Subgroups — C₃³×C₃²⋊(C₃₇⋊C₃) #41008243

Order 26973.

11744 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
2	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	117	yes	maximal, direct product factor
3	C₃⁵×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₇⋊C₃	—	3	81	no	—
5	C₃³×C₃₇⋊C₃	C₃²	1	117	yes	complemented
6	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1053	yes	direct product factor
7	C₃⁴×C₃₇	—	3	27	no	—
8	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
9	C₃⁴×C₃₇	C₃²	1	39	yes	complemented
10	C₃³×C₃₇	—	3	351	no	—
11	C₃³×C₃₇	H(3,𝔽₃)	1	27	yes	complemented
12	C₃³×C₃₇	C₃³	1	13	yes	—
13	C₃³×C₃₇	C₃³	1	117	yes	complemented
14	C₃²×C₃₇⋊C₃	—	3	1053	no	—
15	C₃²×C₃₇⋊C₃	C₃³	1	351	yes	complemented
16	C₃²⋊(C₃₇⋊C₃)	C₃³	1	729	yes	direct product factor
17	C₃³×H(3,𝔽₃)	—	37	1	no	maximal, Sylow
18	C₃²×C₃₇	—	3	351	no	—
19	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	117	yes	complemented
20	C₃²×C₃₇	C₃⁴	1	13	yes	—
21	C₃²×C₃₇	C₃⁴	1	27	yes	complemented
22	C₃×C₃₇⋊C₃	—	3	1053	no	—
23	C₃×C₃₇⋊C₃	C₃⁴	1	81	yes	complemented
24	C₃⁵	—	37	3	no	—
25	C₃⁵	C₃₇⋊C₃	1	1	yes	characteristic, complemented
26	C₃²×H(3,𝔽₃)	—	37	117	no	—
27	C₃×C₃₇	—	3	27	no	—
28	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	39	yes	complemented
29	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
30	C₃₇⋊C₃	—	3	81	no	—
31	C₃⁴	—	3	27	no	—
32	C₃⁴	—	37	117	no	—
33	C₃⁴	—	111	81	no	—
34	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
35	C₃⁴	C₃×C₃₇⋊C₃	1	39	yes	complemented
36	C₃×H(3,𝔽₃)	—	37	1053	no	—
37	C₃₇	C₃³×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
38	C₃³	—	3	351	no	—
39	C₃³	—	37	351	no	—
40	C₃³	—	111	1053	no	—
41	C₃³	C₃²⋊(C₃₇⋊C₃)	1	27	yes	direct product factor
42	C₃³	C₃²×C₃₇⋊C₃	1	13	yes	—
43	C₃³	C₃²×C₃₇⋊C₃	1	117	yes	complemented
44	H(3,𝔽₃)	—	37	729	no	—
45	C₃²	—	3	351	no	—
46	C₃²	—	37	81	no	—
47	C₃²	—	111	1053	no	—
48	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	117	yes	direct product factor
49	C₃²	C₃³×C₃₇⋊C₃	1	13	yes	—
50	C₃²	C₃³×C₃₇⋊C₃	1	27	yes	complemented
51	C₃	—	3	27	no	—
52	C₃	—	111	81	no	—
53	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	39	yes	direct product factor
54	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃⁴×C₃₇⋊C₃	C₃	semidirect
2. C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	direct (N × H)
3. C₃⁵×C₃₇	C₃	semidirect
5. C₃³×C₃₇⋊C₃	C₃²	semidirect
6. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	direct (N × H)
9. C₃⁴×C₃₇	C₃²	semidirect
11. C₃³×C₃₇	H(3,𝔽₃)	semidirect
13. C₃³×C₃₇	C₃³	semidirect
15. C₃²×C₃₇⋊C₃	C₃³	semidirect
16. C₃²⋊(C₃₇⋊C₃)	C₃³	direct (N × H)
19. C₃²×C₃₇	C₃×H(3,𝔽₃)	semidirect
21. C₃²×C₃₇	C₃⁴	semidirect
23. C₃×C₃₇⋊C₃	C₃⁴	semidirect
25. C₃⁵	C₃₇⋊C₃	semidirect
28. C₃×C₃₇	C₃²×H(3,𝔽₃)	semidirect
35. C₃⁴	C₃×C₃₇⋊C₃	semidirect
37. C₃₇	C₃³×H(3,𝔽₃)	semidirect
41. C₃³	C₃²⋊(C₃₇⋊C₃)	direct (N × H)
43. C₃³	C₃²×C₃₇⋊C₃	semidirect
48. C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	direct (N × H)
50. C₃²	C₃³×C₃₇⋊C₃	semidirect
53. C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₃₇⋊C₃	3
C₃²×C₃²⋊(C₃₇⋊C₃)	3
C₃⁵×C₃₇	3
C₃³×H(3,𝔽₃)	37

Subgroups — C33×C32⋊(C37⋊C3) #41008243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×C₃²⋊(C₃₇⋊C₃) #41008243