Subgroups — C₃×C₄³×C₁₀₉ #72198521

Order 20928.

516 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₁₀₉	C₂	1	7	yes	maximal
2	C₂²×C₃×C₄×C₁₀₉	C₂²	1	7	yes	—
3	C₃×C₄²×C₁₀₉	C₄	1	28	yes	direct product factor
4	C₂³×C₃×C₁₀₉	C₂³	1	1	yes	characteristic
5	C₂×C₃×C₄×C₁₀₉	C₂×C₄	1	42	yes	—
6	C₄³×C₁₀₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₄²×C₁₀₉	C₂×C₃	1	7	yes	—
8	C₂²×C₄×C₁₀₉	C₂²×C₃	1	7	yes	—
9	C₄²×C₁₀₉	C₃×C₄	1	28	yes	direct product factor
10	C₂²×C₃×C₁₀₉	C₂²×C₄	1	7	yes	—
11	C₃×C₄×C₁₀₉	C₄²	1	28	yes	direct product factor
12	C₂³×C₁₀₉	C₂³×C₃	1	1	yes	characteristic
13	C₂×C₄×C₁₀₉	C₂×C₃×C₄	1	42	yes	—
14	C₂×C₃×C₁₀₉	C₂×C₄²	1	7	yes	—
15	C₂²×C₁₀₉	C₂²×C₃×C₄	1	7	yes	—
16	C₄×C₁₀₉	C₃×C₄²	1	28	yes	direct product factor
17	C₃×C₁₀₉	C₄³	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₁₀₉	C₂×C₃×C₄²	1	7	yes	—
19	C₃×C₄³	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₁₀₉	C₃×C₄³	1	1	yes	characteristic, direct product factor, Sylow
21	C₂×C₃×C₄²	C₂×C₁₀₉	1	7	yes	—
22	C₄³	C₃×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
23	C₂²×C₃×C₄	C₂²×C₁₀₉	1	7	yes	—
24	C₃×C₄²	C₄×C₁₀₉	1	28	yes	direct product factor
25	C₂×C₄²	C₂×C₃×C₁₀₉	1	7	yes	—
26	C₂³×C₃	C₂³×C₁₀₉	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂×C₄×C₁₀₉	1	42	yes	—
28	C₂²×C₄	C₂²×C₃×C₁₀₉	1	7	yes	—
29	C₄²	C₃×C₄×C₁₀₉	1	28	yes	direct product factor
30	C₂²×C₃	C₂²×C₄×C₁₀₉	1	7	yes	—
31	C₃×C₄	C₄²×C₁₀₉	1	28	yes	direct product factor
32	C₂³	C₂³×C₃×C₁₀₉	1	1	yes	characteristic
33	C₂×C₄	C₂×C₃×C₄×C₁₀₉	1	42	yes	—
34	C₂×C₃	C₂×C₄²×C₁₀₉	1	7	yes	—
35	C₂²	C₂²×C₃×C₄×C₁₀₉	1	7	yes	—
36	C₄	C₃×C₄²×C₁₀₉	1	28	yes	direct product factor
37	C₃	C₄³×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
38	C₂	C₂×C₃×C₄²×C₁₀₉	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³	C₂³×C₃×C₁₀₉
socle, prime radical	C₂³×C₃×C₁₀₉	C₂³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄²×C₁₀₉	C₄	direct (N × H)
6. C₄³×C₁₀₉	C₃	direct (N × H)
9. C₄²×C₁₀₉	C₃×C₄	direct (N × H)
11. C₃×C₄×C₁₀₉	C₄²	direct (N × H)
16. C₄×C₁₀₉	C₃×C₄²	direct (N × H)
17. C₃×C₁₀₉	C₄³	direct (N × H)
19. C₃×C₄³	C₁₀₉	direct (N × H)
20. C₁₀₉	C₃×C₄³	direct (N × H)
22. C₄³	C₃×C₁₀₉	direct (N × H)
24. C₃×C₄²	C₄×C₁₀₉	direct (N × H)
29. C₄²	C₃×C₄×C₁₀₉	direct (N × H)
31. C₃×C₄	C₄²×C₁₀₉	direct (N × H)
36. C₄	C₃×C₄²×C₁₀₉	direct (N × H)
37. C₃	C₄³×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×C₃×C₄²×C₁₀₉	2
C₄³×C₁₀₉	3
C₃×C₄³	109

Subgroups — C3×C43×C109 #72198521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄³×C₁₀₉ #72198521