Subgroups — C₃×#22582680 #41008258

Order 26973.

5588 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582680	C₃	1	81	yes	maximal, direct product factor
2	C₃×#15688530	C₃	1	27	yes	maximal, complemented
3	C₃²×#11758608	C₃	1	12	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
5	#15688530	C₃²	1	729	yes	complemented
6	C₃×#11758608	C₃²	1	9	yes	—
7	C₃×#11758608	C₃²	1	72	yes	complemented
8	C₃×#11758608	C₃²	1	324	yes	complemented
9	C₃²×#66524	C₃²	1	36	yes	complemented
10	C₃⁴×C₃₇	C₃²	1	4	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
12	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	27	yes	—
13	C₃³×C₃₇	—	9	12	no	—
14	C₃³×C₃₇	C₃³	1	12	yes	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃³×C₃₇	C₃³	1	36	yes	—
17	H(3,𝔽₃)×C₃₇	C₃³	1	81	yes	—
18	#11758608	C₃³	1	81	yes	—
19	#11758608	C₃³	1	648	yes	complemented
20	C₃×#66524	C₃³	1	27	yes	—
21	C₃×#66524	C₃³	1	324	yes	complemented
22	C₃×3₋¹⁺⁴	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	36	no	—
25	C₃²×C₃₇	—	9	36	no	—
26	C₃²×C₃₇	C₃⁴	1	36	yes	—
27	C₃²×C₃₇	C₃⁴	1	3	yes	—
28	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
29	#66524	C₃⁴	1	81	yes	—
30	3₋¹⁺⁴	—	37	81	no	—
31	C₃×#737	—	37	27	no	—
32	C₃²×3₋¹⁺²	—	37	12	no	—
33	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
34	C₃×C₃₇	—	3	3	no	—
35	C₃×C₃₇	—	3	36	no	—
36	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	complemented
37	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
38	C₃⁴	C₃×C₃₇⋊C₃	1	4	yes	—
39	#737	—	37	729	no	—
40	C₃×3₋¹⁺²	—	37	9	no	—
41	C₃×3₋¹⁺²	—	37	72	no	—
42	C₃×3₋¹⁺²	—	37	324	no	—
43	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	9	yes	—
44	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	27	yes	—
45	C₃²×C₉	—	37	36	no	—
46	C₃₇	C₃×3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
47	C₃³	—	9	12	no	—
48	C₃³	C₃²×C₃₇⋊C₃	1	12	yes	—
49	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
50	C₃³	C₃²×C₃₇⋊C₃	1	36	yes	—
51	3₋¹⁺²	—	37	81	no	—
52	3₋¹⁺²	—	37	648	no	—
53	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	81	yes	—
54	C₃×C₉	—	37	324	no	—
55	C₃×C₉	—	37	27	no	—
56	C₃²	—	3	3	no	—
57	C₃²	—	3	36	no	—
58	C₃²	—	9	36	no	—
59	C₃²	C₃³×C₃₇⋊C₃	1	36	yes	—
60	C₃²	C₃³×C₃₇⋊C₃	1	3	yes	—
61	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
62	C₉	—	37	81	no	—
63	C₃	—	3	3	no	—
64	C₃	—	3	36	no	—
65	C₃	#22582680	1	3	yes	direct product factor
66	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃³×C₃₇⋊C₃
commutator, cosocle	C₃×C₃₇	C₃⁵
Frattini	C₃	C₃⁴×C₃₇⋊C₃
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃²×C₃₇	C₃⁴
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	C₃×3₋¹⁺⁴
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #22582680	C₃	direct (N × H)
2. C₃×#15688530	C₃	semidirect
3. C₃²×#11758608	C₃	semidirect
5. #15688530	C₃²	semidirect
7. C₃×#11758608	C₃²	semidirect
8. C₃×#11758608	C₃²	semidirect
9. C₃²×#66524	C₃²	semidirect
19. #11758608	C₃³	semidirect
21. C₃×#66524	C₃³	semidirect
36. C₃×C₃₇	3₋¹⁺⁴	semidirect
46. C₃₇	C₃×3₋¹⁺⁴	semidirect
65. C₃	#22582680	direct (N × H)

Maximal subgroups

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

subgroup	index
#22582680	3
C₃×#15688530	3
C₃²×#11758608	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃×3₋¹⁺⁴	37

Subgroups — C3×#22582680 #41008258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582680 #41008258