Subgroups — C₂³×C₄×C₁₃×C₅₃ #72199552

Order 22048.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₁₃×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₃×C₅₃	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₁₃×C₅₃	C₄	1	8	yes	direct product factor
4	C₂³×C₁₃×C₅₃	C₂²	1	7	yes	—
5	C₂×C₄×C₁₃×C₅₃	C₂²	1	28	yes	direct product factor
6	C₂²×C₁₃×C₅₃	C₂×C₄	1	28	yes	direct product factor
7	C₂²×C₁₃×C₅₃	C₂³	1	7	yes	—
8	C₄×C₁₃×C₅₃	C₂³	1	8	yes	direct product factor
9	C₂³×C₄×C₅₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₁₃×C₅₃	C₂²×C₄	1	14	yes	direct product factor
11	C₂×C₁₃×C₅₃	C₂⁴	1	1	yes	characteristic
12	C₂⁴×C₅₃	C₂×C₁₃	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂×C₁₃	1	14	yes	direct product factor
14	C₁₃×C₅₃	C₂³×C₄	1	1	yes	characteristic, direct product factor, Hall
15	C₂³×C₅₃	C₄×C₁₃	1	8	yes	direct product factor
16	C₂³×C₅₃	C₂²×C₁₃	1	7	yes	—
17	C₂×C₄×C₅₃	C₂²×C₁₃	1	28	yes	direct product factor
18	C₂³×C₄×C₁₃	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₅₃	C₂×C₄×C₁₃	1	28	yes	direct product factor
20	C₂²×C₅₃	C₂³×C₁₃	1	7	yes	—
21	C₄×C₅₃	C₂³×C₁₃	1	8	yes	direct product factor
22	C₂⁴×C₁₃	C₂×C₅₃	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	C₂×C₅₃	1	14	yes	direct product factor
24	C₂×C₅₃	C₂²×C₄×C₁₃	1	14	yes	direct product factor
25	C₂×C₅₃	C₂⁴×C₁₃	1	1	yes	characteristic
26	C₂³×C₁₃	C₄×C₅₃	1	8	yes	direct product factor
27	C₂³×C₁₃	C₂²×C₅₃	1	7	yes	—
28	C₂×C₄×C₁₃	C₂²×C₅₃	1	28	yes	direct product factor
29	C₅₃	C₂³×C₄×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²×C₁₃	C₂×C₄×C₅₃	1	28	yes	direct product factor
31	C₂²×C₁₃	C₂³×C₅₃	1	7	yes	—
32	C₄×C₁₃	C₂³×C₅₃	1	8	yes	direct product factor
33	C₂³×C₄	C₁₃×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₁₃	C₂²×C₄×C₅₃	1	14	yes	direct product factor
35	C₂×C₁₃	C₂⁴×C₅₃	1	1	yes	characteristic
36	C₂⁴	C₂×C₁₃×C₅₃	1	1	yes	characteristic
37	C₂²×C₄	C₂×C₁₃×C₅₃	1	14	yes	direct product factor
38	C₁₃	C₂³×C₄×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
39	C₂³	C₄×C₁₃×C₅₃	1	8	yes	direct product factor
40	C₂³	C₂²×C₁₃×C₅₃	1	7	yes	—
41	C₂×C₄	C₂²×C₁₃×C₅₃	1	28	yes	direct product factor
42	C₂²	C₂×C₄×C₁₃×C₅₃	1	28	yes	direct product factor
43	C₂²	C₂³×C₁₃×C₅₃	1	7	yes	—
44	C₄	C₂³×C₁₃×C₅₃	1	8	yes	direct product factor
45	C₂	C₂²×C₄×C₁₃×C₅₃	1	14	yes	direct product factor
46	C₂	C₂⁴×C₁₃×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂⁴×C₁₃×C₅₃
socle, prime radical	C₂⁴×C₁₃×C₅₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₁₃×C₅₃	C₂	direct (N × H)
3. C₂³×C₁₃×C₅₃	C₄	direct (N × H)
5. C₂×C₄×C₁₃×C₅₃	C₂²	direct (N × H)
6. C₂²×C₁₃×C₅₃	C₂×C₄	direct (N × H)
8. C₄×C₁₃×C₅₃	C₂³	direct (N × H)
9. C₂³×C₄×C₅₃	C₁₃	direct (N × H)
10. C₂×C₁₃×C₅₃	C₂²×C₄	direct (N × H)
13. C₂²×C₄×C₅₃	C₂×C₁₃	direct (N × H)
14. C₁₃×C₅₃	C₂³×C₄	direct (N × H)
15. C₂³×C₅₃	C₄×C₁₃	direct (N × H)
17. C₂×C₄×C₅₃	C₂²×C₁₃	direct (N × H)
18. C₂³×C₄×C₁₃	C₅₃	direct (N × H)
19. C₂²×C₅₃	C₂×C₄×C₁₃	direct (N × H)
21. C₄×C₅₃	C₂³×C₁₃	direct (N × H)
23. C₂²×C₄×C₁₃	C₂×C₅₃	direct (N × H)
24. C₂×C₅₃	C₂²×C₄×C₁₃	direct (N × H)
26. C₂³×C₁₃	C₄×C₅₃	direct (N × H)
28. C₂×C₄×C₁₃	C₂²×C₅₃	direct (N × H)
29. C₅₃	C₂³×C₄×C₁₃	direct (N × H)
30. C₂²×C₁₃	C₂×C₄×C₅₃	direct (N × H)
32. C₄×C₁₃	C₂³×C₅₃	direct (N × H)
33. C₂³×C₄	C₁₃×C₅₃	direct (N × H)
34. C₂×C₁₃	C₂²×C₄×C₅₃	direct (N × H)
37. C₂²×C₄	C₂×C₁₃×C₅₃	direct (N × H)
38. C₁₃	C₂³×C₄×C₅₃	direct (N × H)
39. C₂³	C₄×C₁₃×C₅₃	direct (N × H)
41. C₂×C₄	C₂²×C₁₃×C₅₃	direct (N × H)
42. C₂²	C₂×C₄×C₁₃×C₅₃	direct (N × H)
44. C₄	C₂³×C₁₃×C₅₃	direct (N × H)
45. C₂	C₂²×C₄×C₁₃×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₁₃×C₅₃	2
C₂²×C₄×C₁₃×C₅₃	2
C₂³×C₄×C₅₃	13
C₂³×C₄×C₁₃	53

Subgroups — C23×C4×C13×C53 #72199552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×C₁₃×C₅₃ #72199552