Subgroups — C₂×C₃³×C₁₇×C₂₇ #72202078

Order 24786.

2320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₉×C₁₇	C₃	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₁₇×C₂₇	C₃	1	39	yes	maximal, direct product factor
3	C₂×C₃⁴×C₁₇	C₉	1	1	yes	characteristic
4	C₂×C₃²×C₉×C₁₇	C₉	1	26	yes	—
5	C₂×C₃²×C₉×C₁₇	C₃²	1	13	yes	—
6	C₂×C₃×C₁₇×C₂₇	C₃²	1	117	yes	direct product factor
7	C₃³×C₁₇×C₂₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃³×C₉×C₁₇	C₂×C₃	1	1	yes	characteristic
9	C₃²×C₁₇×C₂₇	C₂×C₃	1	39	yes	direct product factor
10	C₂×C₃³×C₂₇	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃⁴×C₁₇	C₂×C₉	1	1	yes	characteristic
12	C₃²×C₉×C₁₇	C₂×C₉	1	26	yes	—
13	C₃²×C₉×C₁₇	C₂×C₃²	1	13	yes	—
14	C₃×C₁₇×C₂₇	C₂×C₃²	1	117	yes	direct product factor
15	C₂×C₃³×C₁₇	C₂₇	1	27	yes	direct product factor
16	C₂×C₃³×C₁₇	C₃×C₉	1	13	yes	—
17	C₂×C₃×C₉×C₁₇	C₃×C₉	1	104	yes	—
18	C₂×C₃×C₉×C₁₇	C₃³	1	13	yes	—
19	C₂×C₁₇×C₂₇	C₃³	1	27	yes	direct product factor
20	C₃³×C₂₇	C₂×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
21	C₂×C₃³×C₉	C₃×C₁₇	1	1	yes	characteristic
22	C₂×C₃²×C₂₇	C₃×C₁₇	1	39	yes	direct product factor
23	C₃³×C₁₇	C₂×C₂₇	1	27	yes	direct product factor
24	C₃³×C₁₇	C₂×C₃×C₉	1	13	yes	—
25	C₃×C₉×C₁₇	C₂×C₃×C₉	1	104	yes	—
26	C₃×C₉×C₁₇	C₂×C₃³	1	13	yes	—
27	C₁₇×C₂₇	C₂×C₃³	1	27	yes	direct product factor
28	C₂×C₃²×C₁₇	C₃×C₂₇	1	117	yes	direct product factor
29	C₂×C₃²×C₁₇	C₃²×C₉	1	13	yes	—
30	C₂×C₉×C₁₇	C₃²×C₉	1	26	yes	—
31	C₂×C₉×C₁₇	C₃⁴	1	1	yes	characteristic
32	C₃³×C₉	C₂×C₃×C₁₇	1	1	yes	characteristic
33	C₃²×C₂₇	C₂×C₃×C₁₇	1	39	yes	direct product factor
34	C₂×C₃⁴	C₉×C₁₇	1	1	yes	characteristic
35	C₂×C₃²×C₉	C₉×C₁₇	1	26	yes	—
36	C₂×C₃²×C₉	C₃²×C₁₇	1	13	yes	—
37	C₂×C₃×C₂₇	C₃²×C₁₇	1	117	yes	direct product factor
38	C₃²×C₁₇	C₂×C₃×C₂₇	1	117	yes	direct product factor
39	C₃²×C₁₇	C₂×C₃²×C₉	1	13	yes	—
40	C₉×C₁₇	C₂×C₃²×C₉	1	26	yes	—
41	C₉×C₁₇	C₂×C₃⁴	1	1	yes	characteristic
42	C₂×C₃×C₁₇	C₃²×C₂₇	1	39	yes	direct product factor
43	C₂×C₃×C₁₇	C₃³×C₉	1	1	yes	characteristic
44	C₃⁴	C₂×C₉×C₁₇	1	1	yes	characteristic
45	C₃²×C₉	C₂×C₉×C₁₇	1	26	yes	—
46	C₃²×C₉	C₂×C₃²×C₁₇	1	13	yes	—
47	C₃×C₂₇	C₂×C₃²×C₁₇	1	117	yes	direct product factor
48	C₂×C₃³	C₁₇×C₂₇	1	27	yes	direct product factor
49	C₂×C₃³	C₃×C₉×C₁₇	1	13	yes	—
50	C₂×C₃×C₉	C₃×C₉×C₁₇	1	104	yes	—
51	C₂×C₃×C₉	C₃³×C₁₇	1	13	yes	—
52	C₂×C₂₇	C₃³×C₁₇	1	27	yes	direct product factor
53	C₃×C₁₇	C₂×C₃²×C₂₇	1	39	yes	direct product factor
54	C₃×C₁₇	C₂×C₃³×C₉	1	1	yes	characteristic
55	C₂×C₁₇	C₃³×C₂₇	1	1	yes	characteristic, direct product factor, Hall
56	C₃³	C₂×C₁₇×C₂₇	1	27	yes	direct product factor
57	C₃³	C₂×C₃×C₉×C₁₇	1	13	yes	—
58	C₃×C₉	C₂×C₃×C₉×C₁₇	1	104	yes	—
59	C₃×C₉	C₂×C₃³×C₁₇	1	13	yes	—
60	C₂₇	C₂×C₃³×C₁₇	1	27	yes	direct product factor
61	C₂×C₃²	C₃×C₁₇×C₂₇	1	117	yes	direct product factor
62	C₂×C₃²	C₃²×C₉×C₁₇	1	13	yes	—
63	C₂×C₉	C₃²×C₉×C₁₇	1	26	yes	—
64	C₂×C₉	C₃⁴×C₁₇	1	1	yes	characteristic
65	C₁₇	C₂×C₃³×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
66	C₃²	C₂×C₃×C₁₇×C₂₇	1	117	yes	direct product factor
67	C₃²	C₂×C₃²×C₉×C₁₇	1	13	yes	—
68	C₉	C₂×C₃²×C₉×C₁₇	1	26	yes	—
69	C₉	C₂×C₃⁴×C₁₇	1	1	yes	characteristic
70	C₂×C₃	C₃²×C₁₇×C₂₇	1	39	yes	direct product factor
71	C₂×C₃	C₃³×C₉×C₁₇	1	1	yes	characteristic
72	C₃	C₂×C₃²×C₁₇×C₂₇	1	39	yes	direct product factor
73	C₃	C₂×C₃³×C₉×C₁₇	1	1	yes	characteristic
74	C₂	C₃³×C₁₇×C₂₇	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂×C₃⁴×C₁₇
socle, prime radical	C₂×C₃⁴×C₁₇	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×C₁₇×C₂₇	C₃	direct (N × H)
6. C₂×C₃×C₁₇×C₂₇	C₃²	direct (N × H)
7. C₃³×C₁₇×C₂₇	C₂	direct (N × H)
9. C₃²×C₁₇×C₂₇	C₂×C₃	direct (N × H)
10. C₂×C₃³×C₂₇	C₁₇	direct (N × H)
14. C₃×C₁₇×C₂₇	C₂×C₃²	direct (N × H)
15. C₂×C₃³×C₁₇	C₂₇	direct (N × H)
19. C₂×C₁₇×C₂₇	C₃³	direct (N × H)
20. C₃³×C₂₇	C₂×C₁₇	direct (N × H)
22. C₂×C₃²×C₂₇	C₃×C₁₇	direct (N × H)
23. C₃³×C₁₇	C₂×C₂₇	direct (N × H)
27. C₁₇×C₂₇	C₂×C₃³	direct (N × H)
28. C₂×C₃²×C₁₇	C₃×C₂₇	direct (N × H)
33. C₃²×C₂₇	C₂×C₃×C₁₇	direct (N × H)
37. C₂×C₃×C₂₇	C₃²×C₁₇	direct (N × H)
38. C₃²×C₁₇	C₂×C₃×C₂₇	direct (N × H)
42. C₂×C₃×C₁₇	C₃²×C₂₇	direct (N × H)
47. C₃×C₂₇	C₂×C₃²×C₁₇	direct (N × H)
48. C₂×C₃³	C₁₇×C₂₇	direct (N × H)
52. C₂×C₂₇	C₃³×C₁₇	direct (N × H)
53. C₃×C₁₇	C₂×C₃²×C₂₇	direct (N × H)
55. C₂×C₁₇	C₃³×C₂₇	direct (N × H)
56. C₃³	C₂×C₁₇×C₂₇	direct (N × H)
60. C₂₇	C₂×C₃³×C₁₇	direct (N × H)
61. C₂×C₃²	C₃×C₁₇×C₂₇	direct (N × H)
65. C₁₇	C₂×C₃³×C₂₇	direct (N × H)
66. C₃²	C₂×C₃×C₁₇×C₂₇	direct (N × H)
70. C₂×C₃	C₃²×C₁₇×C₂₇	direct (N × H)
72. C₃	C₂×C₃²×C₁₇×C₂₇	direct (N × H)
74. C₂	C₃³×C₁₇×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
C₂×C₃³×C₉×C₁₇	3
C₂×C₃²×C₁₇×C₂₇	3
C₃³×C₁₇×C₂₇	2
C₂×C₃³×C₂₇	17

Subgroups — C2×C33×C17×C27 #72202078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×C₁₇×C₂₇ #72202078