Subgroups — C₃×C₅×C₁₆×C₂₇ #72185607

Order 6480.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₈×C₂₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₉×C₁₆	C₃	1	1	yes	maximal, characteristic
3	C₅×C₁₆×C₂₇	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₄×C₅×C₂₇	C₄	1	1	yes	characteristic
5	C₃×C₁₆×C₂₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₅×C₈×C₉	C₂×C₃	1	1	yes	characteristic
7	C₅×C₈×C₂₇	C₂×C₃	1	3	yes	—
8	C₂×C₃×C₅×C₂₇	C₈	1	1	yes	characteristic
9	C₃²×C₅×C₁₆	C₉	1	1	yes	characteristic
10	C₅×C₉×C₁₆	C₉	1	2	yes	—
11	C₅×C₉×C₁₆	C₃²	1	1	yes	characteristic
12	C₃×C₈×C₂₇	C₂×C₅	1	1	yes	characteristic
13	C₃×C₄×C₅×C₉	C₃×C₄	1	1	yes	characteristic
14	C₄×C₅×C₂₇	C₃×C₄	1	3	yes	—
15	C₃×C₉×C₁₆	C₃×C₅	1	1	yes	characteristic
16	C₁₆×C₂₇	C₃×C₅	1	3	yes	direct product factor
17	C₃×C₅×C₂₇	C₁₆	1	1	yes	characteristic, direct product factor, Hall
18	C₃²×C₅×C₈	C₂×C₉	1	1	yes	characteristic
19	C₅×C₈×C₉	C₂×C₉	1	2	yes	—
20	C₅×C₈×C₉	C₂×C₃²	1	1	yes	characteristic
21	C₃×C₄×C₂₇	C₄×C₅	1	1	yes	characteristic
22	C₂×C₃×C₅×C₉	C₃×C₈	1	1	yes	characteristic
23	C₂×C₅×C₂₇	C₃×C₈	1	3	yes	—
24	C₃×C₅×C₁₆	C₂₇	1	3	yes	direct product factor
25	C₃×C₅×C₁₆	C₃×C₉	1	1	yes	characteristic
26	C₃×C₈×C₉	C₂×C₃×C₅	1	1	yes	characteristic
27	C₈×C₂₇	C₂×C₃×C₅	1	3	yes	—
28	C₃²×C₄×C₅	C₄×C₉	1	1	yes	characteristic
29	C₄×C₅×C₉	C₄×C₉	1	2	yes	—
30	C₄×C₅×C₉	C₃²×C₄	1	1	yes	characteristic
31	C₂×C₃×C₂₇	C₅×C₈	1	1	yes	characteristic
32	C₃²×C₁₆	C₅×C₉	1	1	yes	characteristic
33	C₉×C₁₆	C₅×C₉	1	2	yes	—
34	C₉×C₁₆	C₃²×C₅	1	1	yes	characteristic
35	C₃×C₅×C₉	C₃×C₁₆	1	1	yes	characteristic
36	C₅×C₂₇	C₃×C₁₆	1	3	yes	direct product factor
37	C₃×C₅×C₈	C₂×C₂₇	1	3	yes	—
38	C₃×C₅×C₈	C₂×C₃×C₉	1	1	yes	characteristic
39	C₃×C₄×C₉	C₃×C₄×C₅	1	1	yes	characteristic
40	C₄×C₂₇	C₃×C₄×C₅	1	3	yes	—
41	C₂×C₃²×C₅	C₈×C₉	1	1	yes	characteristic
42	C₂×C₅×C₉	C₈×C₉	1	2	yes	—
43	C₂×C₅×C₉	C₃²×C₈	1	1	yes	characteristic
44	C₃×C₂₇	C₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
45	C₅×C₁₆	C₃×C₂₇	1	1	yes	characteristic, direct product factor, Hall
46	C₃²×C₈	C₂×C₅×C₉	1	1	yes	characteristic
47	C₈×C₉	C₂×C₅×C₉	1	2	yes	—
48	C₈×C₉	C₂×C₃²×C₅	1	1	yes	characteristic
49	C₃×C₄×C₅	C₄×C₂₇	1	3	yes	—
50	C₃×C₄×C₅	C₃×C₄×C₉	1	1	yes	characteristic
51	C₂×C₃×C₉	C₃×C₅×C₈	1	1	yes	characteristic
52	C₂×C₂₇	C₃×C₅×C₈	1	3	yes	—
53	C₃×C₁₆	C₅×C₂₇	1	3	yes	direct product factor
54	C₃×C₁₆	C₃×C₅×C₉	1	1	yes	characteristic
55	C₃²×C₅	C₉×C₁₆	1	1	yes	characteristic
56	C₅×C₉	C₉×C₁₆	1	2	yes	—
57	C₅×C₉	C₃²×C₁₆	1	1	yes	characteristic
58	C₅×C₈	C₂×C₃×C₂₇	1	1	yes	characteristic
59	C₃²×C₄	C₄×C₅×C₉	1	1	yes	characteristic
60	C₄×C₉	C₄×C₅×C₉	1	2	yes	—
61	C₄×C₉	C₃²×C₄×C₅	1	1	yes	characteristic
62	C₂×C₃×C₅	C₈×C₂₇	1	3	yes	—
63	C₂×C₃×C₅	C₃×C₈×C₉	1	1	yes	characteristic
64	C₃×C₉	C₃×C₅×C₁₆	1	1	yes	characteristic
65	C₂₇	C₃×C₅×C₁₆	1	3	yes	direct product factor
66	C₃×C₈	C₂×C₅×C₂₇	1	3	yes	—
67	C₃×C₈	C₂×C₃×C₅×C₉	1	1	yes	characteristic
68	C₄×C₅	C₃×C₄×C₂₇	1	1	yes	characteristic
69	C₂×C₃²	C₅×C₈×C₉	1	1	yes	characteristic
70	C₂×C₉	C₅×C₈×C₉	1	2	yes	—
71	C₂×C₉	C₃²×C₅×C₈	1	1	yes	characteristic
72	C₁₆	C₃×C₅×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
73	C₃×C₅	C₁₆×C₂₇	1	3	yes	direct product factor
74	C₃×C₅	C₃×C₉×C₁₆	1	1	yes	characteristic
75	C₃×C₄	C₄×C₅×C₂₇	1	3	yes	—
76	C₃×C₄	C₃×C₄×C₅×C₉	1	1	yes	characteristic
77	C₂×C₅	C₃×C₈×C₂₇	1	1	yes	characteristic
78	C₃²	C₅×C₉×C₁₆	1	1	yes	characteristic
79	C₉	C₅×C₉×C₁₆	1	2	yes	—
80	C₉	C₃²×C₅×C₁₆	1	1	yes	characteristic
81	C₈	C₂×C₃×C₅×C₂₇	1	1	yes	characteristic
82	C₂×C₃	C₅×C₈×C₂₇	1	3	yes	—
83	C₂×C₃	C₃×C₅×C₈×C₉	1	1	yes	characteristic
84	C₅	C₃×C₁₆×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
85	C₄	C₃×C₄×C₅×C₂₇	1	1	yes	characteristic
86	C₃	C₅×C₁₆×C₂₇	1	3	yes	direct product factor
87	C₃	C₃×C₅×C₉×C₁₆	1	1	yes	characteristic
88	C₂	C₃×C₅×C₈×C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈×C₉	C₂×C₃²×C₅
socle, prime radical	C₂×C₃²×C₅	C₈×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×C₁₆×C₂₇	C₃	direct (N × H)
5. C₃×C₁₆×C₂₇	C₅	direct (N × H)
16. C₁₆×C₂₇	C₃×C₅	direct (N × H)
17. C₃×C₅×C₂₇	C₁₆	direct (N × H)
24. C₃×C₅×C₁₆	C₂₇	direct (N × H)
36. C₅×C₂₇	C₃×C₁₆	direct (N × H)
44. C₃×C₂₇	C₅×C₁₆	direct (N × H)
45. C₅×C₁₆	C₃×C₂₇	direct (N × H)
53. C₃×C₁₆	C₅×C₂₇	direct (N × H)
65. C₂₇	C₃×C₅×C₁₆	direct (N × H)
72. C₁₆	C₃×C₅×C₂₇	direct (N × H)
73. C₃×C₅	C₁₆×C₂₇	direct (N × H)
84. C₅	C₃×C₁₆×C₂₇	direct (N × H)
86. C₃	C₅×C₁₆×C₂₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×C₈×C₂₇	2
C₃×C₅×C₉×C₁₆	3
C₅×C₁₆×C₂₇	3
C₃×C₁₆×C₂₇	5

Subgroups — C3×C5×C16×C27 #72185607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₁₆×C₂₇ #72185607