Subgroups — C₂²×C₄×C₅²×C₆₁ #72201693

Order 24400.

432 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅²×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅²×C₆₁	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×C₅×C₆₁	C₅	1	6	yes	maximal, direct product factor
4	C₂³×C₅×C₆₁	C₂×C₅	1	6	yes	—
5	C₂×C₄×C₅×C₆₁	C₂×C₅	1	36	yes	direct product factor
6	C₂²×C₅²×C₆₁	C₄	1	4	yes	direct product factor
7	C₂²×C₅²×C₆₁	C₂²	1	3	yes	—
8	C₄×C₅²×C₆₁	C₂²	1	4	yes	direct product factor
9	C₂×C₅²×C₆₁	C₂×C₄	1	6	yes	direct product factor
10	C₂×C₅²×C₆₁	C₂³	1	1	yes	characteristic
11	C₅²×C₆₁	C₂²×C₄	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₅×C₆₁	C₄×C₅	1	24	yes	direct product factor
13	C₂²×C₅×C₆₁	C₂²×C₅	1	18	yes	—
14	C₄×C₅×C₆₁	C₂²×C₅	1	24	yes	direct product factor
15	C₂²×C₄×C₆₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₅×C₆₁	C₂×C₄×C₅	1	36	yes	direct product factor
17	C₂×C₅×C₆₁	C₂³×C₅	1	6	yes	—
18	C₂³×C₆₁	C₂×C₅²	1	1	yes	characteristic
19	C₂×C₄×C₆₁	C₂×C₅²	1	6	yes	direct product factor
20	C₂²×C₄×C₅²	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×C₆₁	C₂²×C₄×C₅	1	6	yes	direct product factor
22	C₂²×C₆₁	C₄×C₅²	1	4	yes	direct product factor
23	C₂²×C₆₁	C₂²×C₅²	1	3	yes	—
24	C₄×C₆₁	C₂²×C₅²	1	4	yes	direct product factor
25	C₂³×C₅²	C₂×C₆₁	1	1	yes	characteristic
26	C₂×C₄×C₅²	C₂×C₆₁	1	6	yes	direct product factor
27	C₂×C₆₁	C₂×C₄×C₅²	1	6	yes	direct product factor
28	C₂×C₆₁	C₂³×C₅²	1	1	yes	characteristic
29	C₂²×C₅²	C₄×C₆₁	1	4	yes	direct product factor
30	C₂²×C₅²	C₂²×C₆₁	1	3	yes	—
31	C₄×C₅²	C₂²×C₆₁	1	4	yes	direct product factor
32	C₂²×C₄×C₅	C₅×C₆₁	1	6	yes	direct product factor
33	C₆₁	C₂²×C₄×C₅²	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₅²	C₂×C₄×C₆₁	1	6	yes	direct product factor
35	C₂×C₅²	C₂³×C₆₁	1	1	yes	characteristic
36	C₂³×C₅	C₂×C₅×C₆₁	1	6	yes	—
37	C₂×C₄×C₅	C₂×C₅×C₆₁	1	36	yes	direct product factor
38	C₅²	C₂²×C₄×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₅	C₄×C₅×C₆₁	1	24	yes	direct product factor
40	C₂²×C₅	C₂²×C₅×C₆₁	1	18	yes	—
41	C₄×C₅	C₂²×C₅×C₆₁	1	24	yes	direct product factor
42	C₂²×C₄	C₅²×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₅	C₂×C₄×C₅×C₆₁	1	36	yes	direct product factor
44	C₂×C₅	C₂³×C₅×C₆₁	1	6	yes	—
45	C₂³	C₂×C₅²×C₆₁	1	1	yes	characteristic
46	C₂×C₄	C₂×C₅²×C₆₁	1	6	yes	direct product factor
47	C₅	C₂²×C₄×C₅×C₆₁	1	6	yes	direct product factor
48	C₂²	C₄×C₅²×C₆₁	1	4	yes	direct product factor
49	C₂²	C₂²×C₅²×C₆₁	1	3	yes	—
50	C₄	C₂²×C₅²×C₆₁	1	4	yes	direct product factor
51	C₂	C₂×C₄×C₅²×C₆₁	1	6	yes	direct product factor
52	C₂	C₂³×C₅²×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂³×C₅²×C₆₁
socle, prime radical	C₂³×C₅²×C₆₁	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅²×C₆₁	C₂	direct (N × H)
3. C₂²×C₄×C₅×C₆₁	C₅	direct (N × H)
5. C₂×C₄×C₅×C₆₁	C₂×C₅	direct (N × H)
6. C₂²×C₅²×C₆₁	C₄	direct (N × H)
8. C₄×C₅²×C₆₁	C₂²	direct (N × H)
9. C₂×C₅²×C₆₁	C₂×C₄	direct (N × H)
11. C₅²×C₆₁	C₂²×C₄	direct (N × H)
12. C₂²×C₅×C₆₁	C₄×C₅	direct (N × H)
14. C₄×C₅×C₆₁	C₂²×C₅	direct (N × H)
15. C₂²×C₄×C₆₁	C₅²	direct (N × H)
16. C₂×C₅×C₆₁	C₂×C₄×C₅	direct (N × H)
19. C₂×C₄×C₆₁	C₂×C₅²	direct (N × H)
20. C₂²×C₄×C₅²	C₆₁	direct (N × H)
21. C₅×C₆₁	C₂²×C₄×C₅	direct (N × H)
22. C₂²×C₆₁	C₄×C₅²	direct (N × H)
24. C₄×C₆₁	C₂²×C₅²	direct (N × H)
26. C₂×C₄×C₅²	C₂×C₆₁	direct (N × H)
27. C₂×C₆₁	C₂×C₄×C₅²	direct (N × H)
29. C₂²×C₅²	C₄×C₆₁	direct (N × H)
31. C₄×C₅²	C₂²×C₆₁	direct (N × H)
32. C₂²×C₄×C₅	C₅×C₆₁	direct (N × H)
33. C₆₁	C₂²×C₄×C₅²	direct (N × H)
34. C₂×C₅²	C₂×C₄×C₆₁	direct (N × H)
37. C₂×C₄×C₅	C₂×C₅×C₆₁	direct (N × H)
38. C₅²	C₂²×C₄×C₆₁	direct (N × H)
39. C₂²×C₅	C₄×C₅×C₆₁	direct (N × H)
41. C₄×C₅	C₂²×C₅×C₆₁	direct (N × H)
42. C₂²×C₄	C₅²×C₆₁	direct (N × H)
43. C₂×C₅	C₂×C₄×C₅×C₆₁	direct (N × H)
46. C₂×C₄	C₂×C₅²×C₆₁	direct (N × H)
47. C₅	C₂²×C₄×C₅×C₆₁	direct (N × H)
48. C₂²	C₄×C₅²×C₆₁	direct (N × H)
50. C₄	C₂²×C₅²×C₆₁	direct (N × H)
51. C₂	C₂×C₄×C₅²×C₆₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₅²×C₆₁	2
C₂×C₄×C₅²×C₆₁	2
C₂²×C₄×C₅×C₆₁	5
C₂²×C₄×C₅²	61

Subgroups — C22×C4×C52×C61 #72201693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₅²×C₆₁ #72201693