Subgroups — C₂⁵×Q₈×C₅₃ #27270608

Order 13568.

63326 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₅₃	C₂	1	124	yes	maximal, direct product factor
2	C₂⁵×C₄×C₅₃	C₂	1	3	yes	maximal
3	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
4	C₂³×Q₈×C₅₃	C₂²	1	2480	yes	direct product factor
5	C₂⁴×C₄×C₅₃	C₂²	1	186	yes	—
6	C₂⁵×C₅₃	Q₈	1	32	yes	direct product factor
7	C₂⁵×C₅₃	C₂³	1	31	yes	—
8	C₂²×Q₈×C₅₃	C₂³	1	9920	yes	direct product factor
9	C₂³×C₄×C₅₃	C₂³	1	1860	yes	—
10	C₂⁴×C₅₃	C₂×Q₈	1	496	yes	direct product factor
11	C₂⁴×C₅₃	C₂⁴	1	155	yes	—
12	C₂×Q₈×C₅₃	C₂⁴	1	7936	yes	direct product factor
13	C₂²×C₄×C₅₃	C₂⁴	1	3720	yes	—
14	C₂³×C₅₃	C₂²×Q₈	1	1240	yes	direct product factor
15	C₂³×C₅₃	C₂⁵	1	155	yes	—
16	Q₈×C₅₃	C₂⁵	1	1024	yes	direct product factor
17	C₂×C₄×C₅₃	C₂⁵	1	1488	yes	—
18	C₂⁵×Q₈	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₅₃	C₂³×Q₈	1	620	yes	direct product factor
20	C₂²×C₅₃	C₂⁶	1	31	yes	—
21	C₄×C₅₃	C₂⁶	1	96	yes	—
22	C₂⁴×Q₈	C₂×C₅₃	1	124	yes	direct product factor
23	C₂⁵×C₄	C₂×C₅₃	1	3	yes	—
24	C₂×C₅₃	C₂⁴×Q₈	1	62	yes	direct product factor
25	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
26	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
27	C₂³×Q₈	C₂²×C₅₃	1	2480	yes	direct product factor
28	C₂⁴×C₄	C₂²×C₅₃	1	186	yes	—
29	C₅₃	C₂⁵×Q₈	1	1	yes	characteristic, direct product factor, Sylow
30	C₂⁵	Q₈×C₅₃	1	32	yes	direct product factor
31	C₂⁵	C₂³×C₅₃	1	31	yes	—
32	C₂²×Q₈	C₂³×C₅₃	1	9920	yes	direct product factor
33	C₂³×C₄	C₂³×C₅₃	1	1860	yes	—
34	C₂⁴	C₂×Q₈×C₅₃	1	496	yes	direct product factor
35	C₂⁴	C₂⁴×C₅₃	1	155	yes	—
36	C₂×Q₈	C₂⁴×C₅₃	1	7936	yes	direct product factor
37	C₂²×C₄	C₂⁴×C₅₃	1	3720	yes	—
38	C₂³	C₂²×Q₈×C₅₃	1	1240	yes	direct product factor
39	C₂³	C₂⁵×C₅₃	1	155	yes	—
40	Q₈	C₂⁵×C₅₃	1	1024	yes	direct product factor
41	C₂×C₄	C₂⁵×C₅₃	1	1488	yes	—
42	C₂²	C₂³×Q₈×C₅₃	1	620	yes	direct product factor
43	C₂²	C₂⁶×C₅₃	1	31	yes	—
44	C₄	C₂⁶×C₅₃	1	96	yes	—
45	C₂	C₂⁴×Q₈×C₅₃	1	62	yes	direct product factor
46	C₂	C₂⁷×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁶×C₅₃	C₂²
commutator, Frattini, cosocle	C₂	C₂⁷×C₅₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈×C₅₃	C₂	direct (N × H)
4. C₂³×Q₈×C₅₃	C₂²	direct (N × H)
6. C₂⁵×C₅₃	Q₈	direct (N × H)
8. C₂²×Q₈×C₅₃	C₂³	direct (N × H)
10. C₂⁴×C₅₃	C₂×Q₈	direct (N × H)
12. C₂×Q₈×C₅₃	C₂⁴	direct (N × H)
14. C₂³×C₅₃	C₂²×Q₈	direct (N × H)
16. Q₈×C₅₃	C₂⁵	direct (N × H)
18. C₂⁵×Q₈	C₅₃	direct (N × H)
19. C₂²×C₅₃	C₂³×Q₈	direct (N × H)
22. C₂⁴×Q₈	C₂×C₅₃	direct (N × H)
24. C₂×C₅₃	C₂⁴×Q₈	direct (N × H)
27. C₂³×Q₈	C₂²×C₅₃	direct (N × H)
29. C₅₃	C₂⁵×Q₈	direct (N × H)
30. C₂⁵	Q₈×C₅₃	direct (N × H)
32. C₂²×Q₈	C₂³×C₅₃	direct (N × H)
34. C₂⁴	C₂×Q₈×C₅₃	direct (N × H)
36. C₂×Q₈	C₂⁴×C₅₃	direct (N × H)
38. C₂³	C₂²×Q₈×C₅₃	direct (N × H)
40. Q₈	C₂⁵×C₅₃	direct (N × H)
42. C₂²	C₂³×Q₈×C₅₃	direct (N × H)
45. C₂	C₂⁴×Q₈×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×Q₈×C₅₃	2
C₂⁵×C₄×C₅₃	2
C₂⁵×Q₈	53

Subgroups — C25×Q8×C53 #27270608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×Q₈×C₅₃ #27270608