Subgroups — C₃²×#15688541 #41008646

Order 26973.

944 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688541	C₃	1	36	yes	maximal, direct product factor
2	C₃²×#11758603	C₃	1	3	yes	maximal, complemented
3	C₃³×#66524	C₃	1	1	yes	maximal, characteristic
4	#15688541	C₃²	1	81	yes	direct product factor
5	C₃×#11758603	C₃²	1	36	yes	complemented
6	C₃²×#66524	C₉	1	24	yes	—
7	C₃²×#66524	C₉	1	2	yes	—
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	C₃²×#66524	C₃²	1	12	yes	—
10	C₃⁴×C₃₇	C₉	1	1	yes	characteristic
11	C₃³×C₃₇	—	3	9	no	—
12	C₃³×C₃₇	C₃×C₉	1	12	yes	—
13	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₃×#66524	C₃×C₉	1	72	yes	—
15	C₃×#66524	C₃×C₉	1	8	yes	—
16	C₃×#66524	C₃×C₉	1	24	yes	—
17	C₃×#66524	C₃³	1	4	yes	—
18	C₃×#66524	C₃³	1	9	yes	—
19	#11758603	C₃³	1	27	yes	complemented
20	C₃²×C₂₇⋊C₃	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	—	3	36	no	—
22	C₃²×C₃₇	C₂₇⋊C₃	1	9	yes	complemented
23	C₃²×C₃₇	C₃²×C₉	1	9	yes	—
24	C₃²×C₃₇	C₃²×C₉	1	4	yes	—
25	#66524	C₃²×C₉	1	8	yes	—
26	#66524	C₃²×C₉	1	18	yes	—
27	#66524	C₃⁴	1	1	yes	characteristic
28	C₃³×C₉	—	37	1	no	—
29	C₃×C₂₇⋊C₃	—	37	36	no	—
30	C₃²×C₂₇	—	37	3	no	—
31	C₃×C₃₇	—	3	9	no	—
32	C₃×C₃₇	C₃×C₂₇⋊C₃	1	12	yes	complemented
33	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
34	C₃⁴	#66526	1	1	yes	characteristic
35	C₃²×C₉	—	37	24	no	—
36	C₃²×C₉	—	37	12	no	—
37	C₃²×C₉	—	37	2	no	—
38	C₃²×C₉	—	37	1	no	—
39	C₂₇⋊C₃	—	37	81	no	—
40	C₃×C₂₇	—	37	36	no	—
41	C₃₇	C₃²×C₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	3	9	no	—
43	C₃³	C₃×#66526	1	12	yes	—
44	C₃³	C₃×#66526	1	1	yes	characteristic
45	C₃×C₉	—	37	72	no	—
46	C₃×C₉	—	37	8	no	—
47	C₃×C₉	—	37	24	no	—
48	C₃×C₉	—	37	4	no	—
49	C₃×C₉	—	37	9	no	—
50	C₂₇	—	37	27	no	—
51	C₃²	—	3	36	no	—
52	C₃²	#15688541	1	9	yes	direct product factor
53	C₃²	C₃²×#66526	1	9	yes	—
54	C₃²	C₃²×#66526	1	4	yes	—
55	C₉	—	37	1	no	—
56	C₉	—	37	8	no	—
57	C₉	—	37	18	no	—
58	C₃	—	3	9	no	—
59	C₃	C₃×#15688541	1	12	yes	direct product factor
60	C₃	C₃³×#66526	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃×#15688541	C₃	direct (N × H)
2. C₃²×#11758603	C₃	semidirect
4. #15688541	C₃²	direct (N × H)
5. C₃×#11758603	C₃²	semidirect
19. #11758603	C₃³	semidirect
22. C₃²×C₃₇	C₂₇⋊C₃	semidirect
32. C₃×C₃₇	C₃×C₂₇⋊C₃	semidirect
41. C₃₇	C₃²×C₂₇⋊C₃	semidirect
52. C₃²	#15688541	direct (N × H)
59. C₃	C₃×#15688541	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×#15688541	3
C₃²×#11758603	3
C₃³×#66524	3
C₃²×C₂₇⋊C₃	37

Subgroups — C32×#15688541 #41008646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#15688541 #41008646