Subgroups — C₂²×C₅²×#4296 #30685815

Order 17100.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#4296	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₅²×#4296	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×C₅×#4296	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₅²×C₁₉⋊C₃	C₂×C₃	1	3	yes	—
6	C₂²×C₅²×C₁₉	C₉	1	1	yes	characteristic, complemented, Hall
7	C₂×C₅×#4296	C₂×C₅	1	18	yes	direct product factor
8	C₅²×C₁₉⋊C₃	C₂²×C₃	1	1	yes	characteristic
9	C₂²×C₅×C₁₉⋊C₃	C₃×C₅	1	6	yes	—
10	C₂×C₅²×C₁₉	C₂×C₉	1	3	yes	complemented
11	C₂²×C₅²×C₉	—	19	1	no	maximal, Hall
12	C₅×#4296	C₂²×C₅	1	6	yes	direct product factor
13	C₂²×#4296	C₅²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₅×C₁₉⋊C₃	C₂×C₃×C₅	1	18	yes	—
15	C₅²×C₁₉	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
16	C₂×C₅²×C₉	—	19	3	no	—
17	C₂²×C₅×C₁₉	C₅×C₉	1	6	yes	complemented
18	C₂×#4296	C₂×C₅²	1	3	yes	direct product factor
19	C₂²×C₃×C₅²	—	19	1	no	—
20	C₅×C₁₉⋊C₃	C₂²×C₃×C₅	1	6	yes	—
21	C₂²×C₁₉⋊C₃	C₃×C₅²	1	1	yes	characteristic
22	C₅²×C₉	—	19	1	no	Hall
23	C₂×C₅×C₁₉	C₂×C₅×C₉	1	18	yes	complemented
24	C₂²×C₅×C₉	—	19	6	no	—
25	#4296	C₂²×C₅²	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₃×C₅²	—	19	3	no	—
27	C₂×C₁₉⋊C₃	C₂×C₃×C₅²	1	3	yes	—
28	C₂²×C₅²	#4296	1	1	yes	characteristic, direct product factor, Hall
29	C₅×C₁₉	C₂²×C₅×C₉	1	6	yes	complemented
30	C₂×C₅×C₉	—	19	18	no	—
31	C₂²×C₁₉	C₅²×C₉	1	1	yes	characteristic, complemented, Hall
32	C₃×C₅²	—	19	1	no	—
33	C₂²×C₃×C₅	—	19	6	no	—
34	C₁₉⋊C₃	C₂²×C₃×C₅²	1	1	yes	characteristic
35	C₂×C₅²	C₂×#4296	1	3	yes	direct product factor
36	C₅×C₉	—	19	6	no	—
37	C₂×C₁₉	C₂×C₅²×C₉	1	3	yes	complemented
38	C₂²×C₉	—	19	1	no	Hall
39	C₂×C₃×C₅	—	19	18	no	—
40	C₅²	C₂²×#4296	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₅	C₅×#4296	1	6	yes	direct product factor
42	C₁₉	C₂²×C₅²×C₉	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₉	—	19	3	no	—
44	C₃×C₅	—	19	6	no	—
45	C₂²×C₃	—	19	1	no	—
46	C₂×C₅	C₂×C₅×#4296	1	18	yes	direct product factor
47	C₉	—	19	1	no	Sylow
48	C₂×C₃	—	19	3	no	—
49	C₅	C₂²×C₅×#4296	1	6	yes	direct product factor
50	C₂²	C₅²×#4296	1	1	yes	characteristic, direct product factor, Sylow
51	C₃	—	19	1	no	—
52	C₂	C₂×C₅²×#4296	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅²	#4296
commutator, hypocenter	C₁₉	C₂²×C₅²×C₉
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₁₉	C₉
prime radical	C₂²×C₅²×C₁₉⋊C₃	C₃
cosocle	C₁₉⋊C₃	C₂²×C₃×C₅²
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂²×C₅²

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#4296	C₂	direct (N × H)
3. C₅²×#4296	C₂²	direct (N × H)
4. C₂²×C₅×#4296	C₅	direct (N × H)
6. C₂²×C₅²×C₁₉	C₉	semidirect
7. C₂×C₅×#4296	C₂×C₅	direct (N × H)
10. C₂×C₅²×C₁₉	C₂×C₉	semidirect
12. C₅×#4296	C₂²×C₅	direct (N × H)
13. C₂²×#4296	C₅²	direct (N × H)
15. C₅²×C₁₉	C₂²×C₉	semidirect
17. C₂²×C₅×C₁₉	C₅×C₉	semidirect
18. C₂×#4296	C₂×C₅²	direct (N × H)
23. C₂×C₅×C₁₉	C₂×C₅×C₉	semidirect
25. #4296	C₂²×C₅²	direct (N × H)
28. C₂²×C₅²	#4296	direct (N × H)
29. C₅×C₁₉	C₂²×C₅×C₉	semidirect
31. C₂²×C₁₉	C₅²×C₉	semidirect
35. C₂×C₅²	C₂×#4296	direct (N × H)
37. C₂×C₁₉	C₂×C₅²×C₉	semidirect
40. C₅²	C₂²×#4296	direct (N × H)
41. C₂²×C₅	C₅×#4296	direct (N × H)
42. C₁₉	C₂²×C₅²×C₉	semidirect
46. C₂×C₅	C₂×C₅×#4296	direct (N × H)
49. C₅	C₂²×C₅×#4296	direct (N × H)
50. C₂²	C₅²×#4296	direct (N × H)
52. C₂	C₂×C₅²×#4296	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×C₅²×#4296	2
C₂²×C₅²×C₁₉⋊C₃	3
C₂²×C₅×#4296	5
C₂²×C₅²×C₉	19

Subgroups — C22×C52×#4296 #30685815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅²×#4296 #30685815