Subgroups — C₂²×C₅²⋊C₃×C₁₀₉ #45658367

Order 32700.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²⋊C₃×C₁₀₉	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅²×C₁₀₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²⋊C₃×C₁₀₉	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₅²×C₁₀₉	C₂×C₃	1	3	yes	complemented
5	C₅²×C₁₀₉	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
6	C₂²×C₅×C₁₀₉	—	3	2	no	—
7	C₂²×C₃×C₁₀₉	—	25	1	no	maximal, Hall
8	C₂×C₅×C₁₀₉	—	3	6	no	—
9	C₂×C₃×C₁₀₉	—	25	3	no	—
10	C₅×C₁₀₉	—	3	2	no	—
11	C₂²×C₁₀₉	C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Hall
12	C₃×C₁₀₉	—	25	1	no	Hall
13	C₂²×C₅²⋊C₃	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₁₀₉	C₂×C₅²⋊C₃	1	3	yes	direct product factor
15	C₂×C₅²⋊C₃	C₂×C₁₀₉	1	3	yes	direct product factor
16	C₁₀₉	C₂²×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
17	C₂²×C₅²	C₃×C₁₀₉	1	1	yes	characteristic, complemented, Hall
18	C₅²⋊C₃	C₂²×C₁₀₉	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₅²	C₂×C₃×C₁₀₉	1	3	yes	complemented
20	C₅²	C₂²×C₃×C₁₀₉	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₅	—	3	2	no	—
22	C₂²×C₃	—	25	1	no	Hall
23	C₂×C₅	—	3	6	no	—
24	C₂×C₃	—	25	3	no	—
25	C₅	—	3	2	no	—
26	C₂²	C₅²⋊C₃×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
27	C₃	—	25	1	no	Sylow
28	C₂	C₂×C₅²⋊C₃×C₁₀₉	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₁₀₉	C₅²⋊C₃
commutator, cosocle, hypocenter	C₅²	C₂²×C₃×C₁₀₉
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₁₀₉	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂²×C₁₀₉

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²⋊C₃×C₁₀₉	C₂	direct (N × H)
2. C₂²×C₅²×C₁₀₉	C₃	semidirect
3. C₅²⋊C₃×C₁₀₉	C₂²	direct (N × H)
4. C₂×C₅²×C₁₀₉	C₂×C₃	semidirect
5. C₅²×C₁₀₉	C₂²×C₃	semidirect
11. C₂²×C₁₀₉	C₅²⋊C₃	direct (N × H)
13. C₂²×C₅²⋊C₃	C₁₀₉	direct (N × H)
14. C₂×C₁₀₉	C₂×C₅²⋊C₃	direct (N × H)
15. C₂×C₅²⋊C₃	C₂×C₁₀₉	direct (N × H)
16. C₁₀₉	C₂²×C₅²⋊C₃	direct (N × H)
17. C₂²×C₅²	C₃×C₁₀₉	semidirect
18. C₅²⋊C₃	C₂²×C₁₀₉	direct (N × H)
19. C₂×C₅²	C₂×C₃×C₁₀₉	semidirect
20. C₅²	C₂²×C₃×C₁₀₉	semidirect
26. C₂²	C₅²⋊C₃×C₁₀₉	direct (N × H)
28. C₂	C₂×C₅²⋊C₃×C₁₀₉	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×C₅²⋊C₃×C₁₀₉	2
C₂²×C₅²×C₁₀₉	3
C₂²×C₃×C₁₀₉	25
C₂²×C₅²⋊C₃	109

Subgroups — C22×C52⋊C3×C109 #45658367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅²⋊C₃×C₁₀₉ #45658367