Subgroups — C₃²×C₅×C₁₉×C₃₂ #72204396

Order 27360.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₁₉×C₃₂	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×C₈×C₁₉	C₄	1	1	yes	characteristic
4	C₃²×C₁₉×C₃₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×C₁₆×C₁₉	C₂×C₃	1	4	yes	—
6	C₅×C₁₉×C₃₂	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×C₁₆×C₁₉	C₂×C₅	1	1	yes	characteristic
8	C₃×C₅×C₈×C₁₉	C₃×C₄	1	4	yes	—
9	C₃²×C₄×C₅×C₁₉	C₈	1	1	yes	characteristic
10	C₃×C₁₉×C₃₂	C₃×C₅	1	4	yes	direct product factor
11	C₂×C₃²×C₅×C₁₉	C₁₆	1	1	yes	characteristic
12	C₅×C₁₆×C₁₉	C₂×C₃²	1	1	yes	characteristic
13	C₃²×C₅×C₃₂	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₃²×C₈×C₁₉	C₄×C₅	1	1	yes	characteristic
15	C₃×C₄×C₅×C₁₉	C₃×C₈	1	4	yes	—
16	C₃×C₁₆×C₁₉	C₂×C₃×C₅	1	4	yes	—
17	C₃²×C₅×C₁₉	C₃₂	1	1	yes	characteristic, direct product factor, Hall
18	C₅×C₈×C₁₉	C₃²×C₄	1	1	yes	characteristic
19	C₃²×C₅×C₁₆	C₂×C₁₉	1	1	yes	characteristic
20	C₃²×C₄×C₁₉	C₅×C₈	1	1	yes	characteristic
21	C₁₉×C₃₂	C₃²×C₅	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₃×C₅×C₁₉	C₃×C₁₆	1	4	yes	—
23	C₃×C₅×C₃₂	C₃×C₁₉	1	4	yes	direct product factor
24	C₃×C₈×C₁₉	C₃×C₄×C₅	1	4	yes	—
25	C₄×C₅×C₁₉	C₃²×C₈	1	1	yes	characteristic
26	C₃²×C₅×C₈	C₄×C₁₉	1	1	yes	characteristic
27	C₂×C₃²×C₁₉	C₅×C₁₆	1	1	yes	characteristic
28	C₁₆×C₁₉	C₂×C₃²×C₅	1	1	yes	characteristic
29	C₃²×C₃₂	C₅×C₁₉	1	1	yes	characteristic, direct product factor, Hall
30	C₃×C₅×C₁₉	C₃×C₃₂	1	4	yes	direct product factor
31	C₃×C₅×C₁₆	C₂×C₃×C₁₉	1	4	yes	—
32	C₃×C₄×C₁₉	C₃×C₅×C₈	1	4	yes	—
33	C₂×C₅×C₁₉	C₃²×C₁₆	1	1	yes	characteristic
34	C₃²×C₄×C₅	C₈×C₁₉	1	1	yes	characteristic
35	C₃²×C₁₉	C₅×C₃₂	1	1	yes	characteristic, direct product factor, Hall
36	C₅×C₃₂	C₃²×C₁₉	1	1	yes	characteristic, direct product factor, Hall
37	C₈×C₁₉	C₃²×C₄×C₅	1	1	yes	characteristic
38	C₃²×C₁₆	C₂×C₅×C₁₉	1	1	yes	characteristic
39	C₃×C₅×C₈	C₃×C₄×C₁₉	1	4	yes	—
40	C₂×C₃×C₁₉	C₃×C₅×C₁₆	1	4	yes	—
41	C₃×C₃₂	C₃×C₅×C₁₉	1	4	yes	direct product factor
42	C₅×C₁₉	C₃²×C₃₂	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₃²×C₅	C₁₆×C₁₉	1	1	yes	characteristic
44	C₅×C₁₆	C₂×C₃²×C₁₉	1	1	yes	characteristic
45	C₄×C₁₉	C₃²×C₅×C₈	1	1	yes	characteristic
46	C₃²×C₈	C₄×C₅×C₁₉	1	1	yes	characteristic
47	C₃×C₄×C₅	C₃×C₈×C₁₉	1	4	yes	—
48	C₃×C₁₉	C₃×C₅×C₃₂	1	4	yes	direct product factor
49	C₃×C₁₆	C₂×C₃×C₅×C₁₉	1	4	yes	—
50	C₃²×C₅	C₁₉×C₃₂	1	1	yes	characteristic, direct product factor, Hall
51	C₅×C₈	C₃²×C₄×C₁₉	1	1	yes	characteristic
52	C₂×C₁₉	C₃²×C₅×C₁₆	1	1	yes	characteristic
53	C₃²×C₄	C₅×C₈×C₁₉	1	1	yes	characteristic
54	C₃₂	C₃²×C₅×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×C₃×C₅	C₃×C₁₆×C₁₉	1	4	yes	—
56	C₃×C₈	C₃×C₄×C₅×C₁₉	1	4	yes	—
57	C₄×C₅	C₃²×C₈×C₁₉	1	1	yes	characteristic
58	C₁₉	C₃²×C₅×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₃²	C₅×C₁₆×C₁₉	1	1	yes	characteristic
60	C₁₆	C₂×C₃²×C₅×C₁₉	1	1	yes	characteristic
61	C₃×C₅	C₃×C₁₉×C₃₂	1	4	yes	direct product factor
62	C₃×C₄	C₃×C₅×C₈×C₁₉	1	4	yes	—
63	C₂×C₅	C₃²×C₁₆×C₁₉	1	1	yes	characteristic
64	C₃²	C₅×C₁₉×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
65	C₈	C₃²×C₄×C₅×C₁₉	1	1	yes	characteristic
66	C₂×C₃	C₃×C₅×C₁₆×C₁₉	1	4	yes	—
67	C₅	C₃²×C₁₉×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
68	C₄	C₃²×C₅×C₈×C₁₉	1	1	yes	characteristic
69	C₃	C₃×C₅×C₁₉×C₃₂	1	4	yes	direct product factor
70	C₂	C₃²×C₅×C₁₆×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂×C₃²×C₅×C₁₉
socle, prime radical	C₂×C₃²×C₅×C₁₉	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₁₉×C₃₂	C₃	direct (N × H)
4. C₃²×C₁₉×C₃₂	C₅	direct (N × H)
6. C₅×C₁₉×C₃₂	C₃²	direct (N × H)
10. C₃×C₁₉×C₃₂	C₃×C₅	direct (N × H)
13. C₃²×C₅×C₃₂	C₁₉	direct (N × H)
17. C₃²×C₅×C₁₉	C₃₂	direct (N × H)
21. C₁₉×C₃₂	C₃²×C₅	direct (N × H)
23. C₃×C₅×C₃₂	C₃×C₁₉	direct (N × H)
29. C₃²×C₃₂	C₅×C₁₉	direct (N × H)
30. C₃×C₅×C₁₉	C₃×C₃₂	direct (N × H)
35. C₃²×C₁₉	C₅×C₃₂	direct (N × H)
36. C₅×C₃₂	C₃²×C₁₉	direct (N × H)
41. C₃×C₃₂	C₃×C₅×C₁₉	direct (N × H)
42. C₅×C₁₉	C₃²×C₃₂	direct (N × H)
48. C₃×C₁₉	C₃×C₅×C₃₂	direct (N × H)
50. C₃²×C₅	C₁₉×C₃₂	direct (N × H)
54. C₃₂	C₃²×C₅×C₁₉	direct (N × H)
58. C₁₉	C₃²×C₅×C₃₂	direct (N × H)
61. C₃×C₅	C₃×C₁₉×C₃₂	direct (N × H)
64. C₃²	C₅×C₁₉×C₃₂	direct (N × H)
67. C₅	C₃²×C₁₉×C₃₂	direct (N × H)
69. C₃	C₃×C₅×C₁₉×C₃₂	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×C₁₆×C₁₉	2
C₃×C₅×C₁₉×C₃₂	3
C₃²×C₁₉×C₃₂	5
C₃²×C₅×C₃₂	19

Subgroups — C32×C5×C19×C32 #72204396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×C₁₉×C₃₂ #72204396