Subgroups — C₃×C₃₇×H(5,𝔽₃) #41004730

Order 26973.

6776 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×H(5,𝔽₃)	C₃	1	81	yes	maximal, direct product factor
2	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	40	yes	maximal, complemented
3	C₃⁴×C₃₇	C₃²	1	40	yes	complemented
4	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	90	yes	complemented
5	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1080	yes	complemented
6	C₃³×C₃₇	—	9	120	no	—
7	C₃³×C₃₇	C₃³	1	40	yes	—
8	C₃³×C₃₇	C₃³	1	360	yes	complemented
9	H(3,𝔽₃)×C₃₇	C₃³	1	810	yes	complemented
10	C₃×H(5,𝔽₃)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃²×C₃₇	—	3	120	no	—
12	C₃²×C₃₇	—	9	360	no	—
13	C₃²×C₃₇	C₃⁴	1	120	yes	—
14	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
15	H(5,𝔽₃)	C₃×C₃₇	1	81	yes	direct product factor
16	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	40	yes	complemented
17	C₃×C₃₇	—	3	120	no	—
18	C₃×C₃₇	H(5,𝔽₃)	1	3	yes	direct product factor
19	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
20	C₃⁴	C₃²×C₃₇	1	40	yes	complemented
21	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	90	yes	complemented
22	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1080	yes	complemented
23	C₃₇	C₃×H(5,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
24	C₃³	—	9	120	no	—
25	C₃³	C₃³×C₃₇	1	40	yes	—
26	C₃³	C₃³×C₃₇	1	360	yes	complemented
27	H(3,𝔽₃)	C₃³×C₃₇	1	810	yes	complemented
28	C₃²	—	3	120	no	—
29	C₃²	—	9	360	no	—
30	C₃²	C₃⁴×C₃₇	1	120	yes	—
31	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
32	C₃	—	3	120	no	—
33	C₃	C₃₇×H(5,𝔽₃)	1	3	yes	direct product factor
34	C₃	C₃⁵×C₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₇×H(5,𝔽₃)	C₃	direct (N × H)
2. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
3. C₃⁴×C₃₇	C₃²	semidirect
4. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
5. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
8. C₃³×C₃₇	C₃³	semidirect
9. H(3,𝔽₃)×C₃₇	C₃³	semidirect
10. C₃×H(5,𝔽₃)	C₃₇	direct (N × H)
15. H(5,𝔽₃)	C₃×C₃₇	direct (N × H)
16. C₃²×H(3,𝔽₃)	C₃×C₃₇	semidirect
18. C₃×C₃₇	H(5,𝔽₃)	direct (N × H)
20. C₃⁴	C₃²×C₃₇	semidirect
21. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
22. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
23. C₃₇	C₃×H(5,𝔽₃)	direct (N × H)
26. C₃³	C₃³×C₃₇	semidirect
27. H(3,𝔽₃)	C₃³×C₃₇	semidirect
33. C₃	C₃₇×H(5,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×H(5,𝔽₃)	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃×H(5,𝔽₃)	37

Subgroups — C3×C37×H(5,𝔽3) #41004730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×H(5,𝔽₃) #41004730