Subgroups — C₃²×#15688542 #41008648

Order 26973.

944 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688542	C₃	1	36	yes	maximal, direct product factor
2	C₃²×#11758603	C₃	1	3	yes	maximal, complemented
3	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#15688542	C₃²	1	81	yes	direct product factor
5	C₃×#11758603	C₃²	1	36	yes	complemented
6	C₃³×C₃₇⋊C₃	C₉	1	1	yes	characteristic
7	C₃²×#66524	C₉	1	1	yes	characteristic
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	C₃×C₉×C₃₇⋊C₃	C₉	1	24	yes	—
10	C₃×C₉×C₃₇⋊C₃	C₃²	1	12	yes	—
11	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	—	3	9	no	—
14	C₃²×C₃₇⋊C₃	C₃×C₉	1	12	yes	—
15	C₃×C₉×C₃₇	C₃×C₉	1	12	yes	—
16	C₃×#66524	C₃×C₉	1	8	yes	—
17	C₃×#66524	C₃×C₉	1	12	yes	—
18	C₃×#66524	C₃³	1	4	yes	—
19	C₉×C₃₇⋊C₃	C₃×C₉	1	72	yes	—
20	C₉×C₃₇⋊C₃	C₃³	1	9	yes	—
21	#11758603	C₃³	1	27	yes	complemented
22	C₃²×C₂₇⋊C₃	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	C₂₇⋊C₃	1	9	yes	complemented
24	C₃²×C₃₇	C₃²×C₉	1	4	yes	—
25	C₃×C₃₇⋊C₃	—	3	36	no	—
26	C₃×C₃₇⋊C₃	C₃²×C₉	1	9	yes	—
27	C₉×C₃₇	C₃²×C₉	1	9	yes	—
28	#66524	C₃²×C₉	1	8	yes	—
29	#66524	C₃²×C₉	1	9	yes	—
30	#66524	C₃⁴	1	1	yes	characteristic
31	C₃³×C₉	—	37	1	no	—
32	C₃×C₂₇⋊C₃	—	37	36	no	—
33	C₃²×C₂₇	—	37	3	no	—
34	C₃×C₃₇	C₃×C₂₇⋊C₃	1	12	yes	complemented
35	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
36	C₃₇⋊C₃	—	3	9	no	—
37	C₃⁴	—	37	1	no	—
38	C₃²×C₉	—	37	1	no	—
39	C₃²×C₉	—	37	1	no	—
40	C₃²×C₉	—	37	24	no	—
41	C₃²×C₉	—	37	12	no	—
42	C₃²×C₉	#66526	1	1	yes	characteristic
43	C₂₇⋊C₃	—	37	81	no	—
44	C₃×C₂₇	—	37	36	no	—
45	C₃₇	C₃²×C₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
46	C₃³	—	37	12	no	—
47	C₃³	—	111	9	no	—
48	C₃³	C₃×#66526	1	1	yes	characteristic
49	C₃×C₉	—	37	8	no	—
50	C₃×C₉	—	37	12	no	—
51	C₃×C₉	—	37	4	no	—
52	C₃×C₉	—	37	72	no	—
53	C₃×C₉	—	37	9	no	—
54	C₃×C₉	C₃×#66526	1	12	yes	—
55	C₂₇	—	37	27	no	—
56	C₃²	—	37	9	no	—
57	C₃²	—	111	36	no	—
58	C₃²	#15688542	1	9	yes	direct product factor
59	C₃²	C₃²×#66526	1	4	yes	—
60	C₉	—	37	1	no	—
61	C₉	—	37	9	no	—
62	C₉	—	37	8	no	—
63	C₉	C₃²×#66526	1	9	yes	—
64	C₃	—	111	9	no	—
65	C₃	C₃×#15688542	1	12	yes	direct product factor
66	C₃	C₃³×#66526	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃×#66526
commutator	C₃×C₃₇	C₃³×C₉
Frattini	C₃	C₃³×#66526
Fitting	C₃²×C₉×C₃₇	C₉
socle	C₃³×C₃₇	C₃×C₉
prime radical	C₃³×C₃₇⋊C₃	C₉
cosocle	#66524	C₃⁴
hypercenter	C₃²×C₉	#66526
hypocenter	C₃₇	C₃²×C₂₇⋊C₃
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
1. C₃×#15688542	C₃	direct (N × H)
2. C₃²×#11758603	C₃	semidirect
4. #15688542	C₃²	direct (N × H)
5. C₃×#11758603	C₃²	semidirect
21. #11758603	C₃³	semidirect
23. C₃²×C₃₇	C₂₇⋊C₃	semidirect
34. C₃×C₃₇	C₃×C₂₇⋊C₃	semidirect
45. C₃₇	C₃²×C₂₇⋊C₃	semidirect
58. C₃²	#15688542	direct (N × H)
65. C₃	C₃×#15688542	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×#15688542	3
C₃²×#11758603	3
C₃²×C₉×C₃₇⋊C₃	3
C₃²×C₂₇⋊C₃	37

Subgroups — C32×#15688542 #41008648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#15688542 #41008648