Subgroups — C₃²×#15688504 #41007490

Order 26973.

1484 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#66524	C₃	1	1	yes	maximal, characteristic
2	C₃³×#66524	C₃	1	1	yes	maximal, characteristic
3	C₃³×#66524	C₃	1	1	yes	maximal, characteristic
4	C₃×#15688504	C₃	1	36	yes	maximal, direct product factor
5	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	—	3	9	no	—
7	C₃²×#66524	—	3	9	no	—
8	C₃²×#66524	C₉	1	27	yes	complemented
9	C₃²×#66524	C₃²	1	12	yes	—
10	C₃²×#66524	C₃²	1	12	yes	—
11	C₃²×#66524	C₃²	1	12	yes	—
12	#15688504	C₃²	1	81	yes	direct product factor
13	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
14	C₃²×C₉×C₃₇	—	3	9	no	—
15	C₃²×C₉×C₃₇	C₃²	1	12	yes	—
16	C₃³×C₃₇	C₃×C₉	1	9	yes	—
17	C₃³×C₃₇	3₋¹⁺²	1	9	yes	—
18	C₃³×C₃₇	3₋¹⁺²	1	9	yes	—
19	C₃³×C₃₇	3₋¹⁺²	1	9	yes	—
20	C₃³×C₃₇	C₃³	1	4	yes	—
21	C₃×C₉×C₃₇	—	3	36	no	—
22	C₃×C₉×C₃₇	C₃³	1	9	yes	—
23	C₃×#66524	—	3	36	no	—
24	C₃×#66524	—	3	36	no	—
25	C₃×#66524	C₃×C₉	1	108	yes	complemented
26	C₃×#66524	C₃³	1	9	yes	—
27	C₃×#66524	C₃³	1	9	yes	—
28	C₃×#66524	C₃³	1	9	yes	—
29	C₃²×C₉⋊C₉	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	C₉⋊C₉	1	81	yes	complemented
31	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
32	C₃²×C₃₇	C₃×3₋¹⁺²	1	12	yes	—
33	C₃²×C₃₇	C₃×3₋¹⁺²	1	12	yes	—
34	C₃²×C₃₇	C₃×3₋¹⁺²	1	12	yes	—
35	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
36	C₉×C₃₇	—	3	9	no	—
37	#66524	—	3	9	no	—
38	#66524	—	3	9	no	—
39	#66524	C₃²×C₉	1	27	yes	complemented
40	C₃³×C₉	—	37	1	no	—
41	C₃³×C₉	—	37	1	no	—
42	C₃³×C₉	—	37	1	no	—
43	C₃³×C₉	C₃₇⋊C₃	1	1	yes	characteristic
44	C₃×C₉⋊C₉	—	37	36	no	—
45	C₃×C₃₇	C₃×C₉⋊C₉	1	36	yes	complemented
46	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
47	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
48	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
49	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
50	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	C₃²×C₉	—	3	9	no	—
52	C₃²×C₉	—	37	27	no	—
53	C₃²×C₉	—	37	12	no	—
54	C₃²×C₉	—	37	12	no	—
55	C₃²×C₉	—	37	12	no	—
56	C₃²×C₉	—	111	9	no	—
57	C₃²×C₉	—	111	9	no	—
58	C₃²×C₉	C₃×C₃₇⋊C₃	1	12	yes	—
59	C₉⋊C₉	—	37	81	no	—
60	C₃₇	C₃²×C₉⋊C₉	1	1	yes	characteristic, complemented, Sylow
61	C₃³	C₉×C₃₇⋊C₃	1	9	yes	—
62	C₃³	#11758605	1	9	yes	—
63	C₃³	#11758606	1	9	yes	—
64	C₃³	#11758608	1	9	yes	—
65	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
66	C₃×C₉	—	3	36	no	—
67	C₃×C₉	—	37	108	no	—
68	C₃×C₉	—	37	9	no	—
69	C₃×C₉	—	37	9	no	—
70	C₃×C₉	—	37	9	no	—
71	C₃×C₉	—	111	36	no	—
72	C₃×C₉	—	111	36	no	—
73	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
74	C₃²	#15688504	1	81	yes	direct product factor
75	C₃²	C₃×C₉×C₃₇⋊C₃	1	12	yes	—
76	C₃²	C₃×#11758606	1	12	yes	—
77	C₃²	C₃×#11758605	1	12	yes	—
78	C₃²	C₃×#11758608	1	12	yes	—
79	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
80	C₉	—	3	9	no	—
81	C₉	—	37	27	no	—
82	C₉	—	111	9	no	—
83	C₉	—	111	9	no	—
84	C₃	C₃×#15688504	1	36	yes	direct product factor
85	C₃	C₃²×C₉×C₃₇⋊C₃	1	1	yes	characteristic
86	C₃	C₃²×#11758606	1	1	yes	characteristic
87	C₃	C₃²×#11758605	1	1	yes	characteristic
88	C₃	C₃²×#11758608	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃×#15688504	C₃	direct (N × H)
8. C₃²×#66524	C₉	semidirect
12. #15688504	C₃²	direct (N × H)
25. C₃×#66524	C₃×C₉	semidirect
30. C₃²×C₃₇	C₉⋊C₉	semidirect
39. #66524	C₃²×C₉	semidirect
45. C₃×C₃₇	C₃×C₉⋊C₉	semidirect
60. C₃₇	C₃²×C₉⋊C₉	semidirect
74. C₃²	#15688504	direct (N × H)
84. C₃	C₃×#15688504	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×#66524	3
C₃×#15688504	3
C₃³×C₉×C₃₇	3
C₃²×C₉⋊C₉	37

Subgroups — C32×#15688504 #41007490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#15688504 #41007490