Subgroups — C₃×C₄²×C₅×C₁₀₁ #72201528

Order 24240.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₁₀₁	C₂	1	3	yes	maximal
2	C₄²×C₅×C₁₀₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₄²×C₁₀₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₄×C₅×C₁₀₁	C₂×C₃	1	3	yes	—
5	C₂×C₃×C₄×C₁₀₁	C₂×C₅	1	3	yes	—
6	C₂²×C₃×C₅×C₁₀₁	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₅×C₁₀₁	C₄	1	6	yes	direct product factor
8	C₂×C₃×C₅×C₁₀₁	C₂×C₄	1	3	yes	—
9	C₂²×C₅×C₁₀₁	C₂²×C₃	1	1	yes	characteristic
10	C₄×C₅×C₁₀₁	C₃×C₄	1	6	yes	direct product factor
11	C₄²×C₁₀₁	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
12	C₃×C₅×C₁₀₁	C₄²	1	1	yes	characteristic, direct product factor, Hall
13	C₂²×C₃×C₁₀₁	C₂²×C₅	1	1	yes	characteristic
14	C₃×C₄×C₁₀₁	C₄×C₅	1	6	yes	direct product factor
15	C₂×C₅×C₁₀₁	C₂×C₃×C₄	1	3	yes	—
16	C₂×C₄×C₁₀₁	C₂×C₃×C₅	1	3	yes	—
17	C₂×C₃×C₁₀₁	C₂×C₄×C₅	1	3	yes	—
18	C₅×C₁₀₁	C₃×C₄²	1	1	yes	characteristic, direct product factor, Hall
19	C₂²×C₁₀₁	C₂²×C₃×C₅	1	1	yes	characteristic
20	C₄×C₁₀₁	C₃×C₄×C₅	1	6	yes	direct product factor
21	C₃×C₁₀₁	C₄²×C₅	1	1	yes	characteristic, direct product factor, Hall
22	C₃×C₄²×C₅	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₁₀₁	C₂×C₃×C₄×C₅	1	3	yes	—
24	C₂×C₃×C₄×C₅	C₂×C₁₀₁	1	3	yes	—
25	C₁₀₁	C₃×C₄²×C₅	1	1	yes	characteristic, direct product factor, Sylow
26	C₄²×C₅	C₃×C₁₀₁	1	1	yes	characteristic, direct product factor, Hall
27	C₂²×C₃×C₅	C₂²×C₁₀₁	1	1	yes	characteristic
28	C₃×C₄×C₅	C₄×C₁₀₁	1	6	yes	direct product factor
29	C₃×C₄²	C₅×C₁₀₁	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₄×C₅	C₂×C₃×C₁₀₁	1	3	yes	—
31	C₂×C₃×C₅	C₂×C₄×C₁₀₁	1	3	yes	—
32	C₂×C₃×C₄	C₂×C₅×C₁₀₁	1	3	yes	—
33	C₂²×C₅	C₂²×C₃×C₁₀₁	1	1	yes	characteristic
34	C₄×C₅	C₃×C₄×C₁₀₁	1	6	yes	direct product factor
35	C₄²	C₃×C₅×C₁₀₁	1	1	yes	characteristic, direct product factor, Sylow
36	C₃×C₅	C₄²×C₁₀₁	1	1	yes	characteristic, direct product factor, Hall
37	C₂²×C₃	C₂²×C₅×C₁₀₁	1	1	yes	characteristic
38	C₃×C₄	C₄×C₅×C₁₀₁	1	6	yes	direct product factor
39	C₂×C₅	C₂×C₃×C₄×C₁₀₁	1	3	yes	—
40	C₂×C₄	C₂×C₃×C₅×C₁₀₁	1	3	yes	—
41	C₂×C₃	C₂×C₄×C₅×C₁₀₁	1	3	yes	—
42	C₅	C₃×C₄²×C₁₀₁	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²	C₂²×C₃×C₅×C₁₀₁	1	1	yes	characteristic
44	C₄	C₃×C₄×C₅×C₁₀₁	1	6	yes	direct product factor
45	C₃	C₄²×C₅×C₁₀₁	1	1	yes	characteristic, direct product factor, Sylow
46	C₂	C₂×C₃×C₄×C₅×C₁₀₁	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂²×C₃×C₅×C₁₀₁
socle, prime radical	C₂²×C₃×C₅×C₁₀₁	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×C₅×C₁₀₁	C₃	direct (N × H)
3. C₃×C₄²×C₁₀₁	C₅	direct (N × H)
7. C₃×C₄×C₅×C₁₀₁	C₄	direct (N × H)
10. C₄×C₅×C₁₀₁	C₃×C₄	direct (N × H)
11. C₄²×C₁₀₁	C₃×C₅	direct (N × H)
12. C₃×C₅×C₁₀₁	C₄²	direct (N × H)
14. C₃×C₄×C₁₀₁	C₄×C₅	direct (N × H)
18. C₅×C₁₀₁	C₃×C₄²	direct (N × H)
20. C₄×C₁₀₁	C₃×C₄×C₅	direct (N × H)
21. C₃×C₁₀₁	C₄²×C₅	direct (N × H)
22. C₃×C₄²×C₅	C₁₀₁	direct (N × H)
25. C₁₀₁	C₃×C₄²×C₅	direct (N × H)
26. C₄²×C₅	C₃×C₁₀₁	direct (N × H)
28. C₃×C₄×C₅	C₄×C₁₀₁	direct (N × H)
29. C₃×C₄²	C₅×C₁₀₁	direct (N × H)
34. C₄×C₅	C₃×C₄×C₁₀₁	direct (N × H)
35. C₄²	C₃×C₅×C₁₀₁	direct (N × H)
36. C₃×C₅	C₄²×C₁₀₁	direct (N × H)
38. C₃×C₄	C₄×C₅×C₁₀₁	direct (N × H)
42. C₅	C₃×C₄²×C₁₀₁	direct (N × H)
44. C₄	C₃×C₄×C₅×C₁₀₁	direct (N × H)
45. C₃	C₄²×C₅×C₁₀₁	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×C₄×C₅×C₁₀₁	2
C₄²×C₅×C₁₀₁	3
C₃×C₄²×C₁₀₁	5
C₃×C₄²×C₅	101

Subgroups — C3×C42×C5×C101 #72201528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×C₅×C₁₀₁ #72201528