Subgroups — C₃×C₁₃×D₃₄₅ #41002127

Order 26910.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₁₃×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₃×D₃₄₅	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₁₃×D₁₁₅	—	3	1	no	maximal
4	C₃×C₁₃×D₆₉	—	5	1	no	maximal, Hall
5	C₃×C₅×C₁₃×C₂₃	—	2	1	no	—
6	C₃×C₅×C₁₃×C₂₃	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅×C₁₃×C₂₃	C₂×C₃	1	1	yes	characteristic, complemented
8	C₁₃×D₁₁₅	—	3	1	no	Hall
9	C₃²×C₁₃×C₂₃	D₅	1	1	yes	characteristic, complemented, Hall
10	C₃×D₃₄₅	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₁₃×D₆₉	—	5	1	no	—
12	C₃×C₁₃×D₂₃	—	15	1	no	—
13	C₅×C₁₃×C₂₃	C₃×D₃	1	1	yes	characteristic, complemented, Hall
14	C₃×C₁₃×D₁₅	—	23	1	no	maximal, Hall
15	C₃²×C₅×C₂₃	C₂×C₁₃	1	1	yes	characteristic, complemented, Hall
16	C₃×C₁₃×C₂₃	—	2	1	no	—
17	C₃×C₁₃×C₂₃	C₃×D₅	1	1	yes	characteristic, complemented
18	C₃×C₁₃×C₂₃	D₁₅	1	1	yes	characteristic, complemented
19	D₃₄₅	C₃×C₁₃	1	1	yes	characteristic, direct product factor
20	C₃×D₁₁₅	—	3	1	no	—
21	C₁₃×D₂₃	—	15	1	no	Hall
22	C₃²×C₅×C₁₃	D₂₃	1	1	yes	characteristic, complemented, Hall
23	C₃×D₆₉	—	5	1	no	Hall
24	C₁₃×D₁₅	—	23	1	no	—
25	C₃×D₅×C₁₃	—	69	1	no	—
26	C₃×C₅×C₂₃	—	2	1	no	—
27	C₃×C₅×C₂₃	D₃×C₁₃	1	1	yes	characteristic, complemented
28	C₃×C₅×C₂₃	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented
29	C₁₃×C₂₃	C₃×D₁₅	1	1	yes	characteristic, complemented, Hall
30	C₃×D₃×C₁₃	—	115	1	no	Hall
31	D₁₁₅	—	3	1	no	Hall
32	C₃²×C₂₃	D₅×C₁₃	1	1	yes	characteristic, complemented, Hall
33	C₃×C₅×C₁₃	—	2	1	no	—
34	C₃×C₅×C₁₃	C₃×D₂₃	1	1	yes	characteristic, complemented
35	C₃×C₅×C₁₃	D₆₉	1	1	yes	characteristic, complemented
36	D₆₉	—	5	1	no	—
37	C₃×D₂₃	—	15	1	no	—
38	D₅×C₁₃	—	69	1	no	Hall
39	C₃²×C₁₃	D₁₁₅	1	1	yes	characteristic, complemented, Hall
40	C₅×C₂₃	C₃×D₃×C₁₃	1	1	yes	characteristic, complemented, Hall
41	C₃×D₁₅	—	23	1	no	Hall
42	C₂×C₃×C₁₃	—	345	1	no	—
43	D₃×C₁₃	—	115	1	no	—
44	C₃×C₂₃	—	2	1	no	—
45	C₃×C₂₃	C₃×D₅×C₁₃	1	1	yes	characteristic, complemented
46	C₃×C₂₃	C₁₃×D₁₅	1	1	yes	characteristic, complemented
47	C₅×C₁₃	C₃×D₆₉	1	1	yes	characteristic, complemented, Hall
48	D₂₃	—	15	1	no	Hall
49	C₃²×C₅	C₁₃×D₂₃	1	1	yes	characteristic, complemented, Hall
50	C₃×C₁₃	—	2	1	no	—
51	C₃×C₁₃	C₃×D₁₁₅	1	1	yes	characteristic, complemented
52	C₃×C₁₃	D₃₄₅	1	1	yes	characteristic, direct product factor
53	D₁₅	—	23	1	no	—
54	C₃×D₅	—	69	1	no	—
55	C₂×C₁₃	—	345	1	no	Hall
56	C₂₃	C₃×C₁₃×D₁₅	1	1	yes	characteristic, complemented, Sylow
57	C₃×D₃	—	115	1	no	Hall
58	C₃×C₅	—	2	1	no	—
59	C₃×C₅	C₃×C₁₃×D₂₃	1	1	yes	characteristic, complemented
60	C₃×C₅	C₁₃×D₆₉	1	1	yes	characteristic, complemented
61	C₁₃	C₃×D₃₄₅	1	1	yes	characteristic, direct product factor, Sylow
62	D₅	—	69	1	no	Hall
63	C₃²	C₁₃×D₁₁₅	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	345	1	no	—
65	D₃	—	115	1	no	—
66	C₅	C₃×C₁₃×D₆₉	1	1	yes	characteristic, complemented, Sylow
67	C₃	—	2	1	no	—
68	C₃	C₃×C₁₃×D₁₁₅	1	1	yes	characteristic, complemented
69	C₃	C₁₃×D₃₄₅	1	1	yes	characteristic, direct product factor
70	C₂	—	345	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₁₃	D₃₄₅
commutator, cosocle, hypocenter	C₃×C₅×C₂₃	C₂×C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₃×C₂₃	C₂
LCS	G ▷ C₃×C₅×C₂₃
UCS	C₁ ◁ C₃×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₁₃×C₂₃	C₂	semidirect
2. C₁₃×D₃₄₅	C₃	direct (N × H)
6. C₃×C₅×C₁₃×C₂₃	D₃	semidirect
7. C₃×C₅×C₁₃×C₂₃	C₂×C₃	semidirect
9. C₃²×C₁₃×C₂₃	D₅	semidirect
10. C₃×D₃₄₅	C₁₃	direct (N × H)
13. C₅×C₁₃×C₂₃	C₃×D₃	semidirect
15. C₃²×C₅×C₂₃	C₂×C₁₃	semidirect
17. C₃×C₁₃×C₂₃	C₃×D₅	semidirect
18. C₃×C₁₃×C₂₃	D₁₅	semidirect
19. D₃₄₅	C₃×C₁₃	direct (N × H)
22. C₃²×C₅×C₁₃	D₂₃	semidirect
27. C₃×C₅×C₂₃	D₃×C₁₃	semidirect
28. C₃×C₅×C₂₃	C₂×C₃×C₁₃	semidirect
29. C₁₃×C₂₃	C₃×D₁₅	semidirect
32. C₃²×C₂₃	D₅×C₁₃	semidirect
34. C₃×C₅×C₁₃	C₃×D₂₃	semidirect
35. C₃×C₅×C₁₃	D₆₉	semidirect
39. C₃²×C₁₃	D₁₁₅	semidirect
40. C₅×C₂₃	C₃×D₃×C₁₃	semidirect
45. C₃×C₂₃	C₃×D₅×C₁₃	semidirect
46. C₃×C₂₃	C₁₃×D₁₅	semidirect
47. C₅×C₁₃	C₃×D₆₉	semidirect
49. C₃²×C₅	C₁₃×D₂₃	semidirect
51. C₃×C₁₃	C₃×D₁₁₅	semidirect
52. C₃×C₁₃	D₃₄₅	direct (N × H)
56. C₂₃	C₃×C₁₃×D₁₅	semidirect
59. C₃×C₅	C₃×C₁₃×D₂₃	semidirect
60. C₃×C₅	C₁₃×D₆₉	semidirect
61. C₁₃	C₃×D₃₄₅	direct (N × H)
63. C₃²	C₁₃×D₁₁₅	semidirect
66. C₅	C₃×C₁₃×D₆₉	semidirect
68. C₃	C₃×C₁₃×D₁₁₅	semidirect
69. C₃	C₁₃×D₃₄₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₁₃×C₂₃	2
C₁₃×D₃₄₅	3
C₃×C₁₃×D₁₁₅	3
C₃×C₁₃×D₆₉	5
C₃×D₃₄₅	13
C₃×C₁₃×D₁₅	23

Subgroups — C3×C13×D345 #41002127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₁₃×D₃₄₅ #41002127