Subgroups — C₂²×C₃×C₁₇×C₅²⋊C₃ #29506070

Order 15300.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₇×C₅²⋊C₃	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₁₇×C₅²⋊C₃	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₅²×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₇×C₅²⋊C₃	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂×C₁₇×C₅²⋊C₃	C₂×C₃	1	9	yes	direct product factor
6	C₂×C₃×C₅²×C₁₇	C₂×C₃	1	3	yes	complemented
7	C₂²×C₅²×C₁₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₅²×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented
9	C₁₇×C₅²⋊C₃	C₂²×C₃	1	3	yes	direct product factor
10	C₂²×C₃×C₅×C₁₇	—	3	2	no	—
11	C₂²×C₃×C₅²⋊C₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₅²×C₁₇	C₂×C₃²	1	3	yes	complemented
13	C₂²×C₃²×C₁₇	—	25	1	no	maximal, Hall
14	C₂×C₃×C₅×C₁₇	—	3	6	no	—
15	C₂×C₃×C₅²⋊C₃	C₂×C₁₇	1	3	yes	direct product factor
16	C₅²×C₁₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
17	C₂²×C₅×C₁₇	—	3	2	no	—
18	C₂×C₃²×C₁₇	—	25	3	no	—
19	C₂²×C₃×C₅²	C₃×C₁₇	1	1	yes	characteristic, complemented
20	C₂²×C₅²⋊C₃	C₃×C₁₇	1	3	yes	direct product factor
21	C₃×C₅×C₁₇	—	3	2	no	—
22	C₃×C₅²⋊C₃	C₂²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
23	C₂²×C₃×C₁₇	—	25	3	no	—
24	C₂²×C₃×C₁₇	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
25	C₂×C₅×C₁₇	—	3	6	no	—
26	C₃²×C₁₇	—	25	1	no	Hall
27	C₂×C₃×C₅²	C₂×C₃×C₁₇	1	3	yes	complemented
28	C₂×C₅²⋊C₃	C₂×C₃×C₁₇	1	9	yes	direct product factor
29	C₂×C₃×C₁₇	—	25	9	no	—
30	C₂×C₃×C₁₇	C₂×C₅²⋊C₃	1	3	yes	direct product factor
31	C₂²×C₅²	C₃²×C₁₇	1	1	yes	characteristic, complemented, Hall
32	C₅×C₁₇	—	3	2	no	—
33	C₃×C₅²	C₂²×C₃×C₁₇	1	1	yes	characteristic, complemented
34	C₅²⋊C₃	C₂²×C₃×C₁₇	1	3	yes	direct product factor
35	C₂²×C₁₇	C₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Hall
36	C₂²×C₃×C₅	—	3	2	no	—
37	C₃×C₁₇	—	25	3	no	—
38	C₃×C₁₇	C₂²×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
39	C₂×C₅²	C₂×C₃²×C₁₇	1	3	yes	complemented
40	C₂²×C₃²	—	25	1	no	Hall
41	C₂×C₁₇	C₂×C₃×C₅²⋊C₃	1	3	yes	direct product factor
42	C₂×C₃×C₅	—	3	6	no	—
43	C₅²	C₂²×C₃²×C₁₇	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₅	—	3	2	no	—
45	C₂×C₃²	—	25	3	no	—
46	C₁₇	C₂²×C₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
47	C₃×C₅	—	3	2	no	—
48	C₂²×C₃	—	25	3	no	—
49	C₂²×C₃	C₁₇×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
50	C₂×C₅	—	3	6	no	—
51	C₃²	—	25	1	no	Sylow
52	C₂×C₃	—	25	9	no	—
53	C₂×C₃	C₂×C₁₇×C₅²⋊C₃	1	3	yes	direct product factor
54	C₅	—	3	2	no	—
55	C₂²	C₃×C₁₇×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
56	C₃	—	25	3	no	—
57	C₃	C₂²×C₁₇×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
58	C₂	C₂×C₃×C₁₇×C₅²⋊C₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₁₇	C₅²⋊C₃
commutator, cosocle, hypocenter	C₅²	C₂²×C₃²×C₁₇
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₁₇	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂²×C₃×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₁₇×C₅²⋊C₃	C₂	direct (N × H)
2. C₂²×C₁₇×C₅²⋊C₃	C₃	direct (N × H)
3. C₂²×C₃×C₅²×C₁₇	C₃	semidirect
4. C₃×C₁₇×C₅²⋊C₃	C₂²	direct (N × H)
5. C₂×C₁₇×C₅²⋊C₃	C₂×C₃	direct (N × H)
6. C₂×C₃×C₅²×C₁₇	C₂×C₃	semidirect
7. C₂²×C₅²×C₁₇	C₃²	semidirect
8. C₃×C₅²×C₁₇	C₂²×C₃	semidirect
9. C₁₇×C₅²⋊C₃	C₂²×C₃	direct (N × H)
11. C₂²×C₃×C₅²⋊C₃	C₁₇	direct (N × H)
12. C₂×C₅²×C₁₇	C₂×C₃²	semidirect
15. C₂×C₃×C₅²⋊C₃	C₂×C₁₇	direct (N × H)
16. C₅²×C₁₇	C₂²×C₃²	semidirect
19. C₂²×C₃×C₅²	C₃×C₁₇	semidirect
20. C₂²×C₅²⋊C₃	C₃×C₁₇	direct (N × H)
22. C₃×C₅²⋊C₃	C₂²×C₁₇	direct (N × H)
24. C₂²×C₃×C₁₇	C₅²⋊C₃	direct (N × H)
27. C₂×C₃×C₅²	C₂×C₃×C₁₇	semidirect
28. C₂×C₅²⋊C₃	C₂×C₃×C₁₇	direct (N × H)
30. C₂×C₃×C₁₇	C₂×C₅²⋊C₃	direct (N × H)
31. C₂²×C₅²	C₃²×C₁₇	semidirect
33. C₃×C₅²	C₂²×C₃×C₁₇	semidirect
34. C₅²⋊C₃	C₂²×C₃×C₁₇	direct (N × H)
35. C₂²×C₁₇	C₃×C₅²⋊C₃	direct (N × H)
38. C₃×C₁₇	C₂²×C₅²⋊C₃	direct (N × H)
39. C₂×C₅²	C₂×C₃²×C₁₇	semidirect
41. C₂×C₁₇	C₂×C₃×C₅²⋊C₃	direct (N × H)
43. C₅²	C₂²×C₃²×C₁₇	semidirect
46. C₁₇	C₂²×C₃×C₅²⋊C₃	direct (N × H)
49. C₂²×C₃	C₁₇×C₅²⋊C₃	direct (N × H)
53. C₂×C₃	C₂×C₁₇×C₅²⋊C₃	direct (N × H)
55. C₂²	C₃×C₁₇×C₅²⋊C₃	direct (N × H)
57. C₃	C₂²×C₁₇×C₅²⋊C₃	direct (N × H)
58. C₂	C₂×C₃×C₁₇×C₅²⋊C₃	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×C₃×C₁₇×C₅²⋊C₃	2
C₂²×C₁₇×C₅²⋊C₃	3
C₂²×C₃×C₅²×C₁₇	3
C₂²×C₃×C₅²⋊C₃	17
C₂²×C₃²×C₁₇	25

Subgroups — C22×C3×C17×C52⋊C3 #29506070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₁₇×C₅²⋊C₃ #29506070