Subgroups — C₄²×C₉×C₁₆₉ #72201641

Order 24336.

135 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₉×C₁₆₉	C₂	1	3	yes	maximal
2	C₃×C₄²×C₁₆₉	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₁₆₉	C₂×C₃	1	3	yes	—
4	C₄²×C₁₆₉	C₉	1	1	yes	characteristic, direct product factor, Hall
5	C₂²×C₉×C₁₆₉	C₂²	1	1	yes	characteristic
6	C₄×C₉×C₁₆₉	C₄	1	6	yes	direct product factor
7	C₂×C₉×C₁₆₉	C₂×C₄	1	3	yes	—
8	C₂²×C₃×C₁₆₉	C₂²×C₃	1	1	yes	characteristic
9	C₃×C₄×C₁₆₉	C₃×C₄	1	6	yes	—
10	C₄²×C₉×C₁₃	C₁₃	1	1	yes	maximal, characteristic
11	C₉×C₁₆₉	C₄²	1	1	yes	characteristic, direct product factor, Hall
12	C₂×C₄×C₁₆₉	C₂×C₉	1	3	yes	—
13	C₂×C₃×C₁₆₉	C₂×C₃×C₄	1	3	yes	—
14	C₂×C₄×C₉×C₁₃	C₂×C₁₃	1	3	yes	—
15	C₂²×C₁₆₉	C₂²×C₉	1	1	yes	characteristic
16	C₄×C₁₆₉	C₄×C₉	1	6	yes	direct product factor
17	C₃×C₄²×C₁₃	C₃×C₁₃	1	1	yes	characteristic
18	C₃×C₁₆₉	C₃×C₄²	1	1	yes	characteristic
19	C₂²×C₉×C₁₃	C₂²×C₁₃	1	1	yes	characteristic
20	C₄×C₉×C₁₃	C₄×C₁₃	1	6	yes	—
21	C₂×C₁₆₉	C₂×C₄×C₉	1	3	yes	—
22	C₂×C₃×C₄×C₁₃	C₂×C₃×C₁₃	1	3	yes	—
23	C₂×C₉×C₁₃	C₂×C₄×C₁₃	1	3	yes	—
24	C₄²×C₁₃	C₉×C₁₃	1	1	yes	characteristic
25	C₁₆₉	C₄²×C₉	1	1	yes	characteristic, direct product factor, Sylow
26	C₂²×C₃×C₁₃	C₂²×C₃×C₁₃	1	1	yes	characteristic
27	C₃×C₄×C₁₃	C₃×C₄×C₁₃	1	6	yes	—
28	C₄²×C₉	C₁₆₉	1	1	yes	characteristic, direct product factor, Hall
29	C₉×C₁₃	C₄²×C₁₃	1	1	yes	characteristic
30	C₂×C₄×C₁₃	C₂×C₉×C₁₃	1	3	yes	—
31	C₂×C₃×C₁₃	C₂×C₃×C₄×C₁₃	1	3	yes	—
32	C₂×C₄×C₉	C₂×C₁₆₉	1	3	yes	—
33	C₂²×C₁₃	C₂²×C₉×C₁₃	1	1	yes	characteristic
34	C₄×C₁₃	C₄×C₉×C₁₃	1	6	yes	—
35	C₃×C₄²	C₃×C₁₆₉	1	1	yes	characteristic
36	C₃×C₁₃	C₃×C₄²×C₁₃	1	1	yes	characteristic
37	C₂²×C₉	C₂²×C₁₆₉	1	1	yes	characteristic
38	C₄×C₉	C₄×C₁₆₉	1	6	yes	direct product factor
39	C₂×C₁₃	C₂×C₄×C₉×C₁₃	1	3	yes	—
40	C₂×C₃×C₄	C₂×C₃×C₁₆₉	1	3	yes	—
41	C₂×C₉	C₂×C₄×C₁₆₉	1	3	yes	—
42	C₄²	C₉×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
43	C₁₃	C₄²×C₉×C₁₃	1	1	yes	characteristic
44	C₂²×C₃	C₂²×C₃×C₁₆₉	1	1	yes	characteristic
45	C₃×C₄	C₃×C₄×C₁₆₉	1	6	yes	—
46	C₉	C₄²×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×C₄	C₂×C₉×C₁₆₉	1	3	yes	—
48	C₂×C₃	C₂×C₃×C₄×C₁₆₉	1	3	yes	—
49	C₂²	C₂²×C₉×C₁₆₉	1	1	yes	characteristic
50	C₄	C₄×C₉×C₁₆₉	1	6	yes	direct product factor
51	C₃	C₃×C₄²×C₁₆₉	1	1	yes	characteristic
52	C₂	C₂×C₄×C₉×C₁₆₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, socle, prime radical, cosocle	C₂²×C₃×C₁₃	C₂²×C₃×C₁₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄²×C₁₆₉	C₉	direct (N × H)
6. C₄×C₉×C₁₆₉	C₄	direct (N × H)
11. C₉×C₁₆₉	C₄²	direct (N × H)
16. C₄×C₁₆₉	C₄×C₉	direct (N × H)
25. C₁₆₉	C₄²×C₉	direct (N × H)
28. C₄²×C₉	C₁₆₉	direct (N × H)
38. C₄×C₉	C₄×C₁₆₉	direct (N × H)
42. C₄²	C₉×C₁₆₉	direct (N × H)
46. C₉	C₄²×C₁₆₉	direct (N × H)
50. C₄	C₄×C₉×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
C₂×C₄×C₉×C₁₆₉	2
C₃×C₄²×C₁₆₉	3
C₄²×C₉×C₁₃	13

Subgroups — C42×C9×C169 #72201641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₉×C₁₆₉ #72201641