Subgroups — C₂×C₃×C₅₃×C₅²⋊C₃ #37558722

Order 23850.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×C₅²⋊C₃	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₅₃×C₅²⋊C₃	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₅²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₅₃×C₅²⋊C₃	C₂×C₃	1	3	yes	direct product factor
5	C₃×C₅²×C₅₃	C₂×C₃	1	1	yes	characteristic, complemented
6	C₂×C₅²×C₅₃	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₂×C₃×C₅×C₅₃	—	3	2	no	—
8	C₅²×C₅₃	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃²×C₅₃	—	25	1	no	maximal, Hall
10	C₃×C₅×C₅₃	—	3	2	no	—
11	C₂×C₅×C₅₃	—	3	2	no	—
12	C₃²×C₅₃	—	25	1	no	Hall
13	C₂×C₃×C₅²⋊C₃	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₅₃	—	25	3	no	—
15	C₂×C₃×C₅₃	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
16	C₅×C₅₃	—	3	2	no	—
17	C₃×C₅²⋊C₃	C₂×C₅₃	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₅₃	—	25	3	no	—
19	C₃×C₅₃	C₂×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
20	C₂×C₃×C₅²	C₃×C₅₃	1	1	yes	characteristic, complemented
21	C₂×C₅²⋊C₃	C₃×C₅₃	1	3	yes	direct product factor
22	C₂×C₅₃	C₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₅²	C₂×C₃×C₅₃	1	1	yes	characteristic, complemented
24	C₅²⋊C₃	C₂×C₃×C₅₃	1	3	yes	direct product factor
25	C₅₃	C₂×C₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
26	C₂×C₅²	C₃²×C₅₃	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃×C₅	—	3	2	no	—
28	C₅²	C₂×C₃²×C₅₃	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃²	—	25	1	no	Hall
30	C₃×C₅	—	3	2	no	—
31	C₂×C₅	—	3	2	no	—
32	C₃²	—	25	1	no	Sylow
33	C₂×C₃	—	25	3	no	—
34	C₂×C₃	C₅₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
35	C₅	—	3	2	no	—
36	C₃	—	25	3	no	—
37	C₃	C₂×C₅₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
38	C₂	C₃×C₅₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅₃	C₅²⋊C₃
commutator, cosocle, hypocenter	C₅²	C₂×C₃²×C₅₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₅₃	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₃×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅₃×C₅²⋊C₃	C₂	direct (N × H)
2. C₂×C₅₃×C₅²⋊C₃	C₃	direct (N × H)
3. C₂×C₃×C₅²×C₅₃	C₃	semidirect
4. C₅₃×C₅²⋊C₃	C₂×C₃	direct (N × H)
5. C₃×C₅²×C₅₃	C₂×C₃	semidirect
6. C₂×C₅²×C₅₃	C₃²	semidirect
8. C₅²×C₅₃	C₂×C₃²	semidirect
13. C₂×C₃×C₅²⋊C₃	C₅₃	direct (N × H)
15. C₂×C₃×C₅₃	C₅²⋊C₃	direct (N × H)
17. C₃×C₅²⋊C₃	C₂×C₅₃	direct (N × H)
19. C₃×C₅₃	C₂×C₅²⋊C₃	direct (N × H)
20. C₂×C₃×C₅²	C₃×C₅₃	semidirect
21. C₂×C₅²⋊C₃	C₃×C₅₃	direct (N × H)
22. C₂×C₅₃	C₃×C₅²⋊C₃	direct (N × H)
23. C₃×C₅²	C₂×C₃×C₅₃	semidirect
24. C₅²⋊C₃	C₂×C₃×C₅₃	direct (N × H)
25. C₅₃	C₂×C₃×C₅²⋊C₃	direct (N × H)
26. C₂×C₅²	C₃²×C₅₃	semidirect
28. C₅²	C₂×C₃²×C₅₃	semidirect
34. C₂×C₃	C₅₃×C₅²⋊C₃	direct (N × H)
37. C₃	C₂×C₅₃×C₅²⋊C₃	direct (N × H)
38. C₂	C₃×C₅₃×C₅²⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅₃×C₅²⋊C₃	2
C₂×C₅₃×C₅²⋊C₃	3
C₂×C₃×C₅²×C₅₃	3
C₂×C₃²×C₅₃	25
C₂×C₃×C₅²⋊C₃	53

Subgroups — C2×C3×C53×C52⋊C3 #37558722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅₃×C₅²⋊C₃ #37558722