Subgroups — C₃²×C₄²×C₁₆₉ #72201643

Order 24336.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₁₆₉	C₂	1	3	yes	maximal
2	C₃×C₄²×C₁₆₉	C₃	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₄×C₁₆₉	C₂×C₃	1	12	yes	—
4	C₄²×C₁₆₉	C₃²	1	1	yes	characteristic, direct product factor, Hall
5	C₂²×C₃²×C₁₆₉	C₂²	1	1	yes	characteristic
6	C₃²×C₄×C₁₆₉	C₄	1	6	yes	direct product factor
7	C₂×C₃²×C₁₆₉	C₂×C₄	1	3	yes	—
8	C₂²×C₃×C₁₆₉	C₂²×C₃	1	4	yes	—
9	C₃×C₄×C₁₆₉	C₃×C₄	1	24	yes	direct product factor
10	C₃²×C₄²×C₁₃	C₁₃	1	1	yes	maximal, characteristic
11	C₃²×C₁₆₉	C₄²	1	1	yes	characteristic, direct product factor, Hall
12	C₂×C₄×C₁₆₉	C₂×C₃²	1	3	yes	—
13	C₂×C₃×C₁₆₉	C₂×C₃×C₄	1	12	yes	—
14	C₂×C₃²×C₄×C₁₃	C₂×C₁₃	1	3	yes	—
15	C₂²×C₁₆₉	C₂²×C₃²	1	1	yes	characteristic
16	C₄×C₁₆₉	C₃²×C₄	1	6	yes	direct product factor
17	C₃×C₄²×C₁₃	C₃×C₁₃	1	4	yes	—
18	C₃×C₁₆₉	C₃×C₄²	1	4	yes	direct product factor
19	C₂²×C₃²×C₁₃	C₂²×C₁₃	1	1	yes	characteristic
20	C₃²×C₄×C₁₃	C₄×C₁₃	1	6	yes	—
21	C₂×C₁₆₉	C₂×C₃²×C₄	1	3	yes	—
22	C₂×C₃×C₄×C₁₃	C₂×C₃×C₁₃	1	12	yes	—
23	C₂×C₃²×C₁₃	C₂×C₄×C₁₃	1	3	yes	—
24	C₄²×C₁₃	C₃²×C₁₃	1	1	yes	characteristic
25	C₁₆₉	C₃²×C₄²	1	1	yes	characteristic, direct product factor, Sylow
26	C₂²×C₃×C₁₃	C₂²×C₃×C₁₃	1	4	yes	—
27	C₃×C₄×C₁₃	C₃×C₄×C₁₃	1	24	yes	—
28	C₃²×C₄²	C₁₆₉	1	1	yes	characteristic, direct product factor, Hall
29	C₃²×C₁₃	C₄²×C₁₃	1	1	yes	characteristic
30	C₂×C₄×C₁₃	C₂×C₃²×C₁₃	1	3	yes	—
31	C₂×C₃×C₁₃	C₂×C₃×C₄×C₁₃	1	12	yes	—
32	C₂×C₃²×C₄	C₂×C₁₆₉	1	3	yes	—
33	C₂²×C₁₃	C₂²×C₃²×C₁₃	1	1	yes	characteristic
34	C₄×C₁₃	C₃²×C₄×C₁₃	1	6	yes	—
35	C₃×C₄²	C₃×C₁₆₉	1	4	yes	direct product factor
36	C₃×C₁₃	C₃×C₄²×C₁₃	1	4	yes	—
37	C₂²×C₃²	C₂²×C₁₆₉	1	1	yes	characteristic
38	C₃²×C₄	C₄×C₁₆₉	1	6	yes	direct product factor
39	C₂×C₁₃	C₂×C₃²×C₄×C₁₃	1	3	yes	—
40	C₂×C₃×C₄	C₂×C₃×C₁₆₉	1	12	yes	—
41	C₂×C₃²	C₂×C₄×C₁₆₉	1	3	yes	—
42	C₄²	C₃²×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
43	C₁₃	C₃²×C₄²×C₁₃	1	1	yes	characteristic
44	C₂²×C₃	C₂²×C₃×C₁₆₉	1	4	yes	—
45	C₃×C₄	C₃×C₄×C₁₆₉	1	24	yes	direct product factor
46	C₃²	C₄²×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×C₄	C₂×C₃²×C₁₆₉	1	3	yes	—
48	C₂×C₃	C₂×C₃×C₄×C₁₆₉	1	12	yes	—
49	C₂²	C₂²×C₃²×C₁₆₉	1	1	yes	characteristic
50	C₄	C₃²×C₄×C₁₆₉	1	6	yes	direct product factor
51	C₃	C₃×C₄²×C₁₆₉	1	4	yes	direct product factor
52	C₂	C₂×C₃²×C₄×C₁₆₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₁₃	C₂²×C₃²×C₁₃
socle, prime radical	C₂²×C₃²×C₁₃	C₂²×C₁₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄²×C₁₆₉	C₃	direct (N × H)
4. C₄²×C₁₆₉	C₃²	direct (N × H)
6. C₃²×C₄×C₁₆₉	C₄	direct (N × H)
9. C₃×C₄×C₁₆₉	C₃×C₄	direct (N × H)
11. C₃²×C₁₆₉	C₄²	direct (N × H)
16. C₄×C₁₆₉	C₃²×C₄	direct (N × H)
18. C₃×C₁₆₉	C₃×C₄²	direct (N × H)
25. C₁₆₉	C₃²×C₄²	direct (N × H)
28. C₃²×C₄²	C₁₆₉	direct (N × H)
35. C₃×C₄²	C₃×C₁₆₉	direct (N × H)
38. C₃²×C₄	C₄×C₁₆₉	direct (N × H)
42. C₄²	C₃²×C₁₆₉	direct (N × H)
45. C₃×C₄	C₃×C₄×C₁₆₉	direct (N × H)
46. C₃²	C₄²×C₁₆₉	direct (N × H)
50. C₄	C₃²×C₄×C₁₆₉	direct (N × H)
51. C₃	C₃×C₄²×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×C₃²×C₄×C₁₆₉	2
C₃×C₄²×C₁₆₉	3
C₃²×C₄²×C₁₃	13

Subgroups — C32×C42×C169 #72201643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄²×C₁₆₉ #72201643