Subgroups — C₂×C₃×D₃₀₀₉ #49115705

Order 36108.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃₀₀₉	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₁₇×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×D₁₀₀₃	—	3	1	no	maximal
4	C₂×D₃₀₀₉	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃²×C₁₇×C₅₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₃₀₀₉	C₂×C₃	1	2	yes	direct product factor
7	C₃×D₁₀₀₃	—	3	2	no	—
8	C₂×C₃×C₁₇×C₅₉	—	2	1	no	—
9	C₂×C₃×C₁₇×C₅₉	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₁₇×C₅₉	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×D₁₀₀₃	—	3	1	no	Hall
12	C₃×C₁₇×C₅₉	—	2	1	no	—
13	C₃×C₁₇×C₅₉	C₂×D₃	1	1	yes	characteristic, complemented
14	C₃×C₁₇×C₅₉	C₂²×C₃	1	1	yes	characteristic, complemented
15	C₂×C₃×D₁₇₇	—	17	1	no	maximal, Hall
16	C₂×C₁₇×C₅₉	C₃×D₃	1	1	yes	characteristic, complemented
17	D₁₀₀₃	—	3	2	no	—
18	C₂×C₃²×C₅₉	D₁₇	1	1	yes	characteristic, complemented
19	C₃×D₁₇₇	—	17	2	no	—
20	C₁₇×C₅₉	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
21	C₂×D₁₇₇	—	17	1	no	—
22	C₂×C₃×D₅₉	—	51	1	no	—
23	C₂×C₃×D₅₁	—	59	1	no	maximal, Hall
24	C₃²×C₅₉	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₅₉	—	2	1	no	—
26	C₂×C₃×C₅₉	C₃×D₁₇	1	1	yes	characteristic, complemented
27	C₂×C₃×C₅₉	D₅₁	1	1	yes	characteristic, complemented
28	D₁₇₇	—	17	2	no	—
29	C₃×D₅₉	—	51	2	no	—
30	C₂×C₃²×C₁₇	D₅₉	1	1	yes	characteristic, complemented
31	C₃×D₅₁	—	59	2	no	—
32	C₂×D₅₉	—	51	1	no	Hall
33	C₂×D₅₁	—	59	1	no	—
34	C₂×C₃×D₁₇	—	177	1	no	—
35	C₃×C₅₉	—	2	1	no	—
36	C₃×C₅₉	C₂×C₃×D₁₇	1	1	yes	characteristic, complemented
37	C₃×C₅₉	C₂×D₅₁	1	1	yes	characteristic, complemented
38	C₃²×C₁₇	C₂×D₅₉	1	1	yes	characteristic, complemented, Hall
39	C₂×C₅₉	C₃×D₅₁	1	1	yes	characteristic, complemented
40	D₅₉	—	51	2	no	—
41	C₂×C₃×C₁₇	—	2	1	no	—
42	C₂×C₃×C₁₇	C₃×D₅₉	1	1	yes	characteristic, complemented
43	C₂×C₃×C₁₇	D₁₇₇	1	1	yes	characteristic, complemented
44	D₅₁	—	59	2	no	—
45	C₃×D₁₇	—	177	2	no	—
46	C₂×D₁₇	—	177	1	no	Hall
47	C₅₉	C₂×C₃×D₅₁	1	1	yes	characteristic, complemented, Sylow
48	C₃×C₁₇	—	2	1	no	—
49	C₃×C₁₇	C₂×C₃×D₅₉	1	1	yes	characteristic, complemented
50	C₃×C₁₇	C₂×D₁₇₇	1	1	yes	characteristic, complemented
51	C₂×C₃×D₃	—	1003	1	no	Hall
52	C₂×C₁₇	C₃×D₁₇₇	1	1	yes	characteristic, complemented
53	D₁₇	—	177	2	no	—
54	C₂×C₃²	D₁₀₀₃	1	1	yes	characteristic, complemented
55	C₃×D₃	—	1003	2	no	—
56	C₁₇	C₂×C₃×D₁₇₇	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₃	—	3009	1	no	—
58	C₂×D₃	—	1003	1	no	—
59	C₃²	C₂×D₁₀₀₃	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	3009	2	no	—
62	C₂×C₃	C₃×D₁₀₀₃	1	1	yes	characteristic, complemented
63	C₂×C₃	D₃₀₀₉	1	1	yes	characteristic, direct product factor
64	D₃	—	1003	2	no	—
65	C₂²	—	3009	1	no	Sylow
66	C₃	—	2	1	no	—
67	C₃	C₂×D₃₀₀₉	1	1	yes	characteristic, direct product factor
68	C₃	C₂×C₃×D₁₀₀₃	1	1	yes	characteristic, complemented
69	C₂	—	3009	2	no	—
70	C₂	C₃×D₃₀₀₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃₀₀₉
commutator, cosocle, hypocenter	C₃×C₁₇×C₅₉	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₁₇×C₅₉	C₂
LCS	G ▷ C₃×C₁₇×C₅₉
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₃×D₃₀₀₉	C₂	direct (N × H)
2. C₂×C₃²×C₁₇×C₅₉	C₂	semidirect
4. C₂×D₃₀₀₉	C₃	direct (N × H)
5. C₃²×C₁₇×C₅₉	C₂²	semidirect
6. D₃₀₀₉	C₂×C₃	direct (N × H)
9. C₂×C₃×C₁₇×C₅₉	D₃	semidirect
10. C₂×C₃×C₁₇×C₅₉	C₂×C₃	semidirect
13. C₃×C₁₇×C₅₉	C₂×D₃	semidirect
14. C₃×C₁₇×C₅₉	C₂²×C₃	semidirect
16. C₂×C₁₇×C₅₉	C₃×D₃	semidirect
18. C₂×C₃²×C₅₉	D₁₇	semidirect
20. C₁₇×C₅₉	C₂×C₃×D₃	semidirect
24. C₃²×C₅₉	C₂×D₁₇	semidirect
26. C₂×C₃×C₅₉	C₃×D₁₇	semidirect
27. C₂×C₃×C₅₉	D₅₁	semidirect
30. C₂×C₃²×C₁₇	D₅₉	semidirect
36. C₃×C₅₉	C₂×C₃×D₁₇	semidirect
37. C₃×C₅₉	C₂×D₅₁	semidirect
38. C₃²×C₁₇	C₂×D₅₉	semidirect
39. C₂×C₅₉	C₃×D₅₁	semidirect
42. C₂×C₃×C₁₇	C₃×D₅₉	semidirect
43. C₂×C₃×C₁₇	D₁₇₇	semidirect
47. C₅₉	C₂×C₃×D₅₁	semidirect
49. C₃×C₁₇	C₂×C₃×D₅₉	semidirect
50. C₃×C₁₇	C₂×D₁₇₇	semidirect
52. C₂×C₁₇	C₃×D₁₇₇	semidirect
54. C₂×C₃²	D₁₀₀₃	semidirect
56. C₁₇	C₂×C₃×D₁₇₇	semidirect
59. C₃²	C₂×D₁₀₀₃	semidirect
62. C₂×C₃	C₃×D₁₀₀₃	semidirect
63. C₂×C₃	D₃₀₀₉	direct (N × H)
67. C₃	C₂×D₃₀₀₉	direct (N × H)
68. C₃	C₂×C₃×D₁₀₀₃	semidirect
70. C₂	C₃×D₃₀₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×D₃₀₀₉	2
C₂×C₃²×C₁₇×C₅₉	2
C₂×C₃×D₁₀₀₃	3
C₂×D₃₀₀₉	3
C₂×C₃×D₁₇₇	17
C₂×C₃×D₅₁	59

Subgroups — C2×C3×D3009 #49115705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₃₀₀₉ #49115705