Subgroups — C₃×C₅×C₁₇×C₈₁ #72198239

Order 20655.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₁₇×C₂₇	C₃	1	1	yes	maximal, characteristic
2	C₅×C₁₇×C₈₁	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₅×C₉×C₁₇	C₉	1	1	yes	characteristic
4	C₅×C₁₇×C₂₇	C₉	1	2	yes	—
5	C₅×C₁₇×C₂₇	C₃²	1	1	yes	characteristic
6	C₃×C₁₇×C₈₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₁₇×C₂₇	C₃×C₅	1	1	yes	characteristic
8	C₁₇×C₈₁	C₃×C₅	1	3	yes	direct product factor
9	C₃×C₅×C₈₁	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃²×C₅×C₁₇	C₂₇	1	1	yes	characteristic
11	C₅×C₉×C₁₇	C₂₇	1	2	yes	—
12	C₅×C₉×C₁₇	C₃×C₉	1	1	yes	characteristic
13	C₃×C₉×C₁₇	C₅×C₉	1	1	yes	characteristic
14	C₁₇×C₂₇	C₅×C₉	1	2	yes	—
15	C₁₇×C₂₇	C₃²×C₅	1	1	yes	characteristic
16	C₃×C₅×C₂₇	C₃×C₁₇	1	1	yes	characteristic
17	C₅×C₈₁	C₃×C₁₇	1	3	yes	direct product factor
18	C₃×C₅×C₁₇	C₈₁	1	3	yes	direct product factor
19	C₃×C₅×C₁₇	C₃×C₂₇	1	1	yes	characteristic
20	C₃×C₈₁	C₅×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
21	C₃²×C₁₇	C₅×C₂₇	1	1	yes	characteristic
22	C₉×C₁₇	C₅×C₂₇	1	2	yes	—
23	C₉×C₁₇	C₃×C₅×C₉	1	1	yes	characteristic
24	C₃×C₅×C₉	C₉×C₁₇	1	1	yes	characteristic
25	C₅×C₂₇	C₉×C₁₇	1	2	yes	—
26	C₅×C₂₇	C₃²×C₁₇	1	1	yes	characteristic
27	C₅×C₁₇	C₃×C₈₁	1	1	yes	characteristic, direct product factor, Hall
28	C₃×C₂₇	C₃×C₅×C₁₇	1	1	yes	characteristic
29	C₈₁	C₃×C₅×C₁₇	1	3	yes	direct product factor
30	C₃×C₁₇	C₅×C₈₁	1	3	yes	direct product factor
31	C₃×C₁₇	C₃×C₅×C₂₇	1	1	yes	characteristic
32	C₃²×C₅	C₁₇×C₂₇	1	1	yes	characteristic
33	C₅×C₉	C₁₇×C₂₇	1	2	yes	—
34	C₅×C₉	C₃×C₉×C₁₇	1	1	yes	characteristic
35	C₃×C₉	C₅×C₉×C₁₇	1	1	yes	characteristic
36	C₂₇	C₅×C₉×C₁₇	1	2	yes	—
37	C₂₇	C₃²×C₅×C₁₇	1	1	yes	characteristic
38	C₁₇	C₃×C₅×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
39	C₃×C₅	C₁₇×C₈₁	1	3	yes	direct product factor
40	C₃×C₅	C₃×C₁₇×C₂₇	1	1	yes	characteristic
41	C₃²	C₅×C₁₇×C₂₇	1	1	yes	characteristic
42	C₉	C₅×C₁₇×C₂₇	1	2	yes	—
43	C₉	C₃×C₅×C₉×C₁₇	1	1	yes	characteristic
44	C₅	C₃×C₁₇×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
45	C₃	C₅×C₁₇×C₈₁	1	3	yes	direct product factor
46	C₃	C₃×C₅×C₁₇×C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₇	C₃²×C₅×C₁₇
socle, prime radical	C₃²×C₅×C₁₇	C₂₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₇×C₈₁	C₃	direct (N × H)
6. C₃×C₁₇×C₈₁	C₅	direct (N × H)
8. C₁₇×C₈₁	C₃×C₅	direct (N × H)
9. C₃×C₅×C₈₁	C₁₇	direct (N × H)
17. C₅×C₈₁	C₃×C₁₇	direct (N × H)
18. C₃×C₅×C₁₇	C₈₁	direct (N × H)
20. C₃×C₈₁	C₅×C₁₇	direct (N × H)
27. C₅×C₁₇	C₃×C₈₁	direct (N × H)
29. C₈₁	C₃×C₅×C₁₇	direct (N × H)
30. C₃×C₁₇	C₅×C₈₁	direct (N × H)
38. C₁₇	C₃×C₅×C₈₁	direct (N × H)
39. C₃×C₅	C₁₇×C₈₁	direct (N × H)
44. C₅	C₃×C₁₇×C₈₁	direct (N × H)
45. C₃	C₅×C₁₇×C₈₁	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×C₁₇×C₂₇	3
C₅×C₁₇×C₈₁	3
C₃×C₁₇×C₈₁	5
C₃×C₅×C₈₁	17

Subgroups — C3×C5×C17×C81 #72198239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₁₇×C₈₁ #72198239