Subgroups — C₃²×C₉×#66526 #41008632

Order 26973.

2132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#66526	C₃	1	3	yes	maximal
2	C₃×C₉×#66526	C₃	1	36	yes	maximal, direct product factor
3	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66526	C₉	1	81	yes	direct product factor
5	C₃²×#66526	C₃²	1	36	yes	—
6	C₉×#66526	C₃²	1	81	yes	direct product factor
7	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	C₃²×#66524	C₉	1	2	yes	complemented
9	C₃×C₉×C₃₇⋊C₃	C₉	1	24	yes	complemented
10	C₃×C₉×C₃₇⋊C₃	C₃²	1	12	yes	—
11	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic, complemented
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	C₃×C₉	1	8	yes	—
14	C₃²×C₃₇⋊C₃	C₃×C₉	1	27	yes	—
15	C₃²×C₃₇⋊C₃	C₃³	1	4	yes	—
16	C₃×#66526	C₃×C₉	1	324	yes	direct product factor
17	C₃×#66526	C₃³	1	27	yes	—
18	C₃×C₉×C₃₇	C₃×C₉	1	12	yes	complemented
19	C₃×#66524	C₃×C₉	1	24	yes	complemented
20	C₉×C₃₇⋊C₃	C₃×C₉	1	72	yes	complemented
21	C₉×C₃₇⋊C₃	C₃³	1	9	yes	—
22	C₃²×C₉²	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	C₉²	1	9	yes	complemented
24	C₃²×C₃₇	C₃²×C₉	1	4	yes	—
25	C₃×C₃₇⋊C₃	C₉²	1	72	yes	complemented
26	C₃×C₃₇⋊C₃	C₃²×C₉	1	36	yes	—
27	C₃×C₃₇⋊C₃	C₃²×C₉	1	8	yes	—
28	C₃×C₃₇⋊C₃	C₃⁴	1	1	yes	characteristic
29	#66526	C₃²×C₉	1	81	yes	direct product factor
30	C₉×C₃₇	C₃²×C₉	1	9	yes	complemented
31	#66524	C₃²×C₉	1	18	yes	complemented
32	C₃³×C₉	—	37	3	no	—
33	C₃³×C₉	—	37	1	no	—
34	C₃×C₉²	—	37	36	no	—
35	C₃×C₃₇	C₃×C₉²	1	12	yes	complemented
36	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
37	C₃₇⋊C₃	C₃×C₉²	1	24	yes	complemented
38	C₃₇⋊C₃	C₃³×C₉	1	3	yes	—
39	C₃⁴	—	37	1	no	—
40	C₃²×C₉	—	37	81	no	—
41	C₃²×C₉	—	37	36	no	—
42	C₃²×C₉	—	37	2	no	—
43	C₃²×C₉	—	37	24	no	—
44	C₃²×C₉	—	37	12	no	—
45	C₃²×C₉	#66526	1	1	yes	characteristic, direct product factor
46	C₉²	—	37	81	no	—
47	C₃₇	C₃²×C₉²	1	1	yes	characteristic, complemented, Sylow
48	C₃³	—	37	27	no	—
49	C₃³	—	37	8	no	—
50	C₃³	—	37	4	no	—
51	C₃³	C₃×#66526	1	1	yes	characteristic
52	C₃×C₉	—	37	324	no	—
53	C₃×C₉	—	37	27	no	—
54	C₃×C₉	—	37	24	no	—
55	C₃×C₉	—	37	72	no	—
56	C₃×C₉	—	37	9	no	—
57	C₃×C₉	C₃×#66526	1	12	yes	direct product factor
58	C₃²	—	37	36	no	—
59	C₃²	—	37	72	no	—
60	C₃²	—	37	1	no	—
61	C₃²	—	37	8	no	—
62	C₃²	C₉×#66526	1	9	yes	direct product factor
63	C₃²	C₃²×#66526	1	4	yes	—
64	C₉	—	37	81	no	—
65	C₉	—	37	18	no	—
66	C₉	C₃²×#66526	1	9	yes	direct product factor
67	C₃	—	37	3	no	—
68	C₃	—	37	24	no	—
69	C₃	C₃×C₉×#66526	1	12	yes	direct product factor
70	C₃	C₃³×#66526	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃²×C₉	#66526
commutator, hypocenter	C₃₇	C₃²×C₉²
Frattini	C₃	C₃³×#66526
Fitting	C₃²×C₉×C₃₇	C₉
socle	C₃³×C₃₇	C₃×C₉
prime radical	C₃³×C₃₇⋊C₃	C₃²
cosocle	C₃×C₃₇⋊C₃	C₃⁴
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×#66526	C₃	direct (N × H)
4. C₃²×#66526	C₉	direct (N × H)
6. C₉×#66526	C₃²	direct (N × H)
8. C₃²×#66524	C₉	semidirect
9. C₃×C₉×C₃₇⋊C₃	C₉	semidirect
11. C₃²×C₉×C₃₇	C₉	semidirect
16. C₃×#66526	C₃×C₉	direct (N × H)
18. C₃×C₉×C₃₇	C₃×C₉	semidirect
19. C₃×#66524	C₃×C₉	semidirect
20. C₉×C₃₇⋊C₃	C₃×C₉	semidirect
23. C₃²×C₃₇	C₉²	semidirect
25. C₃×C₃₇⋊C₃	C₉²	semidirect
29. #66526	C₃²×C₉	direct (N × H)
30. C₉×C₃₇	C₃²×C₉	semidirect
31. #66524	C₃²×C₉	semidirect
35. C₃×C₃₇	C₃×C₉²	semidirect
37. C₃₇⋊C₃	C₃×C₉²	semidirect
45. C₃²×C₉	#66526	direct (N × H)
47. C₃₇	C₃²×C₉²	semidirect
57. C₃×C₉	C₃×#66526	direct (N × H)
62. C₃²	C₉×#66526	direct (N × H)
66. C₉	C₃²×#66526	direct (N × H)
69. C₃	C₃×C₉×#66526	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×#66526	3
C₃×C₉×#66526	3
C₃²×C₉×C₃₇⋊C₃	3
C₃²×C₉²	37

Subgroups — C32×C9×#66526 #41008632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₉×#66526 #41008632