Subgroups — C₂⁴×C₁₇×C₈₁ #72199539

Order 22032.

670 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₇×C₈₁	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×C₁₇×C₂₇	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₁₇×C₈₁	C₂²	1	35	yes	direct product factor
4	C₂³×C₁₇×C₂₇	C₂×C₃	1	15	yes	—
5	C₂×C₁₇×C₈₁	C₂³	1	15	yes	direct product factor
6	C₂⁴×C₉×C₁₇	C₉	1	1	yes	characteristic
7	C₂²×C₁₇×C₂₇	C₂²×C₃	1	35	yes	—
8	C₁₇×C₈₁	C₂⁴	1	1	yes	characteristic, direct product factor, Hall
9	C₂⁴×C₈₁	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂³×C₉×C₁₇	C₂×C₉	1	15	yes	—
11	C₂×C₁₇×C₂₇	C₂³×C₃	1	15	yes	—
12	C₂⁴×C₃×C₁₇	C₂₇	1	1	yes	characteristic
13	C₂³×C₈₁	C₂×C₁₇	1	15	yes	direct product factor
14	C₂²×C₉×C₁₇	C₂²×C₉	1	35	yes	—
15	C₁₇×C₂₇	C₂⁴×C₃	1	1	yes	characteristic
16	C₂⁴×C₂₇	C₃×C₁₇	1	1	yes	characteristic
17	C₂³×C₃×C₁₇	C₂×C₂₇	1	15	yes	—
18	C₂²×C₈₁	C₂²×C₁₇	1	35	yes	direct product factor
19	C₂×C₉×C₁₇	C₂³×C₉	1	15	yes	—
20	C₂⁴×C₁₇	C₈₁	1	1	yes	characteristic, direct product factor, Hall
21	C₂³×C₂₇	C₂×C₃×C₁₇	1	15	yes	—
22	C₂²×C₃×C₁₇	C₂²×C₂₇	1	35	yes	—
23	C₂×C₈₁	C₂³×C₁₇	1	15	yes	direct product factor
24	C₉×C₁₇	C₂⁴×C₉	1	1	yes	characteristic
25	C₂⁴×C₉	C₉×C₁₇	1	1	yes	characteristic
26	C₂³×C₁₇	C₂×C₈₁	1	15	yes	direct product factor
27	C₂²×C₂₇	C₂²×C₃×C₁₇	1	35	yes	—
28	C₂×C₃×C₁₇	C₂³×C₂₇	1	15	yes	—
29	C₈₁	C₂⁴×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
30	C₂³×C₉	C₂×C₉×C₁₇	1	15	yes	—
31	C₂²×C₁₇	C₂²×C₈₁	1	35	yes	direct product factor
32	C₂×C₂₇	C₂³×C₃×C₁₇	1	15	yes	—
33	C₃×C₁₇	C₂⁴×C₂₇	1	1	yes	characteristic
34	C₂⁴×C₃	C₁₇×C₂₇	1	1	yes	characteristic
35	C₂²×C₉	C₂²×C₉×C₁₇	1	35	yes	—
36	C₂×C₁₇	C₂³×C₈₁	1	15	yes	direct product factor
37	C₂₇	C₂⁴×C₃×C₁₇	1	1	yes	characteristic
38	C₂³×C₃	C₂×C₁₇×C₂₇	1	15	yes	—
39	C₂×C₉	C₂³×C₉×C₁₇	1	15	yes	—
40	C₁₇	C₂⁴×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
41	C₂⁴	C₁₇×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×C₃	C₂²×C₁₇×C₂₇	1	35	yes	—
43	C₉	C₂⁴×C₉×C₁₇	1	1	yes	characteristic
44	C₂³	C₂×C₁₇×C₈₁	1	15	yes	direct product factor
45	C₂×C₃	C₂³×C₁₇×C₂₇	1	15	yes	—
46	C₂²	C₂²×C₁₇×C₈₁	1	35	yes	direct product factor
47	C₃	C₂⁴×C₁₇×C₂₇	1	1	yes	characteristic
48	C₂	C₂³×C₁₇×C₈₁	1	15	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₇	C₂⁴×C₃×C₁₇
socle, prime radical	C₂⁴×C₃×C₁₇	C₂₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₇×C₈₁	C₂	direct (N × H)
3. C₂²×C₁₇×C₈₁	C₂²	direct (N × H)
5. C₂×C₁₇×C₈₁	C₂³	direct (N × H)
8. C₁₇×C₈₁	C₂⁴	direct (N × H)
9. C₂⁴×C₈₁	C₁₇	direct (N × H)
13. C₂³×C₈₁	C₂×C₁₇	direct (N × H)
18. C₂²×C₈₁	C₂²×C₁₇	direct (N × H)
20. C₂⁴×C₁₇	C₈₁	direct (N × H)
23. C₂×C₈₁	C₂³×C₁₇	direct (N × H)
26. C₂³×C₁₇	C₂×C₈₁	direct (N × H)
29. C₈₁	C₂⁴×C₁₇	direct (N × H)
31. C₂²×C₁₇	C₂²×C₈₁	direct (N × H)
36. C₂×C₁₇	C₂³×C₈₁	direct (N × H)
40. C₁₇	C₂⁴×C₈₁	direct (N × H)
41. C₂⁴	C₁₇×C₈₁	direct (N × H)
44. C₂³	C₂×C₁₇×C₈₁	direct (N × H)
46. C₂²	C₂²×C₁₇×C₈₁	direct (N × H)
48. C₂	C₂³×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
C₂³×C₁₇×C₈₁	2
C₂⁴×C₁₇×C₂₇	3
C₂⁴×C₈₁	17

Subgroups — C24×C17×C81 #72199539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₁₇×C₈₁ #72199539