Subgroups — C₃²×C₅×C₉×C₁₆ #72185608

Order 6480.

500 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₅×C₁₆	C₃	1	1	yes	maximal, characteristic
3	C₃×C₅×C₉×C₁₆	C₃	1	12	yes	maximal, direct product factor
4	C₃²×C₄×C₅×C₉	C₄	1	1	yes	characteristic
5	C₃²×C₉×C₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃³×C₅×C₈	C₂×C₃	1	1	yes	characteristic
7	C₃×C₅×C₈×C₉	C₂×C₃	1	12	yes	—
8	C₂×C₃²×C₅×C₉	C₈	1	1	yes	characteristic
9	C₃²×C₅×C₁₆	C₉	1	9	yes	direct product factor
10	C₃²×C₅×C₁₆	C₃²	1	4	yes	—
11	C₅×C₉×C₁₆	C₃²	1	9	yes	direct product factor
12	C₃²×C₈×C₉	C₂×C₅	1	1	yes	characteristic
13	C₃³×C₄×C₅	C₃×C₄	1	1	yes	characteristic
14	C₃×C₄×C₅×C₉	C₃×C₄	1	12	yes	—
15	C₃³×C₁₆	C₃×C₅	1	1	yes	characteristic
16	C₃×C₉×C₁₆	C₃×C₅	1	12	yes	direct product factor
17	C₃²×C₅×C₉	C₁₆	1	1	yes	characteristic, direct product factor, Hall
18	C₃²×C₅×C₈	C₂×C₉	1	9	yes	—
19	C₃²×C₅×C₈	C₂×C₃²	1	4	yes	—
20	C₅×C₈×C₉	C₂×C₃²	1	9	yes	—
21	C₃²×C₄×C₉	C₄×C₅	1	1	yes	characteristic
22	C₂×C₃³×C₅	C₃×C₈	1	1	yes	characteristic
23	C₂×C₃×C₅×C₉	C₃×C₈	1	12	yes	—
24	C₃×C₅×C₁₆	C₃×C₉	1	12	yes	direct product factor
25	C₃×C₅×C₁₆	C₃³	1	1	yes	characteristic
26	C₃³×C₈	C₂×C₃×C₅	1	1	yes	characteristic
27	C₃×C₈×C₉	C₂×C₃×C₅	1	12	yes	—
28	C₃²×C₄×C₅	C₄×C₉	1	9	yes	—
29	C₃²×C₄×C₅	C₃²×C₄	1	4	yes	—
30	C₄×C₅×C₉	C₃²×C₄	1	9	yes	—
31	C₂×C₃²×C₉	C₅×C₈	1	1	yes	characteristic
32	C₃²×C₁₆	C₅×C₉	1	9	yes	direct product factor
33	C₃²×C₁₆	C₃²×C₅	1	4	yes	—
34	C₉×C₁₆	C₃²×C₅	1	9	yes	direct product factor
35	C₃³×C₅	C₃×C₁₆	1	1	yes	characteristic
36	C₃×C₅×C₉	C₃×C₁₆	1	12	yes	direct product factor
37	C₃×C₅×C₈	C₂×C₃×C₉	1	12	yes	—
38	C₃×C₅×C₈	C₂×C₃³	1	1	yes	characteristic
39	C₃³×C₄	C₃×C₄×C₅	1	1	yes	characteristic
40	C₃×C₄×C₉	C₃×C₄×C₅	1	12	yes	—
41	C₂×C₃²×C₅	C₈×C₉	1	9	yes	—
42	C₂×C₃²×C₅	C₃²×C₈	1	4	yes	—
43	C₂×C₅×C₉	C₃²×C₈	1	9	yes	—
44	C₃²×C₉	C₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
45	C₅×C₁₆	C₃²×C₉	1	1	yes	characteristic, direct product factor, Hall
46	C₃²×C₈	C₂×C₅×C₉	1	9	yes	—
47	C₃²×C₈	C₂×C₃²×C₅	1	4	yes	—
48	C₈×C₉	C₂×C₃²×C₅	1	9	yes	—
49	C₃×C₄×C₅	C₃×C₄×C₉	1	12	yes	—
50	C₃×C₄×C₅	C₃³×C₄	1	1	yes	characteristic
51	C₂×C₃³	C₃×C₅×C₈	1	1	yes	characteristic
52	C₂×C₃×C₉	C₃×C₅×C₈	1	12	yes	—
53	C₃×C₁₆	C₃×C₅×C₉	1	12	yes	direct product factor
54	C₃×C₁₆	C₃³×C₅	1	1	yes	characteristic
55	C₃²×C₅	C₉×C₁₆	1	9	yes	direct product factor
56	C₃²×C₅	C₃²×C₁₆	1	4	yes	—
57	C₅×C₉	C₃²×C₁₆	1	9	yes	direct product factor
58	C₅×C₈	C₂×C₃²×C₉	1	1	yes	characteristic
59	C₃²×C₄	C₄×C₅×C₉	1	9	yes	—
60	C₃²×C₄	C₃²×C₄×C₅	1	4	yes	—
61	C₄×C₉	C₃²×C₄×C₅	1	9	yes	—
62	C₂×C₃×C₅	C₃×C₈×C₉	1	12	yes	—
63	C₂×C₃×C₅	C₃³×C₈	1	1	yes	characteristic
64	C₃³	C₃×C₅×C₁₆	1	1	yes	characteristic
65	C₃×C₉	C₃×C₅×C₁₆	1	12	yes	direct product factor
66	C₃×C₈	C₂×C₃×C₅×C₉	1	12	yes	—
67	C₃×C₈	C₂×C₃³×C₅	1	1	yes	characteristic
68	C₄×C₅	C₃²×C₄×C₉	1	1	yes	characteristic
69	C₂×C₃²	C₅×C₈×C₉	1	9	yes	—
70	C₂×C₃²	C₃²×C₅×C₈	1	4	yes	—
71	C₂×C₉	C₃²×C₅×C₈	1	9	yes	—
72	C₁₆	C₃²×C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
73	C₃×C₅	C₃×C₉×C₁₆	1	12	yes	direct product factor
74	C₃×C₅	C₃³×C₁₆	1	1	yes	characteristic
75	C₃×C₄	C₃×C₄×C₅×C₉	1	12	yes	—
76	C₃×C₄	C₃³×C₄×C₅	1	1	yes	characteristic
77	C₂×C₅	C₃²×C₈×C₉	1	1	yes	characteristic
78	C₃²	C₅×C₉×C₁₆	1	9	yes	direct product factor
79	C₃²	C₃²×C₅×C₁₆	1	4	yes	—
80	C₉	C₃²×C₅×C₁₆	1	9	yes	direct product factor
81	C₈	C₂×C₃²×C₅×C₉	1	1	yes	characteristic
82	C₂×C₃	C₃×C₅×C₈×C₉	1	12	yes	—
83	C₂×C₃	C₃³×C₅×C₈	1	1	yes	characteristic
84	C₅	C₃²×C₉×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
85	C₄	C₃²×C₄×C₅×C₉	1	1	yes	characteristic
86	C₃	C₃×C₅×C₉×C₁₆	1	12	yes	direct product factor
87	C₃	C₃³×C₅×C₁₆	1	1	yes	characteristic
88	C₂	C₃²×C₅×C₈×C₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₈	C₂×C₃³×C₅
socle, prime radical	C₂×C₃³×C₅	C₃×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₉×C₁₆	C₃	direct (N × H)
5. C₃²×C₉×C₁₆	C₅	direct (N × H)
9. C₃²×C₅×C₁₆	C₉	direct (N × H)
11. C₅×C₉×C₁₆	C₃²	direct (N × H)
16. C₃×C₉×C₁₆	C₃×C₅	direct (N × H)
17. C₃²×C₅×C₉	C₁₆	direct (N × H)
24. C₃×C₅×C₁₆	C₃×C₉	direct (N × H)
32. C₃²×C₁₆	C₅×C₉	direct (N × H)
34. C₉×C₁₆	C₃²×C₅	direct (N × H)
36. C₃×C₅×C₉	C₃×C₁₆	direct (N × H)
44. C₃²×C₉	C₅×C₁₆	direct (N × H)
45. C₅×C₁₆	C₃²×C₉	direct (N × H)
53. C₃×C₁₆	C₃×C₅×C₉	direct (N × H)
55. C₃²×C₅	C₉×C₁₆	direct (N × H)
57. C₅×C₉	C₃²×C₁₆	direct (N × H)
65. C₃×C₉	C₃×C₅×C₁₆	direct (N × H)
72. C₁₆	C₃²×C₅×C₉	direct (N × H)
73. C₃×C₅	C₃×C₉×C₁₆	direct (N × H)
78. C₃²	C₅×C₉×C₁₆	direct (N × H)
80. C₉	C₃²×C₅×C₁₆	direct (N × H)
84. C₅	C₃²×C₉×C₁₆	direct (N × H)
86. C₃	C₃×C₅×C₉×C₁₆	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×C₅×C₈×C₉	2
C₃³×C₅×C₁₆	3
C₃×C₅×C₉×C₁₆	3
C₃²×C₉×C₁₆	5

Subgroups — C32×C5×C9×C16 #72185608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×C₉×C₁₆ #72185608