Subgroups — C₂²×C₃×C₇×C₁₀₉⋊C₃ #42117608

Order 27468.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×C₁₀₉⋊C₃	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₇×C₁₀₉⋊C₃	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₇×C₁₀₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×C₁₀₉⋊C₃	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂×C₇×C₁₀₉⋊C₃	C₂×C₃	1	9	yes	direct product factor
6	C₂×C₃×C₇×C₁₀₉	C₂×C₃	1	3	yes	complemented
7	C₂²×C₃×C₁₀₉⋊C₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₇×C₁₀₉	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇×C₁₀₉	C₂²×C₃	1	1	yes	characteristic, complemented
10	C₇×C₁₀₉⋊C₃	C₂²×C₃	1	3	yes	direct product factor
11	C₂×C₃×C₁₀₉⋊C₃	C₂×C₇	1	3	yes	direct product factor
12	C₂×C₇×C₁₀₉	C₂×C₃²	1	3	yes	complemented
13	C₂²×C₃×C₁₀₉	C₃×C₇	1	1	yes	characteristic, complemented
14	C₂²×C₁₀₉⋊C₃	C₃×C₇	1	3	yes	direct product factor
15	C₃×C₁₀₉⋊C₃	C₂²×C₇	1	1	yes	characteristic, direct product factor, Hall
16	C₇×C₁₀₉	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₁₀₉	C₂×C₃×C₇	1	3	yes	complemented
18	C₂×C₁₀₉⋊C₃	C₂×C₃×C₇	1	9	yes	direct product factor
19	C₂²×C₁₀₉	C₃²×C₇	1	1	yes	characteristic, complemented, Hall
20	C₃×C₁₀₉	C₂²×C₃×C₇	1	1	yes	characteristic, complemented
21	C₁₀₉⋊C₃	C₂²×C₃×C₇	1	3	yes	direct product factor
22	C₂²×C₃²×C₇	—	109	1	no	maximal, Hall
23	C₂×C₁₀₉	C₂×C₃²×C₇	1	3	yes	complemented
24	C₂×C₃²×C₇	—	109	3	no	—
25	C₁₀₉	C₂²×C₃²×C₇	1	1	yes	characteristic, complemented, Sylow
26	C₂²×C₃×C₇	—	109	3	no	—
27	C₂²×C₃×C₇	C₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor
28	C₃²×C₇	—	109	1	no	Hall
29	C₂×C₃×C₇	—	109	9	no	—
30	C₂×C₃×C₇	C₂×C₁₀₉⋊C₃	1	3	yes	direct product factor
31	C₂²×C₃²	—	109	1	no	Hall
32	C₂²×C₇	C₃×C₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor, Hall
33	C₃×C₇	—	109	3	no	—
34	C₃×C₇	C₂²×C₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor
35	C₂×C₃²	—	109	3	no	—
36	C₂×C₇	C₂×C₃×C₁₀₉⋊C₃	1	3	yes	direct product factor
37	C₂²×C₃	—	109	3	no	—
38	C₂²×C₃	C₇×C₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor
39	C₃²	—	109	1	no	Sylow
40	C₇	C₂²×C₃×C₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₃	—	109	9	no	—
42	C₂×C₃	C₂×C₇×C₁₀₉⋊C₃	1	3	yes	direct product factor
43	C₂²	C₃×C₇×C₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
44	C₃	—	109	3	no	—
45	C₃	C₂²×C₇×C₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor
46	C₂	C₂×C₃×C₇×C₁₀₉⋊C₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₇	C₁₀₉⋊C₃
commutator, cosocle, hypocenter	C₁₀₉	C₂²×C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇×C₁₀₉	C₃
LCS	G ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₃×C₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×C₁₀₉⋊C₃	C₂	direct (N × H)
2. C₂²×C₇×C₁₀₉⋊C₃	C₃	direct (N × H)
3. C₂²×C₃×C₇×C₁₀₉	C₃	semidirect
4. C₃×C₇×C₁₀₉⋊C₃	C₂²	direct (N × H)
5. C₂×C₇×C₁₀₉⋊C₃	C₂×C₃	direct (N × H)
6. C₂×C₃×C₇×C₁₀₉	C₂×C₃	semidirect
7. C₂²×C₃×C₁₀₉⋊C₃	C₇	direct (N × H)
8. C₂²×C₇×C₁₀₉	C₃²	semidirect
9. C₃×C₇×C₁₀₉	C₂²×C₃	semidirect
10. C₇×C₁₀₉⋊C₃	C₂²×C₃	direct (N × H)
11. C₂×C₃×C₁₀₉⋊C₃	C₂×C₇	direct (N × H)
12. C₂×C₇×C₁₀₉	C₂×C₃²	semidirect
13. C₂²×C₃×C₁₀₉	C₃×C₇	semidirect
14. C₂²×C₁₀₉⋊C₃	C₃×C₇	direct (N × H)
15. C₃×C₁₀₉⋊C₃	C₂²×C₇	direct (N × H)
16. C₇×C₁₀₉	C₂²×C₃²	semidirect
17. C₂×C₃×C₁₀₉	C₂×C₃×C₇	semidirect
18. C₂×C₁₀₉⋊C₃	C₂×C₃×C₇	direct (N × H)
19. C₂²×C₁₀₉	C₃²×C₇	semidirect
20. C₃×C₁₀₉	C₂²×C₃×C₇	semidirect
21. C₁₀₉⋊C₃	C₂²×C₃×C₇	direct (N × H)
23. C₂×C₁₀₉	C₂×C₃²×C₇	semidirect
25. C₁₀₉	C₂²×C₃²×C₇	semidirect
27. C₂²×C₃×C₇	C₁₀₉⋊C₃	direct (N × H)
30. C₂×C₃×C₇	C₂×C₁₀₉⋊C₃	direct (N × H)
32. C₂²×C₇	C₃×C₁₀₉⋊C₃	direct (N × H)
34. C₃×C₇	C₂²×C₁₀₉⋊C₃	direct (N × H)
36. C₂×C₇	C₂×C₃×C₁₀₉⋊C₃	direct (N × H)
38. C₂²×C₃	C₇×C₁₀₉⋊C₃	direct (N × H)
40. C₇	C₂²×C₃×C₁₀₉⋊C₃	direct (N × H)
42. C₂×C₃	C₂×C₇×C₁₀₉⋊C₃	direct (N × H)
43. C₂²	C₃×C₇×C₁₀₉⋊C₃	direct (N × H)
45. C₃	C₂²×C₇×C₁₀₉⋊C₃	direct (N × H)
46. C₂	C₂×C₃×C₇×C₁₀₉⋊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
C₂×C₃×C₇×C₁₀₉⋊C₃	2
C₂²×C₇×C₁₀₉⋊C₃	3
C₂²×C₃×C₇×C₁₀₉	3
C₂²×C₃×C₁₀₉⋊C₃	7
C₂²×C₃²×C₇	109

Subgroups — C22×C3×C7×C109⋊C3 #42117608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₇×C₁₀₉⋊C₃ #42117608