Subgroups — C₂⁴×C₃²×C₁₆₉ #72201651

Order 24336.

1206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₁₆₉	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×C₃×C₁₆₉	C₃	1	4	yes	maximal, direct product factor
3	C₂³×C₃×C₁₆₉	C₂×C₃	1	60	yes	direct product factor
4	C₂⁴×C₁₆₉	C₃²	1	1	yes	characteristic, direct product factor, Hall
5	C₂²×C₃²×C₁₆₉	C₂²	1	35	yes	direct product factor
6	C₂×C₃²×C₁₆₉	C₂³	1	15	yes	direct product factor
7	C₂²×C₃×C₁₆₉	C₂²×C₃	1	140	yes	direct product factor
8	C₂⁴×C₃²×C₁₃	C₁₃	1	1	yes	maximal, characteristic
9	C₃²×C₁₆₉	C₂⁴	1	1	yes	characteristic, direct product factor, Hall
10	C₂³×C₁₆₉	C₂×C₃²	1	15	yes	direct product factor
11	C₂×C₃×C₁₆₉	C₂³×C₃	1	60	yes	direct product factor
12	C₂³×C₃²×C₁₃	C₂×C₁₃	1	15	yes	—
13	C₂²×C₁₆₉	C₂²×C₃²	1	35	yes	direct product factor
14	C₂⁴×C₃×C₁₃	C₃×C₁₃	1	4	yes	—
15	C₃×C₁₆₉	C₂⁴×C₃	1	4	yes	direct product factor
16	C₂²×C₃²×C₁₃	C₂²×C₁₃	1	35	yes	—
17	C₂×C₁₆₉	C₂³×C₃²	1	15	yes	direct product factor
18	C₂³×C₃×C₁₃	C₂×C₃×C₁₃	1	60	yes	—
19	C₂×C₃²×C₁₃	C₂³×C₁₃	1	15	yes	—
20	C₂⁴×C₁₃	C₃²×C₁₃	1	1	yes	characteristic
21	C₁₆₉	C₂⁴×C₃²	1	1	yes	characteristic, direct product factor, Sylow
22	C₂²×C₃×C₁₃	C₂²×C₃×C₁₃	1	140	yes	—
23	C₂⁴×C₃²	C₁₆₉	1	1	yes	characteristic, direct product factor, Hall
24	C₃²×C₁₃	C₂⁴×C₁₃	1	1	yes	characteristic
25	C₂³×C₁₃	C₂×C₃²×C₁₃	1	15	yes	—
26	C₂×C₃×C₁₃	C₂³×C₃×C₁₃	1	60	yes	—
27	C₂³×C₃²	C₂×C₁₆₉	1	15	yes	direct product factor
28	C₂²×C₁₃	C₂²×C₃²×C₁₃	1	35	yes	—
29	C₂⁴×C₃	C₃×C₁₆₉	1	4	yes	direct product factor
30	C₃×C₁₃	C₂⁴×C₃×C₁₃	1	4	yes	—
31	C₂²×C₃²	C₂²×C₁₆₉	1	35	yes	direct product factor
32	C₂×C₁₃	C₂³×C₃²×C₁₃	1	15	yes	—
33	C₂³×C₃	C₂×C₃×C₁₆₉	1	60	yes	direct product factor
34	C₂×C₃²	C₂³×C₁₆₉	1	15	yes	direct product factor
35	C₂⁴	C₃²×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
36	C₁₃	C₂⁴×C₃²×C₁₃	1	1	yes	characteristic
37	C₂²×C₃	C₂²×C₃×C₁₆₉	1	140	yes	direct product factor
38	C₃²	C₂⁴×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂³	C₂×C₃²×C₁₆₉	1	15	yes	direct product factor
40	C₂×C₃	C₂³×C₃×C₁₆₉	1	60	yes	direct product factor
41	C₂²	C₂²×C₃²×C₁₆₉	1	35	yes	direct product factor
42	C₃	C₂⁴×C₃×C₁₆₉	1	4	yes	direct product factor
43	C₂	C₂³×C₃²×C₁₆₉	1	15	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₃	C₂⁴×C₃²×C₁₃
socle, prime radical	C₂⁴×C₃²×C₁₃	C₁₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃²×C₁₆₉	C₂	direct (N × H)
2. C₂⁴×C₃×C₁₆₉	C₃	direct (N × H)
3. C₂³×C₃×C₁₆₉	C₂×C₃	direct (N × H)
4. C₂⁴×C₁₆₉	C₃²	direct (N × H)
5. C₂²×C₃²×C₁₆₉	C₂²	direct (N × H)
6. C₂×C₃²×C₁₆₉	C₂³	direct (N × H)
7. C₂²×C₃×C₁₆₉	C₂²×C₃	direct (N × H)
9. C₃²×C₁₆₉	C₂⁴	direct (N × H)
10. C₂³×C₁₆₉	C₂×C₃²	direct (N × H)
11. C₂×C₃×C₁₆₉	C₂³×C₃	direct (N × H)
13. C₂²×C₁₆₉	C₂²×C₃²	direct (N × H)
15. C₃×C₁₆₉	C₂⁴×C₃	direct (N × H)
17. C₂×C₁₆₉	C₂³×C₃²	direct (N × H)
21. C₁₆₉	C₂⁴×C₃²	direct (N × H)
23. C₂⁴×C₃²	C₁₆₉	direct (N × H)
27. C₂³×C₃²	C₂×C₁₆₉	direct (N × H)
29. C₂⁴×C₃	C₃×C₁₆₉	direct (N × H)
31. C₂²×C₃²	C₂²×C₁₆₉	direct (N × H)
33. C₂³×C₃	C₂×C₃×C₁₆₉	direct (N × H)
34. C₂×C₃²	C₂³×C₁₆₉	direct (N × H)
35. C₂⁴	C₃²×C₁₆₉	direct (N × H)
37. C₂²×C₃	C₂²×C₃×C₁₆₉	direct (N × H)
38. C₃²	C₂⁴×C₁₆₉	direct (N × H)
39. C₂³	C₂×C₃²×C₁₆₉	direct (N × H)
40. C₂×C₃	C₂³×C₃×C₁₆₉	direct (N × H)
41. C₂²	C₂²×C₃²×C₁₆₉	direct (N × H)
42. C₃	C₂⁴×C₃×C₁₆₉	direct (N × H)
43. C₂	C₂³×C₃²×C₁₆₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₁₆₉	2
C₂⁴×C₃×C₁₆₉	3
C₂⁴×C₃²×C₁₃	13

Subgroups — C24×C32×C169 #72201651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃²×C₁₆₉ #72201651