Subgroups — C₂²×C₄×C₃₁×C₅₃ #72203459

Order 26288.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃₁×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃₁×C₅₃	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃₁×C₅₃	C₄	1	4	yes	direct product factor
4	C₂²×C₃₁×C₅₃	C₂²	1	3	yes	—
5	C₄×C₃₁×C₅₃	C₂²	1	4	yes	direct product factor
6	C₂×C₃₁×C₅₃	C₂×C₄	1	6	yes	direct product factor
7	C₂×C₃₁×C₅₃	C₂³	1	1	yes	characteristic
8	C₃₁×C₅₃	C₂²×C₄	1	1	yes	characteristic, direct product factor, Hall
9	C₂²×C₄×C₅₃	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₄×C₃₁	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂³×C₅₃	C₂×C₃₁	1	1	yes	characteristic
12	C₂×C₄×C₅₃	C₂×C₃₁	1	6	yes	direct product factor
13	C₂³×C₃₁	C₂×C₅₃	1	1	yes	characteristic
14	C₂×C₄×C₃₁	C₂×C₅₃	1	6	yes	direct product factor
15	C₂²×C₅₃	C₄×C₃₁	1	4	yes	direct product factor
16	C₂²×C₅₃	C₂²×C₃₁	1	3	yes	—
17	C₄×C₅₃	C₂²×C₃₁	1	4	yes	direct product factor
18	C₂²×C₃₁	C₄×C₅₃	1	4	yes	direct product factor
19	C₂²×C₃₁	C₂²×C₅₃	1	3	yes	—
20	C₄×C₃₁	C₂²×C₅₃	1	4	yes	direct product factor
21	C₂×C₅₃	C₂×C₄×C₃₁	1	6	yes	direct product factor
22	C₂×C₅₃	C₂³×C₃₁	1	1	yes	characteristic
23	C₂×C₃₁	C₂×C₄×C₅₃	1	6	yes	direct product factor
24	C₂×C₃₁	C₂³×C₅₃	1	1	yes	characteristic
25	C₅₃	C₂²×C₄×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
26	C₃₁	C₂²×C₄×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
27	C₂²×C₄	C₃₁×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
28	C₂³	C₂×C₃₁×C₅₃	1	1	yes	characteristic
29	C₂×C₄	C₂×C₃₁×C₅₃	1	6	yes	direct product factor
30	C₂²	C₄×C₃₁×C₅₃	1	4	yes	direct product factor
31	C₂²	C₂²×C₃₁×C₅₃	1	3	yes	—
32	C₄	C₂²×C₃₁×C₅₃	1	4	yes	direct product factor
33	C₂	C₂×C₄×C₃₁×C₅₃	1	6	yes	direct product factor
34	C₂	C₂³×C₃₁×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂³×C₃₁×C₅₃
socle, prime radical	C₂³×C₃₁×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. C₂×C₄×C₃₁×C₅₃	C₂	direct (N × H)
3. C₂²×C₃₁×C₅₃	C₄	direct (N × H)
5. C₄×C₃₁×C₅₃	C₂²	direct (N × H)
6. C₂×C₃₁×C₅₃	C₂×C₄	direct (N × H)
8. C₃₁×C₅₃	C₂²×C₄	direct (N × H)
9. C₂²×C₄×C₅₃	C₃₁	direct (N × H)
10. C₂²×C₄×C₃₁	C₅₃	direct (N × H)
12. C₂×C₄×C₅₃	C₂×C₃₁	direct (N × H)
14. C₂×C₄×C₃₁	C₂×C₅₃	direct (N × H)
15. C₂²×C₅₃	C₄×C₃₁	direct (N × H)
17. C₄×C₅₃	C₂²×C₃₁	direct (N × H)
18. C₂²×C₃₁	C₄×C₅₃	direct (N × H)
20. C₄×C₃₁	C₂²×C₅₃	direct (N × H)
21. C₂×C₅₃	C₂×C₄×C₃₁	direct (N × H)
23. C₂×C₃₁	C₂×C₄×C₅₃	direct (N × H)
25. C₅₃	C₂²×C₄×C₃₁	direct (N × H)
26. C₃₁	C₂²×C₄×C₅₃	direct (N × H)
27. C₂²×C₄	C₃₁×C₅₃	direct (N × H)
29. C₂×C₄	C₂×C₃₁×C₅₃	direct (N × H)
30. C₂²	C₄×C₃₁×C₅₃	direct (N × H)
32. C₄	C₂²×C₃₁×C₅₃	direct (N × H)
33. C₂	C₂×C₄×C₃₁×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
C₂³×C₃₁×C₅₃	2
C₂×C₄×C₃₁×C₅₃	2
C₂²×C₄×C₅₃	31
C₂²×C₄×C₃₁	53

Subgroups — C22×C4×C31×C53 #72203459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₃₁×C₅₃ #72203459