Subgroups — C₃²×#15688530 #41008249

Order 26973.

5804 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688530	C₃	1	108	yes	maximal, direct product factor
2	C₃²×#11758608	C₃	1	8	yes	maximal, complemented
3	C₃³×#66524	C₃	1	4	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
5	#15688530	C₃²	1	729	yes	direct product factor
6	C₃×#11758608	C₃²	1	288	yes	complemented
7	C₃²×#66524	C₃²	1	1	yes	characteristic
8	C₃²×#66524	C₃²	1	8	yes	—
9	C₃²×#66524	C₃²	1	144	yes	complemented
10	C₃⁴×C₃₇	C₃²	1	4	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	36	yes	—
12	C₃³×C₃₇	—	3	36	no	—
13	C₃³×C₃₇	C₃³	1	48	yes	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	H(3,𝔽₃)×C₃₇	C₃³	1	81	yes	—
16	#11758608	C₃³	1	648	yes	complemented
17	C₃×#66524	C₃³	1	12	yes	—
18	C₃×#66524	C₃³	1	96	yes	—
19	C₃×#66524	C₃³	1	324	yes	complemented
20	C₃²×#737	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	—	3	144	no	—
22	C₃²×C₃₇	#737	1	9	yes	complemented
23	C₃²×C₃₇	C₃⁴	1	36	yes	—
24	C₃²×C₃₇	C₃⁴	1	4	yes	—
25	#66524	C₃⁴	1	9	yes	—
26	#66524	C₃⁴	1	72	yes	—
27	C₃×#737	—	37	108	no	—
28	C₃²×3₋¹⁺²	—	37	8	no	—
29	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
30	C₃³×C₉	—	37	4	no	—
31	C₃×C₃₇	—	3	36	no	—
32	C₃×C₃₇	C₃×#737	1	12	yes	complemented
33	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
34	C₃⁴	C₃×C₃₇⋊C₃	1	4	yes	—
35	#737	—	37	729	no	—
36	C₃×3₋¹⁺²	—	37	288	no	—
37	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	36	yes	—
38	C₃²×C₉	—	37	144	no	—
39	C₃²×C₉	—	37	1	no	—
40	C₃²×C₉	—	37	8	no	—
41	C₃₇	C₃²×#737	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	3	36	no	—
43	C₃³	C₃²×C₃₇⋊C₃	1	48	yes	—
44	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
45	3₋¹⁺²	—	37	648	no	—
46	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	81	yes	—
47	C₃×C₉	—	37	12	no	—
48	C₃×C₉	—	37	96	no	—
49	C₃×C₉	—	37	324	no	—
50	C₃²	—	3	144	no	—
51	C₃²	#15688530	1	9	yes	direct product factor
52	C₃²	C₃³×C₃₇⋊C₃	1	36	yes	—
53	C₃²	C₃³×C₃₇⋊C₃	1	4	yes	—
54	C₉	—	37	9	no	—
55	C₉	—	37	72	no	—
56	C₃	—	3	36	no	—
57	C₃	C₃×#15688530	1	12	yes	direct product factor
58	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃²×C₃₇⋊C₃
commutator, cosocle	C₃×C₃₇	C₃⁵
Frattini	C₃	C₃⁴×C₃₇⋊C₃
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃³×C₃₇	C₃³
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	C₃²×#737
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#15688530	C₃	direct (N × H)
2. C₃²×#11758608	C₃	semidirect
3. C₃³×#66524	C₃	semidirect
5. #15688530	C₃²	direct (N × H)
6. C₃×#11758608	C₃²	semidirect
9. C₃²×#66524	C₃²	semidirect
16. #11758608	C₃³	semidirect
19. C₃×#66524	C₃³	semidirect
22. C₃²×C₃₇	#737	semidirect
32. C₃×C₃₇	C₃×#737	semidirect
41. C₃₇	C₃²×#737	semidirect
51. C₃²	#15688530	direct (N × H)
57. C₃	C₃×#15688530	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×#15688530	3
C₃²×#11758608	3
C₃³×#66524	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃²×#737	37

Subgroups — C32×#15688530 #41008249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#15688530 #41008249