Subgroups — C₂²×C₄×C₉×C₅₉ #72187366

Order 8496.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₉×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₉×C₅₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₄×C₅₉	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₉×C₅₉	C₄	1	4	yes	direct product factor
5	C₂²×C₉×C₅₉	C₂²	1	3	yes	—
6	C₄×C₉×C₅₉	C₂²	1	4	yes	direct product factor
7	C₂³×C₃×C₅₉	C₂×C₃	1	1	yes	characteristic
8	C₂×C₃×C₄×C₅₉	C₂×C₃	1	6	yes	—
9	C₂×C₉×C₅₉	C₂×C₄	1	6	yes	direct product factor
10	C₂×C₉×C₅₉	C₂³	1	1	yes	characteristic
11	C₂²×C₄×C₅₉	C₉	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₃×C₅₉	C₃×C₄	1	4	yes	—
13	C₂²×C₃×C₅₉	C₂²×C₃	1	3	yes	—
14	C₃×C₄×C₅₉	C₂²×C₃	1	4	yes	—
15	C₉×C₅₉	C₂²×C₄	1	1	yes	characteristic, direct product factor, Hall
16	C₂³×C₅₉	C₂×C₉	1	1	yes	characteristic
17	C₂×C₄×C₅₉	C₂×C₉	1	6	yes	direct product factor
18	C₂×C₃×C₅₉	C₂×C₃×C₄	1	6	yes	—
19	C₂×C₃×C₅₉	C₂³×C₃	1	1	yes	characteristic
20	C₂²×C₅₉	C₄×C₉	1	4	yes	direct product factor
21	C₂²×C₅₉	C₂²×C₉	1	3	yes	—
22	C₄×C₅₉	C₂²×C₉	1	4	yes	direct product factor
23	C₃×C₅₉	C₂²×C₃×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅₉	C₂×C₄×C₉	1	6	yes	direct product factor
26	C₂×C₅₉	C₂³×C₉	1	1	yes	characteristic
27	C₂³×C₉	C₂×C₅₉	1	1	yes	characteristic
28	C₂×C₄×C₉	C₂×C₅₉	1	6	yes	direct product factor
29	C₅₉	C₂²×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²×C₃×C₄	C₃×C₅₉	1	1	yes	characteristic
31	C₂²×C₉	C₄×C₅₉	1	4	yes	direct product factor
32	C₂²×C₉	C₂²×C₅₉	1	3	yes	—
33	C₄×C₉	C₂²×C₅₉	1	4	yes	direct product factor
34	C₂³×C₃	C₂×C₃×C₅₉	1	1	yes	characteristic
35	C₂×C₃×C₄	C₂×C₃×C₅₉	1	6	yes	—
36	C₂×C₉	C₂×C₄×C₅₉	1	6	yes	direct product factor
37	C₂×C₉	C₂³×C₅₉	1	1	yes	characteristic
38	C₂²×C₄	C₉×C₅₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₃	C₃×C₄×C₅₉	1	4	yes	—
40	C₂²×C₃	C₂²×C₃×C₅₉	1	3	yes	—
41	C₃×C₄	C₂²×C₃×C₅₉	1	4	yes	—
42	C₉	C₂²×C₄×C₅₉	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³	C₂×C₉×C₅₉	1	1	yes	characteristic
44	C₂×C₄	C₂×C₉×C₅₉	1	6	yes	direct product factor
45	C₂×C₃	C₂×C₃×C₄×C₅₉	1	6	yes	—
46	C₂×C₃	C₂³×C₃×C₅₉	1	1	yes	characteristic
47	C₂²	C₄×C₉×C₅₉	1	4	yes	direct product factor
48	C₂²	C₂²×C₉×C₅₉	1	3	yes	—
49	C₄	C₂²×C₉×C₅₉	1	4	yes	direct product factor
50	C₃	C₂²×C₃×C₄×C₅₉	1	1	yes	characteristic
51	C₂	C₂×C₄×C₉×C₅₉	1	6	yes	direct product factor
52	C₂	C₂³×C₉×C₅₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂³×C₃×C₅₉
socle, prime radical	C₂³×C₃×C₅₉	C₂×C₃
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×C₉×C₅₉	C₂	direct (N × H)
4. C₂²×C₉×C₅₉	C₄	direct (N × H)
6. C₄×C₉×C₅₉	C₂²	direct (N × H)
9. C₂×C₉×C₅₉	C₂×C₄	direct (N × H)
11. C₂²×C₄×C₅₉	C₉	direct (N × H)
15. C₉×C₅₉	C₂²×C₄	direct (N × H)
17. C₂×C₄×C₅₉	C₂×C₉	direct (N × H)
20. C₂²×C₅₉	C₄×C₉	direct (N × H)
22. C₄×C₅₉	C₂²×C₉	direct (N × H)
24. C₂²×C₄×C₉	C₅₉	direct (N × H)
25. C₂×C₅₉	C₂×C₄×C₉	direct (N × H)
28. C₂×C₄×C₉	C₂×C₅₉	direct (N × H)
29. C₅₉	C₂²×C₄×C₉	direct (N × H)
31. C₂²×C₉	C₄×C₅₉	direct (N × H)
33. C₄×C₉	C₂²×C₅₉	direct (N × H)
36. C₂×C₉	C₂×C₄×C₅₉	direct (N × H)
38. C₂²×C₄	C₉×C₅₉	direct (N × H)
42. C₉	C₂²×C₄×C₅₉	direct (N × H)
44. C₂×C₄	C₂×C₉×C₅₉	direct (N × H)
47. C₂²	C₄×C₉×C₅₉	direct (N × H)
49. C₄	C₂²×C₉×C₅₉	direct (N × H)
51. C₂	C₂×C₄×C₉×C₅₉	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×C₉×C₅₉	2
C₂×C₄×C₉×C₅₉	2
C₂²×C₃×C₄×C₅₉	3
C₂²×C₄×C₉	59

Subgroups — C22×C4×C9×C59 #72187366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₉×C₅₉ #72187366