Subgroups — C₅²×D₃²×C₁₃ #26043812

Order 11700.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₁₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅²×D₃×C₁₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₅²×D₃×C₁₃	—	3	2	no	maximal
4	C₃²×C₅²×C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅×D₃²×C₁₃	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₃×C₅²×C₁₃	—	3	2	no	—
7	C₅²×D₃×C₁₃	—	3	2	no	—
8	C₅²×D₃×C₁₃	—	6	1	no	—
9	C₅²×D₃×C₁₃	D₃	1	2	yes	direct product factor
10	C₂²×C₅²×C₁₃	—	9	1	no	Hall
11	C₅×C₁₃×C₃⋊D₃	C₂×C₅	1	6	yes	complemented
12	C₃×C₅×D₃×C₁₃	C₂×C₅	1	12	yes	complemented
13	C₃×C₅²×C₁₃	—	2	1	no	—
14	C₃×C₅²×C₁₃	C₂×D₃	1	2	yes	complemented
15	C₅²×D₃²	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×D₃×C₁₃	—	3	12	no	—
17	C₂×C₅²×C₁₃	—	3	2	no	—
18	C₂×C₅²×C₁₃	—	9	1	no	—
19	C₃²×C₅×C₁₃	C₂²×C₅	1	6	yes	complemented
20	D₃²×C₁₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₅²×C₃⋊D₃	C₂×C₁₃	1	1	yes	characteristic, complemented
22	C₃×C₅²×D₃	C₂×C₁₃	1	2	yes	complemented
23	C₂×C₃×C₅×C₁₃	—	3	12	no	—
24	C₅×D₃×C₁₃	—	3	12	no	—
25	C₅×D₃×C₁₃	—	6	6	no	—
26	C₅×D₃×C₁₃	C₅×D₃	1	12	yes	direct product factor
27	C₅²×C₁₃	D₃²	1	1	yes	characteristic, direct product factor, Hall
28	C₂×C₅²×D₃	—	3	2	no	—
29	C₂²×C₅×C₁₃	—	9	6	no	—
30	C₁₃×C₃⋊D₃	C₂×C₅²	1	1	yes	characteristic, complemented
31	C₃×D₃×C₁₃	C₂×C₅²	1	2	yes	complemented
32	C₃²×C₅²	C₂²×C₁₃	1	1	yes	characteristic, complemented, Hall
33	C₃×C₅×C₁₃	—	2	6	no	—
34	C₃×C₅×C₁₃	C₂×C₅×D₃	1	12	yes	complemented
35	C₅×D₃²	C₅×C₁₃	1	6	yes	direct product factor
36	C₂×D₃×C₁₃	—	3	2	no	—
37	C₂×C₃×C₅²	—	3	2	no	—
38	C₅²×D₃	—	3	2	no	—
39	C₅²×D₃	—	6	1	no	—
40	C₅²×D₃	D₃×C₁₃	1	2	yes	direct product factor
41	C₂×C₅×C₁₃	—	3	12	no	—
42	C₂×C₅×C₁₃	—	9	6	no	—
43	C₃²×C₁₃	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
44	C₂²×C₅²	—	9	1	no	Hall
45	C₅×C₃⋊D₃	C₂×C₅×C₁₃	1	6	yes	complemented
46	C₃×C₅×D₃	C₂×C₅×C₁₃	1	12	yes	complemented
47	C₂×C₃×C₁₃	—	3	2	no	—
48	D₃×C₁₃	—	3	2	no	—
49	D₃×C₁₃	—	6	1	no	—
50	D₃×C₁₃	C₅²×D₃	1	2	yes	direct product factor
51	C₃×C₅²	—	2	1	no	—
52	C₃×C₅²	C₂×D₃×C₁₃	1	2	yes	complemented
53	C₅×C₁₃	C₅×D₃²	1	6	yes	direct product factor
54	C₂×C₅×D₃	—	3	12	no	—
55	C₂²×C₁₃	—	9	1	no	Hall
56	C₂×C₅²	—	3	2	no	—
57	C₂×C₅²	—	9	1	no	—
58	C₃²×C₅	C₂²×C₅×C₁₃	1	6	yes	complemented
59	C₃×C₁₃	—	2	1	no	—
60	C₃×C₁₃	C₂×C₅²×D₃	1	2	yes	complemented
61	D₃²	C₅²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
62	C₂×C₃×C₅	—	3	12	no	—
63	C₅×D₃	—	3	12	no	—
64	C₅×D₃	—	6	6	no	—
65	C₅×D₃	C₅×D₃×C₁₃	1	12	yes	direct product factor
66	C₂×C₁₃	—	3	2	no	—
67	C₂×C₁₃	—	9	1	no	—
68	C₅²	D₃²×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₅	—	9	6	no	—
70	C₃⋊D₃	C₂×C₅²×C₁₃	1	1	yes	characteristic, complemented
71	C₃×D₃	C₂×C₅²×C₁₃	1	2	yes	complemented
72	C₃×C₅	—	2	6	no	—
73	C₃×C₅	C₂×C₅×D₃×C₁₃	1	12	yes	complemented
74	C₁₃	C₅²×D₃²	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×D₃	—	3	2	no	—
76	C₂×C₅	—	3	12	no	—
77	C₂×C₅	—	9	6	no	—
78	C₃²	C₂²×C₅²×C₁₃	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	3	2	no	—
80	D₃	—	3	2	no	—
81	D₃	—	6	1	no	—
82	D₃	C₅²×D₃×C₁₃	1	2	yes	direct product factor
83	C₅	C₅×D₃²×C₁₃	1	6	yes	direct product factor
84	C₂²	—	9	1	no	Sylow
85	C₃	—	2	1	no	—
86	C₃	C₂×C₅²×D₃×C₁₃	1	2	yes	complemented
87	C₂	—	3	2	no	—
88	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₅²×C₁₃	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₅²×C₁₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₃	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₅²×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₅²×C₁₃×C₃⋊D₃	C₂	semidirect
2. C₃×C₅²×D₃×C₁₃	C₂	semidirect
4. C₃²×C₅²×C₁₃	C₂²	semidirect
5. C₅×D₃²×C₁₃	C₅	direct (N × H)
9. C₅²×D₃×C₁₃	D₃	direct (N × H)
11. C₅×C₁₃×C₃⋊D₃	C₂×C₅	semidirect
12. C₃×C₅×D₃×C₁₃	C₂×C₅	semidirect
14. C₃×C₅²×C₁₃	C₂×D₃	semidirect
15. C₅²×D₃²	C₁₃	direct (N × H)
19. C₃²×C₅×C₁₃	C₂²×C₅	semidirect
20. D₃²×C₁₃	C₅²	direct (N × H)
21. C₅²×C₃⋊D₃	C₂×C₁₃	semidirect
22. C₃×C₅²×D₃	C₂×C₁₃	semidirect
26. C₅×D₃×C₁₃	C₅×D₃	direct (N × H)
27. C₅²×C₁₃	D₃²	direct (N × H)
30. C₁₃×C₃⋊D₃	C₂×C₅²	semidirect
31. C₃×D₃×C₁₃	C₂×C₅²	semidirect
32. C₃²×C₅²	C₂²×C₁₃	semidirect
34. C₃×C₅×C₁₃	C₂×C₅×D₃	semidirect
35. C₅×D₃²	C₅×C₁₃	direct (N × H)
40. C₅²×D₃	D₃×C₁₃	direct (N × H)
43. C₃²×C₁₃	C₂²×C₅²	semidirect
45. C₅×C₃⋊D₃	C₂×C₅×C₁₃	semidirect
46. C₃×C₅×D₃	C₂×C₅×C₁₃	semidirect
50. D₃×C₁₃	C₅²×D₃	direct (N × H)
52. C₃×C₅²	C₂×D₃×C₁₃	semidirect
53. C₅×C₁₃	C₅×D₃²	direct (N × H)
58. C₃²×C₅	C₂²×C₅×C₁₃	semidirect
60. C₃×C₁₃	C₂×C₅²×D₃	semidirect
61. D₃²	C₅²×C₁₃	direct (N × H)
65. C₅×D₃	C₅×D₃×C₁₃	direct (N × H)
68. C₅²	D₃²×C₁₃	direct (N × H)
70. C₃⋊D₃	C₂×C₅²×C₁₃	semidirect
71. C₃×D₃	C₂×C₅²×C₁₃	semidirect
73. C₃×C₅	C₂×C₅×D₃×C₁₃	semidirect
74. C₁₃	C₅²×D₃²	direct (N × H)
78. C₃²	C₂²×C₅²×C₁₃	semidirect
82. D₃	C₅²×D₃×C₁₃	direct (N × H)
83. C₅	C₅×D₃²×C₁₃	direct (N × H)
86. C₃	C₂×C₅²×D₃×C₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₁₃×C₃⋊D₃	2
C₃×C₅²×D₃×C₁₃	2
C₂×C₅²×D₃×C₁₃	3
C₅×D₃²×C₁₃	5
C₅²×D₃²	13

Subgroups — C52×D32×C13 #26043812

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×D₃²×C₁₃ #26043812