Subgroups — C₃³×H(3,𝔽₃)×C₃₇ #41004727

Order 26973.

11744 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁵×C₃₇	C₃	1	4	yes	maximal, complemented
2	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	117	yes	maximal, direct product factor
3	C₃⁴×C₃₇	—	3	108	no	—
4	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
5	C₃⁴×C₃₇	C₃²	1	156	yes	complemented
6	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1053	yes	direct product factor
7	C₃³×C₃₇	—	3	1404	no	—
8	C₃³×C₃₇	H(3,𝔽₃)	1	27	yes	direct product factor
9	C₃³×C₃₇	C₃³	1	13	yes	—
10	C₃³×C₃₇	C₃³	1	468	yes	complemented
11	H(3,𝔽₃)×C₃₇	C₃³	1	729	yes	direct product factor
12	C₃³×H(3,𝔽₃)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃²×C₃₇	—	3	1404	no	—
14	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	117	yes	direct product factor
15	C₃²×C₃₇	C₃⁴	1	13	yes	—
16	C₃²×C₃₇	C₃⁴	1	108	yes	complemented
17	C₃⁵	C₃×C₃₇	1	4	yes	complemented
18	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	117	yes	direct product factor
19	C₃×C₃₇	—	3	108	no	—
20	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	39	yes	direct product factor
21	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
22	C₃⁴	—	3	108	no	—
23	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
24	C₃⁴	C₃²×C₃₇	1	156	yes	complemented
25	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1053	yes	direct product factor
26	C₃₇	C₃³×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
27	C₃³	—	3	1404	no	—
28	C₃³	H(3,𝔽₃)×C₃₇	1	27	yes	direct product factor
29	C₃³	C₃³×C₃₇	1	13	yes	—
30	C₃³	C₃³×C₃₇	1	468	yes	complemented
31	H(3,𝔽₃)	C₃³×C₃₇	1	729	yes	direct product factor
32	C₃²	—	3	1404	no	—
33	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	117	yes	direct product factor
34	C₃²	C₃⁴×C₃₇	1	13	yes	—
35	C₃²	C₃⁴×C₃₇	1	108	yes	complemented
36	C₃	—	3	108	no	—
37	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	39	yes	direct product factor
38	C₃	C₃⁵×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃⁴×C₃₇	C₃²
commutator, Frattini, cosocle	C₃	C₃⁵×C₃₇
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃⁴×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃⁵×C₃₇	C₃	semidirect
2. C₃²×H(3,𝔽₃)×C₃₇	C₃	direct (N × H)
5. C₃⁴×C₃₇	C₃²	semidirect
6. C₃×H(3,𝔽₃)×C₃₇	C₃²	direct (N × H)
8. C₃³×C₃₇	H(3,𝔽₃)	direct (N × H)
10. C₃³×C₃₇	C₃³	semidirect
11. H(3,𝔽₃)×C₃₇	C₃³	direct (N × H)
12. C₃³×H(3,𝔽₃)	C₃₇	direct (N × H)
14. C₃²×C₃₇	C₃×H(3,𝔽₃)	direct (N × H)
16. C₃²×C₃₇	C₃⁴	semidirect
17. C₃⁵	C₃×C₃₇	semidirect
18. C₃²×H(3,𝔽₃)	C₃×C₃₇	direct (N × H)
20. C₃×C₃₇	C₃²×H(3,𝔽₃)	direct (N × H)
24. C₃⁴	C₃²×C₃₇	semidirect
25. C₃×H(3,𝔽₃)	C₃²×C₃₇	direct (N × H)
26. C₃₇	C₃³×H(3,𝔽₃)	direct (N × H)
28. C₃³	H(3,𝔽₃)×C₃₇	direct (N × H)
30. C₃³	C₃³×C₃₇	semidirect
31. H(3,𝔽₃)	C₃³×C₃₇	direct (N × H)
33. C₃²	C₃×H(3,𝔽₃)×C₃₇	direct (N × H)
35. C₃²	C₃⁴×C₃₇	semidirect
37. C₃	C₃²×H(3,𝔽₃)×C₃₇	direct (N × H)

Maximal subgroups

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

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

Subgroups — C33×H(3,𝔽3)×C37 #41004727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×H(3,𝔽₃)×C₃₇ #41004727