	direct product factor
44	C₃×C₅	C₂²×C₃×C₅×C₃₁	1	6	yes	—
45	C₂²×C₃	C₃×C₅²×C₃₁	1	1	yes	characteristic
46	C₂×C₅	C₂×C₅×C₉×C₃₁	1	18	yes	direct product factor
47	C₉	C₂²×C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃	C₂×C₃×C₅²×C₃₁	1	3	yes	—
49	C₅	C₂²×C₅×C₉×C₃₁	1	6	yes	direct product factor
50	C₂²	C₅²×C₉×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
51	C₃	C₂²×C₃×C₅²×C₃₁	1	1	yes	characteristic
52	C₂	C₂×C₅²×C₉×C₃₁	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂²×C₃×C₅²×C₃₁
socle, prime radical	C₂²×C₃×C₅²×C₃₁	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×C₉×C₃₁	C₂	direct (N × H)
3. C₅²×C₉×C₃₁	C₂²	direct (N × H)
4. C₂²×C₅×C₉×C₃₁	C₅	direct (N × H)
6. C₂²×C₅²×C₃₁	C₉	direct (N × H)
7. C₂×C₅×C₉×C₃₁	C₂×C₅	direct (N × H)
10. C₂×C₅²×C₃₁	C₂×C₉	direct (N × H)
11. C₅×C₉×C₃₁	C₂²×C₅	direct (N × H)
12. C₂²×C₉×C₃₁	C₅²	direct (N × H)
14. C₂²×C₅²×C₉	C₃₁	direct (N × H)
15. C₅²×C₃₁	C₂²×C₉	direct (N × H)
16. C₂²×C₅×C₃₁	C₅×C₉	direct (N × H)
17. C₂×C₉×C₃₁	C₂×C₅²	direct (N × H)
19. C₂×C₅²×C₉	C₂×C₃₁	direct (N × H)
21. C₂×C₅×C₃₁	C₂×C₅×C₉	direct (N × H)
23. C₉×C₃₁	C₂²×C₅²	direct (N × H)
24. C₅²×C₉	C₂²×C₃₁	direct (N × H)
26. C₂²×C₅×C₉	C₅×C₃₁	direct (N × H)
27. C₅×C₃₁	C₂²×C₅×C₉	direct (N × H)
29. C₂²×C₃₁	C₅²×C₉	direct (N × H)
30. C₂²×C₅²	C₉×C₃₁	direct (N × H)
32. C₂×C₅×C₉	C₂×C₅×C₃₁	direct (N × H)
34. C₂×C₃₁	C₂×C₅²×C₉	direct (N × H)
36. C₂×C₅²	C₂×C₉×C₃₁	direct (N × H)
37. C₅×C₉	C₂²×C₅×C₃₁	direct (N × H)
38. C₂²×C₉	C₅²×C₃₁	direct (N × H)
39. C₃₁	C₂²×C₅²×C₉	direct (N × H)
41. C₅²	C₂²×C₉×C₃₁	direct (N × H)
42. C₂²×C₅	C₅×C₉×C₃₁	direct (N × H)
43. C₂×C₉	C₂×C₅²×C₃₁	direct (N × H)
46. C₂×C₅	C₂×C₅×C₉×C₃₁	direct (N × H)
47. C₉	C₂²×C₅²×C₃₁	direct (N × H)
49. C₅	C₂²×C₅×C₉×C₃₁	direct (N × H)
50. C₂²	C₅²×C₉×C₃₁	direct (N × H)
52. C₂	C₂×C₅²×C₉×C₃₁	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×C₅²×C₉×C₃₁	2
C₂²×C₃×C₅²×C₃₁	3
C₂²×C₅×C₉×C₃₁	5
C₂²×C₅²×C₉	31

Subgroups — C22×C52×C9×C31 #42124469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅²×C₉×C₃₁ #42124469