Subgroups — C₃²×C₁₆×C₁₆₉ #72201635

Order 24336.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×C₁₆₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₆×C₁₆₉	C₃	1	4	yes	maximal, direct product factor
3	C₃×C₈×C₁₆₉	C₂×C₃	1	4	yes	—
4	C₁₆×C₁₆₉	C₃²	1	1	yes	characteristic, direct product factor, Hall
5	C₃²×C₄×C₁₆₉	C₄	1	1	yes	characteristic
6	C₂×C₃²×C₁₆₉	C₈	1	1	yes	characteristic
7	C₃×C₄×C₁₆₉	C₃×C₄	1	4	yes	—
8	C₃²×C₁₃×C₁₆	C₁₃	1	1	yes	maximal, characteristic
9	C₃²×C₁₆₉	C₁₆	1	1	yes	characteristic, direct product factor, Hall
10	C₈×C₁₆₉	C₂×C₃²	1	1	yes	characteristic
11	C₂×C₃×C₁₆₉	C₃×C₈	1	4	yes	—
12	C₃²×C₈×C₁₃	C₂×C₁₃	1	1	yes	characteristic
13	C₄×C₁₆₉	C₃²×C₄	1	1	yes	characteristic
14	C₃×C₁₃×C₁₆	C₃×C₁₃	1	4	yes	—
15	C₃×C₁₆₉	C₃×C₁₆	1	4	yes	direct product factor
16	C₃²×C₄×C₁₃	C₄×C₁₃	1	1	yes	characteristic
17	C₂×C₁₆₉	C₃²×C₈	1	1	yes	characteristic
18	C₃×C₈×C₁₃	C₂×C₃×C₁₃	1	4	yes	—
19	C₂×C₃²×C₁₃	C₈×C₁₃	1	1	yes	characteristic
20	C₁₃×C₁₆	C₃²×C₁₃	1	1	yes	characteristic
21	C₁₆₉	C₃²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
22	C₃×C₄×C₁₃	C₃×C₄×C₁₃	1	4	yes	—
23	C₃²×C₁₆	C₁₆₉	1	1	yes	characteristic, direct product factor, Hall
24	C₃²×C₁₃	C₁₃×C₁₆	1	1	yes	characteristic
25	C₈×C₁₃	C₂×C₃²×C₁₃	1	1	yes	characteristic
26	C₂×C₃×C₁₃	C₃×C₈×C₁₃	1	4	yes	—
27	C₃²×C₈	C₂×C₁₆₉	1	1	yes	characteristic
28	C₄×C₁₃	C₃²×C₄×C₁₃	1	1	yes	characteristic
29	C₃×C₁₆	C₃×C₁₆₉	1	4	yes	direct product factor
30	C₃×C₁₃	C₃×C₁₃×C₁₆	1	4	yes	—
31	C₃²×C₄	C₄×C₁₆₉	1	1	yes	characteristic
32	C₂×C₁₃	C₃²×C₈×C₁₃	1	1	yes	characteristic
33	C₃×C₈	C₂×C₃×C₁₆₉	1	4	yes	—
34	C₂×C₃²	C₈×C₁₆₉	1	1	yes	characteristic
35	C₁₆	C₃²×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
36	C₁₃	C₃²×C₁₃×C₁₆	1	1	yes	characteristic
37	C₃×C₄	C₃×C₄×C₁₆₉	1	4	yes	—
38	C₃²	C₁₆×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₈	C₂×C₃²×C₁₆₉	1	1	yes	characteristic
40	C₂×C₃	C₃×C₈×C₁₆₉	1	4	yes	—
41	C₄	C₃²×C₄×C₁₆₉	1	1	yes	characteristic
42	C₃	C₃×C₁₆×C₁₆₉	1	4	yes	direct product factor
43	C₂	C₃²×C₈×C₁₆₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈×C₁₃	C₂×C₃²×C₁₃
socle, prime radical	C₂×C₃²×C₁₃	C₈×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×C₁₆×C₁₆₉	C₃	direct (N × H)
4. C₁₆×C₁₆₉	C₃²	direct (N × H)
9. C₃²×C₁₆₉	C₁₆	direct (N × H)
15. C₃×C₁₆₉	C₃×C₁₆	direct (N × H)
21. C₁₆₉	C₃²×C₁₆	direct (N × H)
23. C₃²×C₁₆	C₁₆₉	direct (N × H)
29. C₃×C₁₆	C₃×C₁₆₉	direct (N × H)
35. C₁₆	C₃²×C₁₆₉	direct (N × H)
38. C₃²	C₁₆×C₁₆₉	direct (N × H)
42. C₃	C₃×C₁₆×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
C₃²×C₈×C₁₆₉	2
C₃×C₁₆×C₁₆₉	3
C₃²×C₁₃×C₁₆	13

Subgroups — C32×C16×C169 #72201635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₁₆×C₁₆₉ #72201635