Subgroups — C₂×C₅×C₁₇×D₁₁₁ #49179915

Order 37740.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₇×D₁₁₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅×C₁₇×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₁₇×D₃₇	—	3	1	no	maximal, Hall
4	C₃×C₅×C₁₇×C₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₁₇×D₁₁₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×C₁₇×D₃₇	—	3	2	no	—
7	C₂×C₅×C₁₇×C₃₇	D₃	1	1	yes	characteristic, complemented
8	C₁₇×D₁₁₁	C₂×C₅	1	2	yes	direct product factor
9	C₂×C₃×C₁₇×C₃₇	C₂×C₅	1	1	yes	characteristic, complemented
10	C₅×C₁₇×C₃₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₁₇×D₃₇	—	3	1	no	Hall
12	C₂×C₅×D₁₁₁	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₃×C₁₇×C₃₇	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
14	C₂×C₁₇×C₃₇	C₅×D₃	1	1	yes	characteristic, complemented
15	C₁₇×D₃₇	—	3	2	no	—
16	C₂×C₃×C₅×C₃₇	C₂×C₁₇	1	1	yes	characteristic, complemented
17	C₅×D₁₁₁	C₂×C₁₇	1	2	yes	direct product factor
18	C₂×C₅×D₃×C₁₇	—	37	1	no	maximal, Hall
19	C₂×C₅×D₃₇	—	3	1	no	Hall
20	C₁₇×C₃₇	C₂×C₅×D₃	1	1	yes	characteristic, complemented, Hall
21	C₃×C₅×C₃₇	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₅×C₁₇	D₃₇	1	1	yes	characteristic, complemented
23	C₅×D₃×C₁₇	—	37	2	no	—
24	C₂×D₁₁₁	C₅×C₁₇	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₅×C₃₇	D₃×C₁₇	1	1	yes	characteristic, complemented
26	C₅×D₃₇	—	3	2	no	—
27	C₂²×C₅×C₁₇	—	111	1	no	Hall
28	C₃×C₅×C₁₇	C₂×D₃₇	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₃₇	C₂×C₅×C₁₇	1	1	yes	characteristic, complemented
30	D₁₁₁	C₂×C₅×C₁₇	1	2	yes	direct product factor
31	C₂×D₃×C₁₇	—	37	1	no	Hall
32	C₅×C₃₇	C₂×D₃×C₁₇	1	1	yes	characteristic, complemented, Hall
33	C₂×C₅×C₁₇	—	111	2	no	—
34	C₂×C₅×C₁₇	D₁₁₁	1	1	yes	characteristic, direct product factor
35	C₂×D₃₇	—	3	1	no	Hall
36	C₃×C₃₇	C₂²×C₅×C₁₇	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₁₇	C₅×D₃₇	1	1	yes	characteristic, complemented
38	D₃×C₁₇	—	37	2	no	—
39	C₅×C₁₇	C₂×D₁₁₁	1	1	yes	characteristic, direct product factor, Hall
40	C₂×C₃₇	C₅×D₃×C₁₇	1	1	yes	characteristic, complemented
41	D₃₇	—	3	2	no	—
42	C₂²×C₁₇	—	111	1	no	Hall
43	C₂×C₅×D₃	—	37	1	no	Hall
44	C₃×C₁₇	C₂×C₅×D₃₇	1	1	yes	characteristic, complemented, Hall
45	C₃₇	C₂×C₅×D₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₁₇	—	111	2	no	—
47	C₂×C₁₇	C₅×D₁₁₁	1	1	yes	characteristic, direct product factor
48	C₂×C₃×C₅	C₁₇×D₃₇	1	1	yes	characteristic, complemented
49	C₅×D₃	—	37	2	no	—
50	C₂²×C₅	—	111	1	no	Hall
51	C₁₇	C₂×C₅×D₁₁₁	1	1	yes	characteristic, direct product factor, Sylow
52	C₃×C₅	C₂×C₁₇×D₃₇	1	1	yes	characteristic, complemented, Hall
53	C₂×D₃	—	37	1	no	Hall
54	C₂×C₅	—	111	2	no	—
55	C₂×C₅	C₁₇×D₁₁₁	1	1	yes	characteristic, direct product factor
56	C₂×C₃	C₅×C₁₇×D₃₇	1	1	yes	characteristic, complemented
57	D₃	—	37	2	no	—
58	C₅	C₂×C₁₇×D₁₁₁	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²	—	111	1	no	Sylow
60	C₃	C₂×C₅×C₁₇×D₃₇	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	111	2	no	—
62	C₂	C₅×C₁₇×D₁₁₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅×C₁₇	D₁₁₁
commutator, cosocle, hypocenter	C₃×C₃₇	C₂²×C₅×C₁₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₁₇×C₃₇	C₂
LCS	G ▷ C₃×C₃₇
UCS	C₁ ◁ C₂×C₅×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₅×C₁₇×D₁₁₁	C₂	direct (N × H)
2. C₂×C₃×C₅×C₁₇×C₃₇	C₂	semidirect
4. C₃×C₅×C₁₇×C₃₇	C₂²	semidirect
5. C₂×C₁₇×D₁₁₁	C₅	direct (N × H)
7. C₂×C₅×C₁₇×C₃₇	D₃	semidirect
8. C₁₇×D₁₁₁	C₂×C₅	direct (N × H)
9. C₂×C₃×C₁₇×C₃₇	C₂×C₅	semidirect
10. C₅×C₁₇×C₃₇	C₂×D₃	semidirect
12. C₂×C₅×D₁₁₁	C₁₇	direct (N × H)
13. C₃×C₁₇×C₃₇	C₂²×C₅	semidirect
14. C₂×C₁₇×C₃₇	C₅×D₃	semidirect
16. C₂×C₃×C₅×C₃₇	C₂×C₁₇	semidirect
17. C₅×D₁₁₁	C₂×C₁₇	direct (N × H)
20. C₁₇×C₃₇	C₂×C₅×D₃	semidirect
21. C₃×C₅×C₃₇	C₂²×C₁₇	semidirect
22. C₂×C₃×C₅×C₁₇	D₃₇	semidirect
24. C₂×D₁₁₁	C₅×C₁₇	direct (N × H)
25. C₂×C₅×C₃₇	D₃×C₁₇	semidirect
28. C₃×C₅×C₁₇	C₂×D₃₇	semidirect
29. C₂×C₃×C₃₇	C₂×C₅×C₁₇	semidirect
30. D₁₁₁	C₂×C₅×C₁₇	direct (N × H)
32. C₅×C₃₇	C₂×D₃×C₁₇	semidirect
34. C₂×C₅×C₁₇	D₁₁₁	direct (N × H)
36. C₃×C₃₇	C₂²×C₅×C₁₇	semidirect
37. C₂×C₃×C₁₇	C₅×D₃₇	semidirect
39. C₅×C₁₇	C₂×D₁₁₁	direct (N × H)
40. C₂×C₃₇	C₅×D₃×C₁₇	semidirect
44. C₃×C₁₇	C₂×C₅×D₃₇	semidirect
45. C₃₇	C₂×C₅×D₃×C₁₇	semidirect
47. C₂×C₁₇	C₅×D₁₁₁	direct (N × H)
48. C₂×C₃×C₅	C₁₇×D₃₇	semidirect
51. C₁₇	C₂×C₅×D₁₁₁	direct (N × H)
52. C₃×C₅	C₂×C₁₇×D₃₇	semidirect
55. C₂×C₅	C₁₇×D₁₁₁	direct (N × H)
56. C₂×C₃	C₅×C₁₇×D₃₇	semidirect
58. C₅	C₂×C₁₇×D₁₁₁	direct (N × H)
60. C₃	C₂×C₅×C₁₇×D₃₇	semidirect
62. C₂	C₅×C₁₇×D₁₁₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₁₇×D₁₁₁	2
C₂×C₃×C₅×C₁₇×C₃₇	2
C₂×C₅×C₁₇×D₃₇	3
C₂×C₁₇×D₁₁₁	5
C₂×C₅×D₁₁₁	17
C₂×C₅×D₃×C₁₇	37

Subgroups — C2×C5×C17×D111 #49179915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₁₇×D₁₁₁ #49179915