Subgroups — C₂×C₅×D₁₂₀₉ #37564009

Order 24180.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₁₂₀₉	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅×C₁₃×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₄₀₃	—	3	1	no	maximal, Hall
4	C₃×C₅×C₁₃×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×D₁₂₀₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×D₄₀₃	—	3	2	no	—
7	C₂×C₅×C₁₃×C₃₁	D₃	1	1	yes	characteristic, complemented
8	D₁₂₀₉	C₂×C₅	1	2	yes	direct product factor
9	C₂×C₃×C₁₃×C₃₁	C₂×C₅	1	1	yes	characteristic, complemented
10	C₅×C₁₃×C₃₁	C₂×D₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₅×D₉₃	—	13	1	no	maximal, Hall
12	C₂×D₄₀₃	—	3	1	no	Hall
13	C₃×C₁₃×C₃₁	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅×C₃₁	D₁₃	1	1	yes	characteristic, complemented
15	C₅×D₉₃	—	13	2	no	—
16	C₂×C₁₃×C₃₁	C₅×D₃	1	1	yes	characteristic, complemented
17	D₄₀₃	—	3	2	no	—
18	C₂×C₅×D₃₉	—	31	1	no	maximal, Hall
19	C₂×C₅×D₃₁	—	39	1	no	Hall
20	C₃×C₅×C₃₁	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
21	C₁₃×C₃₁	C₂×C₅×D₃	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₅×C₁₃	D₃₁	1	1	yes	characteristic, complemented
23	C₅×D₃₉	—	31	2	no	—
24	C₂×D₉₃	—	13	1	no	Hall
25	C₂×C₅×C₃₁	D₃₉	1	1	yes	characteristic, complemented
26	C₅×D₃₁	—	39	2	no	—
27	C₂×C₅×D₁₃	—	93	1	no	Hall
28	C₃×C₅×C₁₃	C₂×D₃₁	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₃₁	C₅×D₁₃	1	1	yes	characteristic, complemented
30	D₉₃	—	13	2	no	—
31	C₂×D₃₉	—	31	1	no	Hall
32	C₅×C₃₁	C₂×D₃₉	1	1	yes	characteristic, complemented, Hall
33	C₂×C₅×C₁₃	D₉₃	1	1	yes	characteristic, complemented
34	C₅×D₁₃	—	93	2	no	—
35	C₂×D₃₁	—	39	1	no	Hall
36	C₃×C₃₁	C₂×C₅×D₁₃	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₁₃	C₅×D₃₁	1	1	yes	characteristic, complemented
38	D₃₉	—	31	2	no	—
39	C₅×C₁₃	C₂×D₉₃	1	1	yes	characteristic, complemented, Hall
40	C₂×C₃₁	C₅×D₃₉	1	1	yes	characteristic, complemented
41	D₃₁	—	39	2	no	—
42	C₂×C₅×D₃	—	403	1	no	Hall
43	C₂×D₁₃	—	93	1	no	Hall
44	C₃×C₁₃	C₂×C₅×D₃₁	1	1	yes	characteristic, complemented, Hall
45	C₃₁	C₂×C₅×D₃₉	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃×C₅	D₄₀₃	1	1	yes	characteristic, complemented
47	C₅×D₃	—	403	2	no	—
48	C₂×C₁₃	C₅×D₉₃	1	1	yes	characteristic, complemented
49	D₁₃	—	93	2	no	—
50	C₂²×C₅	—	1209	1	no	Hall
51	C₃×C₅	C₂×D₄₀₃	1	1	yes	characteristic, complemented, Hall
52	C₁₃	C₂×C₅×D₉₃	1	1	yes	characteristic, complemented, Sylow
53	C₂×D₃	—	403	1	no	Hall
54	C₂×C₅	—	1209	2	no	—
55	C₂×C₅	D₁₂₀₉	1	1	yes	characteristic, direct product factor
56	C₂×C₃	C₅×D₄₀₃	1	1	yes	characteristic, complemented
57	D₃	—	403	2	no	—
58	C₅	C₂×D₁₂₀₉	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²	—	1209	1	no	Sylow
60	C₃	C₂×C₅×D₄₀₃	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	1209	2	no	—
62	C₂	C₅×D₁₂₀₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	D₁₂₀₉
commutator, cosocle, hypocenter	C₃×C₁₃×C₃₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₁₃×C₃₁	C₂
LCS	G ▷ C₃×C₁₃×C₃₁
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₅×D₁₂₀₉	C₂	direct (N × H)
2. C₂×C₃×C₅×C₁₃×C₃₁	C₂	semidirect
4. C₃×C₅×C₁₃×C₃₁	C₂²	semidirect
5. C₂×D₁₂₀₉	C₅	direct (N × H)
7. C₂×C₅×C₁₃×C₃₁	D₃	semidirect
8. D₁₂₀₉	C₂×C₅	direct (N × H)
9. C₂×C₃×C₁₃×C₃₁	C₂×C₅	semidirect
10. C₅×C₁₃×C₃₁	C₂×D₃	semidirect
13. C₃×C₁₃×C₃₁	C₂²×C₅	semidirect
14. C₂×C₃×C₅×C₃₁	D₁₃	semidirect
16. C₂×C₁₃×C₃₁	C₅×D₃	semidirect
20. C₃×C₅×C₃₁	C₂×D₁₃	semidirect
21. C₁₃×C₃₁	C₂×C₅×D₃	semidirect
22. C₂×C₃×C₅×C₁₃	D₃₁	semidirect
25. C₂×C₅×C₃₁	D₃₉	semidirect
28. C₃×C₅×C₁₃	C₂×D₃₁	semidirect
29. C₂×C₃×C₃₁	C₅×D₁₃	semidirect
32. C₅×C₃₁	C₂×D₃₉	semidirect
33. C₂×C₅×C₁₃	D₉₃	semidirect
36. C₃×C₃₁	C₂×C₅×D₁₃	semidirect
37. C₂×C₃×C₁₃	C₅×D₃₁	semidirect
39. C₅×C₁₃	C₂×D₉₃	semidirect
40. C₂×C₃₁	C₅×D₃₉	semidirect
44. C₃×C₁₃	C₂×C₅×D₃₁	semidirect
45. C₃₁	C₂×C₅×D₃₉	semidirect
46. C₂×C₃×C₅	D₄₀₃	semidirect
48. C₂×C₁₃	C₅×D₉₃	semidirect
51. C₃×C₅	C₂×D₄₀₃	semidirect
52. C₁₃	C₂×C₅×D₉₃	semidirect
55. C₂×C₅	D₁₂₀₉	direct (N × H)
56. C₂×C₃	C₅×D₄₀₃	semidirect
58. C₅	C₂×D₁₂₀₉	direct (N × H)
60. C₃	C₂×C₅×D₄₀₃	semidirect
62. C₂	C₅×D₁₂₀₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×D₁₂₀₉	2
C₂×C₃×C₅×C₁₃×C₃₁	2
C₂×C₅×D₄₀₃	3
C₂×D₁₂₀₉	5
C₂×C₅×D₉₃	13
C₂×C₅×D₃₉	31

Subgroups — C2×C5×D1209 #37564009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₁₂₀₉ #37564009