Subgroups — C₂²×C₈×C₁₃×C₅₃ #72199550

Order 22048.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₃×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₁₃×C₅₃	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₁₃×C₅₃	C₄	1	1	yes	characteristic
4	C₂×C₄×C₁₃×C₅₃	C₄	1	3	yes	—
5	C₂×C₄×C₁₃×C₅₃	C₂²	1	3	yes	—
6	C₈×C₁₃×C₅₃	C₂²	1	4	yes	direct product factor
7	C₂²×C₁₃×C₅₃	C₈	1	4	yes	direct product factor
8	C₂²×C₁₃×C₅₃	C₂×C₄	1	3	yes	—
9	C₄×C₁₃×C₅₃	C₂×C₄	1	3	yes	—
10	C₄×C₁₃×C₅₃	C₂³	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₁₃×C₅₃	C₂×C₈	1	6	yes	direct product factor
13	C₂×C₁₃×C₅₃	C₂²×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₁₃	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂×C₁₃	1	6	yes	direct product factor
16	C₁₃×C₅₃	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
17	C₂³×C₅₃	C₄×C₁₃	1	1	yes	characteristic
18	C₂×C₄×C₅₃	C₄×C₁₃	1	3	yes	—
19	C₂×C₄×C₅₃	C₂²×C₁₃	1	3	yes	—
20	C₈×C₅₃	C₂²×C₁₃	1	4	yes	direct product factor
21	C₂²×C₈×C₁₃	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂²×C₅₃	C₈×C₁₃	1	4	yes	direct product factor
23	C₂²×C₅₃	C₂×C₄×C₁₃	1	3	yes	—
24	C₄×C₅₃	C₂×C₄×C₁₃	1	3	yes	—
25	C₄×C₅₃	C₂³×C₁₃	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂×C₅₃	1	1	yes	characteristic
27	C₂×C₈×C₁₃	C₂×C₅₃	1	6	yes	direct product factor
28	C₂×C₅₃	C₂×C₈×C₁₃	1	6	yes	direct product factor
29	C₂×C₅₃	C₂²×C₄×C₁₃	1	1	yes	characteristic
30	C₂³×C₁₃	C₄×C₅₃	1	1	yes	characteristic
31	C₂×C₄×C₁₃	C₄×C₅₃	1	3	yes	—
32	C₂×C₄×C₁₃	C₂²×C₅₃	1	3	yes	—
33	C₈×C₁₃	C₂²×C₅₃	1	4	yes	direct product factor
34	C₅₃	C₂²×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×C₁₃	C₈×C₅₃	1	4	yes	direct product factor
36	C₂²×C₁₃	C₂×C₄×C₅₃	1	3	yes	—
37	C₄×C₁₃	C₂×C₄×C₅₃	1	3	yes	—
38	C₄×C₁₃	C₂³×C₅₃	1	1	yes	characteristic
39	C₂²×C₈	C₁₃×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₁₃	C₂×C₈×C₅₃	1	6	yes	direct product factor
41	C₂×C₁₃	C₂²×C₄×C₅₃	1	1	yes	characteristic
42	C₂²×C₄	C₂×C₁₃×C₅₃	1	1	yes	characteristic
43	C₂×C₈	C₂×C₁₃×C₅₃	1	6	yes	direct product factor
44	C₁₃	C₂²×C₈×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³	C₄×C₁₃×C₅₃	1	1	yes	characteristic
46	C₂×C₄	C₄×C₁₃×C₅₃	1	3	yes	—
47	C₂×C₄	C₂²×C₁₃×C₅₃	1	3	yes	—
48	C₈	C₂²×C₁₃×C₅₃	1	4	yes	direct product factor
49	C₂²	C₈×C₁₃×C₅₃	1	4	yes	direct product factor
50	C₂²	C₂×C₄×C₁₃×C₅₃	1	3	yes	—
51	C₄	C₂×C₄×C₁₃×C₅₃	1	3	yes	—
52	C₄	C₂³×C₁₃×C₅₃	1	1	yes	characteristic
53	C₂	C₂×C₈×C₁₃×C₅₃	1	6	yes	direct product factor
54	C₂	C₂²×C₄×C₁₃×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₁₃×C₅₃
socle, prime radical	C₂³×C₁₃×C₅₃	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×C₁₃×C₅₃	C₂	direct (N × H)
6. C₈×C₁₃×C₅₃	C₂²	direct (N × H)
7. C₂²×C₁₃×C₅₃	C₈	direct (N × H)
11. C₂²×C₈×C₅₃	C₁₃	direct (N × H)
12. C₂×C₁₃×C₅₃	C₂×C₈	direct (N × H)
15. C₂×C₈×C₅₃	C₂×C₁₃	direct (N × H)
16. C₁₃×C₅₃	C₂²×C₈	direct (N × H)
20. C₈×C₅₃	C₂²×C₁₃	direct (N × H)
21. C₂²×C₈×C₁₃	C₅₃	direct (N × H)
22. C₂²×C₅₃	C₈×C₁₃	direct (N × H)
27. C₂×C₈×C₁₃	C₂×C₅₃	direct (N × H)
28. C₂×C₅₃	C₂×C₈×C₁₃	direct (N × H)
33. C₈×C₁₃	C₂²×C₅₃	direct (N × H)
34. C₅₃	C₂²×C₈×C₁₃	direct (N × H)
35. C₂²×C₁₃	C₈×C₅₃	direct (N × H)
39. C₂²×C₈	C₁₃×C₅₃	direct (N × H)
40. C₂×C₁₃	C₂×C₈×C₅₃	direct (N × H)
43. C₂×C₈	C₂×C₁₃×C₅₃	direct (N × H)
44. C₁₃	C₂²×C₈×C₅₃	direct (N × H)
48. C₈	C₂²×C₁₃×C₅₃	direct (N × H)
49. C₂²	C₈×C₁₃×C₅₃	direct (N × H)
53. C₂	C₂×C₈×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
C₂²×C₄×C₁₃×C₅₃	2
C₂×C₈×C₁₃×C₅₃	2
C₂²×C₈×C₅₃	13
C₂²×C₈×C₁₃	53

Subgroups — C22×C8×C13×C53 #72199550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×C₁₃×C₅₃ #72199550