Subgroups — C₂⁵×C₈×C₅₃ #27270583

Order 13568.

15454 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₈×C₅₃	C₂	1	62	yes	maximal, direct product factor
3	C₂⁶×C₅₃	C₄	1	1	yes	characteristic
4	C₂⁴×C₄×C₅₃	C₄	1	31	yes	—
5	C₂⁴×C₄×C₅₃	C₂²	1	31	yes	—
6	C₂³×C₈×C₅₃	C₂²	1	620	yes	direct product factor
7	C₂⁵×C₅₃	C₈	1	32	yes	direct product factor
8	C₂⁵×C₅₃	C₂×C₄	1	31	yes	—
9	C₂³×C₄×C₅₃	C₂×C₄	1	465	yes	—
10	C₂³×C₄×C₅₃	C₂³	1	155	yes	—
11	C₂²×C₈×C₅₃	C₂³	1	1240	yes	direct product factor
12	C₂⁴×C₅₃	C₂×C₈	1	496	yes	direct product factor
13	C₂⁴×C₅₃	C₂²×C₄	1	155	yes	—
14	C₂²×C₄×C₅₃	C₂²×C₄	1	1085	yes	—
15	C₂²×C₄×C₅₃	C₂⁴	1	155	yes	—
16	C₂×C₈×C₅₃	C₂⁴	1	496	yes	direct product factor
17	C₂³×C₅₃	C₂²×C₈	1	1240	yes	direct product factor
18	C₂³×C₅₃	C₂³×C₄	1	155	yes	—
19	C₂×C₄×C₅₃	C₂³×C₄	1	465	yes	—
20	C₂×C₄×C₅₃	C₂⁵	1	31	yes	—
21	C₈×C₅₃	C₂⁵	1	32	yes	direct product factor
22	C₂⁵×C₈	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅₃	C₂³×C₈	1	620	yes	direct product factor
24	C₂²×C₅₃	C₂⁴×C₄	1	31	yes	—
25	C₄×C₅₃	C₂⁴×C₄	1	31	yes	—
26	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
27	C₂⁵×C₄	C₂×C₅₃	1	1	yes	characteristic
28	C₂⁴×C₈	C₂×C₅₃	1	62	yes	direct product factor
29	C₂×C₅₃	C₂⁴×C₈	1	62	yes	direct product factor
30	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
31	C₂⁶	C₄×C₅₃	1	1	yes	characteristic
32	C₂⁴×C₄	C₄×C₅₃	1	31	yes	—
33	C₂⁴×C₄	C₂²×C₅₃	1	31	yes	—
34	C₂³×C₈	C₂²×C₅₃	1	620	yes	direct product factor
35	C₅₃	C₂⁵×C₈	1	1	yes	characteristic, direct product factor, Sylow
36	C₂⁵	C₈×C₅₃	1	32	yes	direct product factor
37	C₂⁵	C₂×C₄×C₅₃	1	31	yes	—
38	C₂³×C₄	C₂×C₄×C₅₃	1	465	yes	—
39	C₂³×C₄	C₂³×C₅₃	1	155	yes	—
40	C₂²×C₈	C₂³×C₅₃	1	1240	yes	direct product factor
41	C₂⁴	C₂×C₈×C₅₃	1	496	yes	direct product factor
42	C₂⁴	C₂²×C₄×C₅₃	1	155	yes	—
43	C₂²×C₄	C₂²×C₄×C₅₃	1	1085	yes	—
44	C₂²×C₄	C₂⁴×C₅₃	1	155	yes	—
45	C₂×C₈	C₂⁴×C₅₃	1	496	yes	direct product factor
46	C₂³	C₂²×C₈×C₅₃	1	1240	yes	direct product factor
47	C₂³	C₂³×C₄×C₅₃	1	155	yes	—
48	C₂×C₄	C₂³×C₄×C₅₃	1	465	yes	—
49	C₂×C₄	C₂⁵×C₅₃	1	31	yes	—
50	C₈	C₂⁵×C₅₃	1	32	yes	direct product factor
51	C₂²	C₂³×C₈×C₅₃	1	620	yes	direct product factor
52	C₂²	C₂⁴×C₄×C₅₃	1	31	yes	—
53	C₄	C₂⁴×C₄×C₅₃	1	31	yes	—
54	C₄	C₂⁶×C₅₃	1	1	yes	characteristic
55	C₂	C₂⁴×C₈×C₅₃	1	62	yes	direct product factor
56	C₂	C₂⁵×C₄×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂⁶×C₅₃
socle, prime radical	C₂⁶×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. C₂⁴×C₈×C₅₃	C₂	direct (N × H)
6. C₂³×C₈×C₅₃	C₂²	direct (N × H)
7. C₂⁵×C₅₃	C₈	direct (N × H)
11. C₂²×C₈×C₅₃	C₂³	direct (N × H)
12. C₂⁴×C₅₃	C₂×C₈	direct (N × H)
16. C₂×C₈×C₅₃	C₂⁴	direct (N × H)
17. C₂³×C₅₃	C₂²×C₈	direct (N × H)
21. C₈×C₅₃	C₂⁵	direct (N × H)
22. C₂⁵×C₈	C₅₃	direct (N × H)
23. C₂²×C₅₃	C₂³×C₈	direct (N × H)
28. C₂⁴×C₈	C₂×C₅₃	direct (N × H)
29. C₂×C₅₃	C₂⁴×C₈	direct (N × H)
34. C₂³×C₈	C₂²×C₅₃	direct (N × H)
35. C₅₃	C₂⁵×C₈	direct (N × H)
36. C₂⁵	C₈×C₅₃	direct (N × H)
40. C₂²×C₈	C₂³×C₅₃	direct (N × H)
41. C₂⁴	C₂×C₈×C₅₃	direct (N × H)
45. C₂×C₈	C₂⁴×C₅₃	direct (N × H)
46. C₂³	C₂²×C₈×C₅₃	direct (N × H)
50. C₈	C₂⁵×C₅₃	direct (N × H)
51. C₂²	C₂³×C₈×C₅₃	direct (N × H)
55. C₂	C₂⁴×C₈×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
C₂⁵×C₄×C₅₃	2
C₂⁴×C₈×C₅₃	2
C₂⁵×C₈	53

Subgroups — C25×C8×C53 #27270583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×C₈×C₅₃ #27270583