Subgroups — C₃×C₅²×C₂²⋊(C₁₃⋊C₃) #26043862

Order 11700.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₂²⋊(C₁₃⋊C₃)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅²×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅²×C₁₃⋊C₃	—	4	1	no	maximal, Hall
4	C₃×C₅×C₂²⋊(C₁₃⋊C₃)	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₃×C₅²×C₁₃	—	3	1	no	—
6	C₂²×C₅²×C₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₅²×C₁₃	A₄	1	1	yes	characteristic, complemented
8	C₅²×C₁₃⋊C₃	—	4	3	no	—
9	C₃×C₅²×A₄	—	13	1	no	maximal, Hall
10	C₂²×C₃×C₅×C₁₃	C₃×C₅	1	6	yes	complemented
11	C₅×C₂²⋊(C₁₃⋊C₃)	C₃×C₅	1	18	yes	direct product factor
12	C₂×C₅²×C₁₃	—	3	1	no	—
13	C₃×C₅×C₁₃⋊C₃	—	4	6	no	—
14	C₃×C₂²⋊(C₁₃⋊C₃)	C₅²	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₃×C₅×C₁₃	—	3	6	no	—
16	C₅²×C₁₃	C₃×A₄	1	1	yes	characteristic, complemented, Hall
17	C₂²×C₃×C₅²	C₁₃⋊C₃	1	1	yes	characteristic, complemented
18	C₅²×A₄	—	13	3	no	—
19	C₂²×C₅×C₁₃	C₃²×C₅	1	6	yes	complemented
20	C₃²×C₅²	—	52	1	no	Hall
21	C₃×C₅×C₁₃	C₅×A₄	1	6	yes	complemented
22	C₅×C₁₃⋊C₃	—	4	18	no	—
23	C₃×C₅×A₄	—	13	6	no	—
24	C₂²×C₃×C₁₃	C₃×C₅²	1	1	yes	characteristic, complemented
25	C₂²⋊(C₁₃⋊C₃)	C₃×C₅²	1	3	yes	direct product factor
26	C₂×C₃×C₅²	—	3	1	no	—
27	C₂×C₅×C₁₃	—	3	6	no	—
28	C₃×C₁₃⋊C₃	—	4	1	no	Hall
29	C₂²×C₅²	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₁₃	—	3	1	no	—
31	C₃×C₅²	—	52	3	no	—
32	C₃×C₅²	C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic, direct product factor
33	C₅×C₁₃	C₃×C₅×A₄	1	6	yes	complemented
34	C₂²×C₃×C₅	C₅×C₁₃⋊C₃	1	6	yes	complemented
35	C₅×A₄	—	13	18	no	—
36	C₂²×C₁₃	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
37	C₂×C₅²	—	3	1	no	—
38	C₃²×C₅	—	52	6	no	—
39	C₃×C₁₃	C₅²×A₄	1	1	yes	characteristic, complemented
40	C₁₃⋊C₃	—	4	3	no	—
41	C₃×A₄	—	13	1	no	Hall
42	C₂×C₃×C₅	—	3	6	no	—
43	C₂×C₁₃	—	3	1	no	—
44	C₅²	C₃×C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²×C₅	C₃×C₅×C₁₃⋊C₃	1	6	yes	complemented
46	C₃×C₅	—	52	18	no	—
47	C₃×C₅	C₅×C₂²⋊(C₁₃⋊C₃)	1	6	yes	direct product factor
48	C₁₃	C₃×C₅²×A₄	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₃	C₅²×C₁₃⋊C₃	1	1	yes	characteristic, complemented
50	A₄	—	13	3	no	—
51	C₂×C₅	—	3	6	no	—
52	C₃²	—	52	1	no	Sylow
53	C₂×C₃	—	3	1	no	—
54	C₅	C₃×C₅×C₂²⋊(C₁₃⋊C₃)	1	6	yes	direct product factor
55	C₂²	C₃×C₅²×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	52	3	no	—
57	C₃	C₅²×C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic, direct product factor
58	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅²	C₂²⋊(C₁₃⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₁₃	C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₁₃	C₃
LCS	G ▷ C₂²×C₁₃
UCS	C₁ ◁ C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₅²×C₂²⋊(C₁₃⋊C₃)	C₃	direct (N × H)
2. C₂²×C₃×C₅²×C₁₃	C₃	semidirect
4. C₃×C₅×C₂²⋊(C₁₃⋊C₃)	C₅	direct (N × H)
6. C₂²×C₅²×C₁₃	C₃²	semidirect
7. C₃×C₅²×C₁₃	A₄	semidirect
10. C₂²×C₃×C₅×C₁₃	C₃×C₅	semidirect
11. C₅×C₂²⋊(C₁₃⋊C₃)	C₃×C₅	direct (N × H)
14. C₃×C₂²⋊(C₁₃⋊C₃)	C₅²	direct (N × H)
16. C₅²×C₁₃	C₃×A₄	semidirect
17. C₂²×C₃×C₅²	C₁₃⋊C₃	semidirect
19. C₂²×C₅×C₁₃	C₃²×C₅	semidirect
21. C₃×C₅×C₁₃	C₅×A₄	semidirect
24. C₂²×C₃×C₁₃	C₃×C₅²	semidirect
25. C₂²⋊(C₁₃⋊C₃)	C₃×C₅²	direct (N × H)
29. C₂²×C₅²	C₃×C₁₃⋊C₃	semidirect
32. C₃×C₅²	C₂²⋊(C₁₃⋊C₃)	direct (N × H)
33. C₅×C₁₃	C₃×C₅×A₄	semidirect
34. C₂²×C₃×C₅	C₅×C₁₃⋊C₃	semidirect
36. C₂²×C₁₃	C₃²×C₅²	semidirect
39. C₃×C₁₃	C₅²×A₄	semidirect
44. C₅²	C₃×C₂²⋊(C₁₃⋊C₃)	direct (N × H)
45. C₂²×C₅	C₃×C₅×C₁₃⋊C₃	semidirect
47. C₃×C₅	C₅×C₂²⋊(C₁₃⋊C₃)	direct (N × H)
48. C₁₃	C₃×C₅²×A₄	semidirect
49. C₂²×C₃	C₅²×C₁₃⋊C₃	semidirect
54. C₅	C₃×C₅×C₂²⋊(C₁₃⋊C₃)	direct (N × H)
55. C₂²	C₃×C₅²×C₁₃⋊C₃	semidirect
57. C₃	C₅²×C₂²⋊(C₁₃⋊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
C₅²×C₂²⋊(C₁₃⋊C₃)	3
C₂²×C₃×C₅²×C₁₃	3
C₃×C₅²×C₁₃⋊C₃	4
C₃×C₅×C₂²⋊(C₁₃⋊C₃)	5
C₃×C₅²×A₄	13

Subgroups — C3×C52×C22⋊(C13⋊C3) #26043862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×C₂²⋊(C₁₃⋊C₃) #26043862