Subgroups — C₂×C₃×C₅×D₃×C₁₃₇ #37590451

Order 24660.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×D₃×C₁₃₇	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₅×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₃×C₁₃₇	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂²×C₃×C₅×C₁₃₇	—	3	1	no	maximal
5	C₃²×C₅×C₁₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃×D₃×C₁₃₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×D₃×C₁₃₇	C₂×C₃	1	2	yes	direct product factor
8	C₂×C₃×C₅×C₁₃₇	—	2	1	no	—
9	C₂×C₃×C₅×C₁₃₇	—	3	2	no	—
10	C₂×C₃×C₅×C₁₃₇	D₃	1	1	yes	characteristic, direct product factor
11	C₂×C₃×C₅×C₁₃₇	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂²×C₅×C₁₃₇	—	3	1	no	Hall
13	C₃×D₃×C₁₃₇	C₂×C₅	1	2	yes	direct product factor
14	C₂×C₃²×C₁₃₇	C₂×C₅	1	1	yes	characteristic, complemented
15	C₃×C₅×C₁₃₇	—	2	1	no	—
16	C₃×C₅×C₁₃₇	C₂×D₃	1	1	yes	characteristic, direct product factor
17	C₃×C₅×C₁₃₇	C₂²×C₃	1	1	yes	characteristic, complemented
18	C₂²×C₃×C₁₃₇	—	3	1	no	—
19	C₂×D₃×C₁₃₇	C₃×C₅	1	1	yes	characteristic, direct product factor
20	C₂×C₅×C₁₃₇	—	3	2	no	—
21	C₂×C₅×C₁₃₇	C₃×D₃	1	1	yes	characteristic, direct product factor
22	C₃²×C₁₃₇	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
23	C₂×C₃×C₁₃₇	—	2	1	no	—
24	C₂×C₃×C₁₃₇	—	3	2	no	—
25	C₂×C₃×C₁₃₇	C₅×D₃	1	1	yes	characteristic, direct product factor
26	C₂×C₃×C₁₃₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
27	D₃×C₁₃₇	C₂×C₃×C₅	1	2	yes	direct product factor
28	C₅×C₁₃₇	C₂×C₃×D₃	1	1	yes	characteristic, direct product factor, Hall
29	C₂²×C₁₃₇	—	3	1	no	Hall
30	C₃×C₁₃₇	—	2	1	no	—
31	C₃×C₁₃₇	C₂×C₅×D₃	1	1	yes	characteristic, direct product factor
32	C₃×C₁₃₇	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
33	C₂×C₁₃₇	—	3	2	no	—
34	C₂×C₁₃₇	C₃×C₅×D₃	1	1	yes	characteristic, direct product factor
35	C₂×C₃×C₅×D₃	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
36	C₁₃₇	C₂×C₃×C₅×D₃	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×C₃²×C₅	C₂×C₁₃₇	1	1	yes	characteristic, complemented
38	C₃×C₅×D₃	C₂×C₁₃₇	1	2	yes	direct product factor
39	C₂²×C₃×C₅	—	3	1	no	—
40	C₂×C₅×D₃	C₃×C₁₃₇	1	1	yes	characteristic, direct product factor
41	C₃²×C₅	C₂²×C₁₃₇	1	1	yes	characteristic, complemented, Hall
42	C₂×C₃×D₃	C₅×C₁₃₇	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₃×C₅	—	2	1	no	—
44	C₂×C₃×C₅	—	3	2	no	—
45	C₂×C₃×C₅	D₃×C₁₃₇	1	1	yes	characteristic, direct product factor
46	C₂×C₃×C₅	C₂×C₃×C₁₃₇	1	1	yes	characteristic, complemented
47	C₅×D₃	C₂×C₃×C₁₃₇	1	2	yes	direct product factor
48	C₂²×C₅	—	3	1	no	Hall
49	C₂×C₃²	C₂×C₅×C₁₃₇	1	1	yes	characteristic, complemented
50	C₃×D₃	C₂×C₅×C₁₃₇	1	2	yes	direct product factor
51	C₃×C₅	—	2	1	no	—
52	C₃×C₅	C₂×D₃×C₁₃₇	1	1	yes	characteristic, direct product factor
53	C₃×C₅	C₂²×C₃×C₁₃₇	1	1	yes	characteristic, complemented
54	C₂²×C₃	—	3	1	no	—
55	C₂×D₃	C₃×C₅×C₁₃₇	1	1	yes	characteristic, direct product factor
56	C₂×C₅	—	3	2	no	—
57	C₂×C₅	C₃×D₃×C₁₃₇	1	1	yes	characteristic, direct product factor
58	C₃²	C₂²×C₅×C₁₃₇	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	—	3	2	no	—
61	C₂×C₃	C₂×C₃×C₅×C₁₃₇	1	1	yes	characteristic, complemented
62	C₂×C₃	C₅×D₃×C₁₃₇	1	1	yes	characteristic, direct product factor
63	D₃	C₂×C₃×C₅×C₁₃₇	1	2	yes	direct product factor
64	C₅	C₂×C₃×D₃×C₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	3	1	no	Sylow
66	C₃	—	2	1	no	—
67	C₃	C₂²×C₃×C₅×C₁₃₇	1	1	yes	characteristic, complemented
68	C₃	C₂×C₅×D₃×C₁₃₇	1	1	yes	characteristic, direct product factor
69	C₂	—	3	2	no	—
70	C₂	C₃×C₅×D₃×C₁₃₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅×C₁₃₇	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₃×C₅×C₁₃₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅×C₁₃₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅×C₁₃₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×D₃×C₁₃₇	C₂	direct (N × H)
2. C₂×C₃²×C₅×C₁₃₇	C₂	semidirect
3. C₂×C₅×D₃×C₁₃₇	C₃	direct (N × H)
5. C₃²×C₅×C₁₃₇	C₂²	semidirect
6. C₂×C₃×D₃×C₁₃₇	C₅	direct (N × H)
7. C₅×D₃×C₁₃₇	C₂×C₃	direct (N × H)
10. C₂×C₃×C₅×C₁₃₇	D₃	direct (N × H)
11. C₂×C₃×C₅×C₁₃₇	C₂×C₃	semidirect
13. C₃×D₃×C₁₃₇	C₂×C₅	direct (N × H)
14. C₂×C₃²×C₁₃₇	C₂×C₅	semidirect
16. C₃×C₅×C₁₃₇	C₂×D₃	direct (N × H)
17. C₃×C₅×C₁₃₇	C₂²×C₃	semidirect
19. C₂×D₃×C₁₃₇	C₃×C₅	direct (N × H)
21. C₂×C₅×C₁₃₇	C₃×D₃	direct (N × H)
22. C₃²×C₁₃₇	C₂²×C₅	semidirect
25. C₂×C₃×C₁₃₇	C₅×D₃	direct (N × H)
26. C₂×C₃×C₁₃₇	C₂×C₃×C₅	semidirect
27. D₃×C₁₃₇	C₂×C₃×C₅	direct (N × H)
28. C₅×C₁₃₇	C₂×C₃×D₃	direct (N × H)
31. C₃×C₁₃₇	C₂×C₅×D₃	direct (N × H)
32. C₃×C₁₃₇	C₂²×C₃×C₅	semidirect
34. C₂×C₁₃₇	C₃×C₅×D₃	direct (N × H)
35. C₂×C₃×C₅×D₃	C₁₃₇	direct (N × H)
36. C₁₃₇	C₂×C₃×C₅×D₃	direct (N × H)
37. C₂×C₃²×C₅	C₂×C₁₃₇	semidirect
38. C₃×C₅×D₃	C₂×C₁₃₇	direct (N × H)
40. C₂×C₅×D₃	C₃×C₁₃₇	direct (N × H)
41. C₃²×C₅	C₂²×C₁₃₇	semidirect
42. C₂×C₃×D₃	C₅×C₁₃₇	direct (N × H)
45. C₂×C₃×C₅	D₃×C₁₃₇	direct (N × H)
46. C₂×C₃×C₅	C₂×C₃×C₁₃₇	semidirect
47. C₅×D₃	C₂×C₃×C₁₃₇	direct (N × H)
49. C₂×C₃²	C₂×C₅×C₁₃₇	semidirect
50. C₃×D₃	C₂×C₅×C₁₃₇	direct (N × H)
52. C₃×C₅	C₂×D₃×C₁₃₇	direct (N × H)
53. C₃×C₅	C₂²×C₃×C₁₃₇	semidirect
55. C₂×D₃	C₃×C₅×C₁₃₇	direct (N × H)
57. C₂×C₅	C₃×D₃×C₁₃₇	direct (N × H)
58. C₃²	C₂²×C₅×C₁₃₇	semidirect
61. C₂×C₃	C₂×C₃×C₅×C₁₃₇	semidirect
62. C₂×C₃	C₅×D₃×C₁₃₇	direct (N × H)
63. D₃	C₂×C₃×C₅×C₁₃₇	direct (N × H)
64. C₅	C₂×C₃×D₃×C₁₃₇	direct (N × H)
67. C₃	C₂²×C₃×C₅×C₁₃₇	semidirect
68. C₃	C₂×C₅×D₃×C₁₃₇	direct (N × H)
70. C₂	C₃×C₅×D₃×C₁₃₇	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×C₅×D₃×C₁₃₇	2
C₂×C₃²×C₅×C₁₃₇	2
C₂×C₅×D₃×C₁₃₇	3
C₂²×C₃×C₅×C₁₃₇	3
C₂×C₃×D₃×C₁₃₇	5
C₂×C₃×C₅×D₃	137

Subgroups — C2×C3×C5×D3×C137 #37590451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×D₃×C₁₃₇ #37590451