Subgroups — C₂⁵×D₃×C₉ #13180119

Order 1728.

17698 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₃×C₉	C₂	1	62	yes	maximal, direct product factor
3	C₂⁵×C₃×D₃	C₃	1	1	yes	maximal, characteristic
4	C₂⁶×C₉	—	3	1	no	maximal
5	C₂⁴×C₃×C₉	C₂²	1	31	yes	complemented
6	C₂³×D₃×C₉	C₂²	1	620	yes	direct product factor
7	C₂⁵×C₃²	C₂×C₃	1	1	yes	characteristic
8	C₂⁴×C₃×D₃	C₂×C₃	1	62	yes	—
9	C₂⁵×C₉	—	2	1	no	—
10	C₂⁵×C₉	—	3	62	no	—
11	C₂⁵×C₉	D₃	1	1	yes	characteristic, direct product factor
12	C₂³×C₃×C₉	C₂³	1	155	yes	complemented
13	C₂²×D₃×C₉	C₂³	1	1240	yes	direct product factor
14	C₂⁶×C₃	—	3	1	no	—
15	C₂⁵×D₃	C₉	1	1	yes	characteristic, direct product factor
16	C₂⁴×C₃²	C₂²×C₃	1	31	yes	—
17	C₂³×C₃×D₃	C₂²×C₃	1	620	yes	—
18	C₂⁴×C₉	—	2	31	no	—
19	C₂⁴×C₉	—	3	620	no	—
20	C₂⁴×C₉	C₂×D₃	1	31	yes	direct product factor
21	C₂²×C₃×C₉	C₂⁴	1	155	yes	complemented
22	C₂×D₃×C₉	C₂⁴	1	496	yes	direct product factor
23	C₂⁵×C₃	—	2	1	no	—
24	C₂⁵×C₃	—	3	62	no	—
25	C₂⁵×C₃	C₂×C₉	1	1	yes	characteristic, complemented
26	C₂⁵×C₃	C₃×D₃	1	1	yes	characteristic
27	C₂⁴×D₃	C₂×C₉	1	62	yes	direct product factor
28	C₂³×C₃²	C₂³×C₃	1	155	yes	—
29	C₂²×C₃×D₃	C₂³×C₃	1	1240	yes	—
30	C₂³×C₉	—	2	155	no	—
31	C₂³×C₉	—	3	1240	no	—
32	C₂³×C₉	C₂²×D₃	1	155	yes	direct product factor
33	C₂⁶	—	3	1	no	Sylow
34	C₂×C₃×C₉	C₂⁵	1	31	yes	complemented
35	D₃×C₉	C₂⁵	1	32	yes	direct product factor
36	C₂⁴×C₃	—	2	31	no	—
37	C₂⁴×C₃	—	3	620	no	—
38	C₂⁴×C₃	C₂²×C₉	1	31	yes	complemented
39	C₂⁴×C₃	C₂×C₃×D₃	1	31	yes	—
40	C₂³×D₃	C₂²×C₉	1	620	yes	direct product factor
41	C₂²×C₃²	C₂⁴×C₃	1	155	yes	—
42	C₂×C₃×D₃	C₂⁴×C₃	1	496	yes	—
43	C₂²×C₉	—	2	155	no	—
44	C₂²×C₉	—	3	496	no	—
45	C₂²×C₉	C₂³×D₃	1	155	yes	direct product factor
46	C₂⁵	—	3	62	no	—
47	C₂⁵	D₃×C₉	1	1	yes	characteristic, direct product factor
48	C₃×C₉	C₂⁶	1	1	yes	characteristic, complemented, Sylow
49	C₂³×C₃	—	2	155	no	—
50	C₂³×C₃	—	3	1240	no	—
51	C₂³×C₃	C₂³×C₉	1	155	yes	complemented
52	C₂³×C₃	C₂²×C₃×D₃	1	155	yes	—
53	C₂²×D₃	C₂³×C₉	1	1240	yes	direct product factor
54	C₂×C₃²	C₂⁵×C₃	1	31	yes	—
55	C₃×D₃	C₂⁵×C₃	1	32	yes	—
56	C₂×C₉	—	2	31	no	—
57	C₂×C₉	—	3	32	no	—
58	C₂×C₉	C₂⁴×D₃	1	31	yes	direct product factor
59	C₂⁴	—	3	620	no	—
60	C₂⁴	C₂×D₃×C₉	1	31	yes	direct product factor
61	C₂²×C₃	—	2	155	no	—
62	C₂²×C₃	—	3	496	no	—
63	C₂²×C₃	C₂⁴×C₉	1	155	yes	complemented
64	C₂²×C₃	C₂³×C₃×D₃	1	155	yes	—
65	C₂×D₃	C₂⁴×C₉	1	496	yes	direct product factor
66	C₃²	C₂⁶×C₃	1	1	yes	characteristic
67	C₉	—	2	1	no	—
68	C₉	C₂⁵×D₃	1	1	yes	characteristic, direct product factor
69	C₂³	—	3	1240	no	—
70	C₂³	C₂²×D₃×C₉	1	155	yes	direct product factor
71	C₂×C₃	—	2	31	no	—
72	C₂×C₃	—	3	32	no	—
73	C₂×C₃	C₂⁵×C₉	1	31	yes	complemented
74	C₂×C₃	C₂⁴×C₃×D₃	1	31	yes	—
75	D₃	C₂⁵×C₉	1	32	yes	direct product factor
76	C₂²	—	3	496	no	—
77	C₂²	C₂³×D₃×C₉	1	155	yes	direct product factor
78	C₃	—	2	1	no	—
79	C₃	C₂⁶×C₉	1	1	yes	characteristic, complemented
80	C₃	C₂⁵×C₃×D₃	1	1	yes	characteristic
81	C₂	—	3	32	no	—
82	C₂	C₂⁴×D₃×C₉	1	31	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁵×C₉	D₃
commutator, hypocenter	C₃	C₂⁶×C₉
Frattini	C₃	C₂⁵×C₃×D₃
Fitting	C₂⁵×C₃×C₉	C₂
socle	C₂⁵×C₃²	C₂×C₃
prime radical	C₂⁵×C₃×D₃	C₃
cosocle	C₃²	C₂⁶×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂⁵×C₉

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₃×C₉	C₂	semidirect
2. C₂⁴×D₃×C₉	C₂	direct (N × H)
5. C₂⁴×C₃×C₉	C₂²	semidirect
6. C₂³×D₃×C₉	C₂²	direct (N × H)
11. C₂⁵×C₉	D₃	direct (N × H)
12. C₂³×C₃×C₉	C₂³	semidirect
13. C₂²×D₃×C₉	C₂³	direct (N × H)
15. C₂⁵×D₃	C₉	direct (N × H)
20. C₂⁴×C₉	C₂×D₃	direct (N × H)
21. C₂²×C₃×C₉	C₂⁴	semidirect
22. C₂×D₃×C₉	C₂⁴	direct (N × H)
25. C₂⁵×C₃	C₂×C₉	semidirect
27. C₂⁴×D₃	C₂×C₉	direct (N × H)
32. C₂³×C₉	C₂²×D₃	direct (N × H)
34. C₂×C₃×C₉	C₂⁵	semidirect
35. D₃×C₉	C₂⁵	direct (N × H)
38. C₂⁴×C₃	C₂²×C₉	semidirect
40. C₂³×D₃	C₂²×C₉	direct (N × H)
45. C₂²×C₉	C₂³×D₃	direct (N × H)
47. C₂⁵	D₃×C₉	direct (N × H)
48. C₃×C₉	C₂⁶	semidirect
51. C₂³×C₃	C₂³×C₉	semidirect
53. C₂²×D₃	C₂³×C₉	direct (N × H)
58. C₂×C₉	C₂⁴×D₃	direct (N × H)
60. C₂⁴	C₂×D₃×C₉	direct (N × H)
63. C₂²×C₃	C₂⁴×C₉	semidirect
65. C₂×D₃	C₂⁴×C₉	direct (N × H)
68. C₉	C₂⁵×D₃	direct (N × H)
70. C₂³	C₂²×D₃×C₉	direct (N × H)
73. C₂×C₃	C₂⁵×C₉	semidirect
75. D₃	C₂⁵×C₉	direct (N × H)
77. C₂²	C₂³×D₃×C₉	direct (N × H)
79. C₃	C₂⁶×C₉	semidirect
82. C₂	C₂⁴×D₃×C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃×C₉	2
C₂⁴×D₃×C₉	2
C₂⁵×C₃×D₃	3
C₂⁶×C₉	3

Subgroups — C25×D3×C9 #13180119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×D₃×C₉ #13180119