Subgroups — C₂×C₅²×C₁₇×C₂₇ #72200293

Order 22950.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₁₇×C₂₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₅×C₁₇×C₂₇	C₅	1	6	yes	maximal, direct product factor
3	C₅×C₁₇×C₂₇	C₂×C₅	1	6	yes	direct product factor
4	C₂×C₅²×C₉×C₁₇	C₃	1	1	yes	maximal, characteristic
5	C₅²×C₉×C₁₇	C₂×C₃	1	1	yes	characteristic
6	C₂×C₃×C₅²×C₁₇	C₉	1	1	yes	characteristic
7	C₂×C₅×C₉×C₁₇	C₃×C₅	1	6	yes	—
8	C₂×C₅²×C₂₇	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅²×C₁₇	C₂×C₉	1	1	yes	characteristic
10	C₂×C₁₇×C₂₇	C₅²	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₅²×C₁₇	C₂₇	1	1	yes	characteristic, direct product factor, Hall
12	C₅×C₉×C₁₇	C₂×C₃×C₅	1	6	yes	—
13	C₅²×C₂₇	C₂×C₁₇	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃×C₅×C₁₇	C₅×C₉	1	6	yes	—
15	C₁₇×C₂₇	C₂×C₅²	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₅²×C₉	C₃×C₁₇	1	1	yes	characteristic
17	C₅²×C₁₇	C₂×C₂₇	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₉×C₁₇	C₃×C₅²	1	1	yes	characteristic
19	C₂×C₅×C₂₇	C₅×C₁₇	1	6	yes	direct product factor
20	C₃×C₅×C₁₇	C₂×C₅×C₉	1	6	yes	—
21	C₅²×C₉	C₂×C₃×C₁₇	1	1	yes	characteristic
22	C₂×C₅×C₁₇	C₅×C₂₇	1	6	yes	direct product factor
23	C₉×C₁₇	C₂×C₃×C₅²	1	1	yes	characteristic
24	C₂×C₃×C₅²	C₉×C₁₇	1	1	yes	characteristic
25	C₅×C₂₇	C₂×C₅×C₁₇	1	6	yes	direct product factor
26	C₂×C₃×C₁₇	C₅²×C₉	1	1	yes	characteristic
27	C₂×C₅×C₉	C₃×C₅×C₁₇	1	6	yes	—
28	C₅×C₁₇	C₂×C₅×C₂₇	1	6	yes	direct product factor
29	C₃×C₅²	C₂×C₉×C₁₇	1	1	yes	characteristic
30	C₂×C₂₇	C₅²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
31	C₃×C₁₇	C₂×C₅²×C₉	1	1	yes	characteristic
32	C₂×C₅²	C₁₇×C₂₇	1	1	yes	characteristic, direct product factor, Hall
33	C₅×C₉	C₂×C₃×C₅×C₁₇	1	6	yes	—
34	C₂×C₁₇	C₅²×C₂₇	1	1	yes	characteristic, direct product factor, Hall
35	C₂×C₃×C₅	C₅×C₉×C₁₇	1	6	yes	—
36	C₂₇	C₂×C₅²×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
37	C₅²	C₂×C₁₇×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×C₉	C₃×C₅²×C₁₇	1	1	yes	characteristic
39	C₁₇	C₂×C₅²×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
40	C₃×C₅	C₂×C₅×C₉×C₁₇	1	6	yes	—
41	C₂×C₅	C₅×C₁₇×C₂₇	1	6	yes	direct product factor
42	C₉	C₂×C₃×C₅²×C₁₇	1	1	yes	characteristic
43	C₂×C₃	C₅²×C₉×C₁₇	1	1	yes	characteristic
44	C₅	C₂×C₅×C₁₇×C₂₇	1	6	yes	direct product factor
45	C₃	C₂×C₅²×C₉×C₁₇	1	1	yes	characteristic
46	C₂	C₅²×C₁₇×C₂₇	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂×C₃×C₅²×C₁₇
socle, prime radical	C₂×C₃×C₅²×C₁₇	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅²×C₁₇×C₂₇	C₂	direct (N × H)
2. C₂×C₅×C₁₇×C₂₇	C₅	direct (N × H)
3. C₅×C₁₇×C₂₇	C₂×C₅	direct (N × H)
8. C₂×C₅²×C₂₇	C₁₇	direct (N × H)
10. C₂×C₁₇×C₂₇	C₅²	direct (N × H)
11. C₂×C₅²×C₁₇	C₂₇	direct (N × H)
13. C₅²×C₂₇	C₂×C₁₇	direct (N × H)
15. C₁₇×C₂₇	C₂×C₅²	direct (N × H)
17. C₅²×C₁₇	C₂×C₂₇	direct (N × H)
19. C₂×C₅×C₂₇	C₅×C₁₇	direct (N × H)
22. C₂×C₅×C₁₇	C₅×C₂₇	direct (N × H)
25. C₅×C₂₇	C₂×C₅×C₁₇	direct (N × H)
28. C₅×C₁₇	C₂×C₅×C₂₇	direct (N × H)
30. C₂×C₂₇	C₅²×C₁₇	direct (N × H)
32. C₂×C₅²	C₁₇×C₂₇	direct (N × H)
34. C₂×C₁₇	C₅²×C₂₇	direct (N × H)
36. C₂₇	C₂×C₅²×C₁₇	direct (N × H)
37. C₅²	C₂×C₁₇×C₂₇	direct (N × H)
39. C₁₇	C₂×C₅²×C₂₇	direct (N × H)
41. C₂×C₅	C₅×C₁₇×C₂₇	direct (N × H)
44. C₅	C₂×C₅×C₁₇×C₂₇	direct (N × H)
46. C₂	C₅²×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
C₅²×C₁₇×C₂₇	2
C₂×C₅×C₁₇×C₂₇	5
C₂×C₅²×C₉×C₁₇	3
C₂×C₅²×C₂₇	17

Subgroups — C2×C52×C17×C27 #72200293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×C₁₇×C₂₇ #72200293