Subgroups — C₃×C₃₇×C₃²⋊H(3,𝔽₃) #41004654

Order 26973.

3824 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁵×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	12	yes	maximal, complemented
3	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	27	yes	maximal, direct product factor
4	C₃⁴×C₃₇	—	3	36	no	—
5	C₃⁴×C₃₇	C₃²	1	4	yes	—
6	C₃⁴×C₃₇	C₃²	1	9	yes	complemented
7	C₃⁴×C₃₇	C₃²	1	9	yes	complemented
8	C₃×H(3,𝔽₃)×C₃₇	—	3	108	no	—
9	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	108	yes	complemented
10	C₃³×C₃₇	—	3	243	no	—
11	C₃³×C₃₇	—	3	36	no	—
12	C₃³×C₃₇	—	3	108	no	—
13	C₃³×C₃₇	—	3	108	no	—
14	C₃³×C₃₇	H(3,𝔽₃)	1	36	yes	complemented
15	C₃³×C₃₇	C₃³	1	12	yes	—
16	C₃³×C₃₇	C₃³	1	1	yes	characteristic
17	C₃³×C₃₇	C₃³	1	27	yes	complemented
18	H(3,𝔽₃)×C₃₇	—	3	324	no	—
19	C₃×C₃²⋊H(3,𝔽₃)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃²×C₃₇	—	3	243	no	—
21	C₃²×C₃₇	—	3	36	no	—
22	C₃²×C₃₇	—	3	108	no	—
23	C₃²×C₃₇	—	3	108	no	—
24	C₃²×C₃₇	—	9	81	no	—
25	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
26	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	36	yes	complemented
27	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
28	C₃⁵	C₃×C₃₇	1	1	yes	characteristic, complemented
29	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	12	yes	complemented
30	C₃²⋊H(3,𝔽₃)	C₃×C₃₇	1	27	yes	direct product factor
31	C₃×C₃₇	—	3	36	no	—
32	C₃×C₃₇	—	9	27	no	—
33	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	9	yes	direct product factor
34	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	4	yes	—
35	C₃⁴	—	3	36	no	—
36	C₃⁴	C₃²×C₃₇	1	4	yes	—
37	C₃⁴	C₃²×C₃₇	1	9	yes	complemented
38	C₃⁴	C₃²×C₃₇	1	9	yes	complemented
39	C₃×H(3,𝔽₃)	—	3	108	no	—
40	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	108	yes	complemented
41	C₃₇	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
42	C₃³	—	3	243	no	—
43	C₃³	—	3	36	no	—
44	C₃³	—	3	108	no	—
45	C₃³	—	3	108	no	—
46	C₃³	H(3,𝔽₃)×C₃₇	1	36	yes	complemented
47	C₃³	C₃³×C₃₇	1	12	yes	—
48	C₃³	C₃³×C₃₇	1	1	yes	characteristic
49	C₃³	C₃³×C₃₇	1	27	yes	complemented
50	H(3,𝔽₃)	—	3	324	no	—
51	C₃²	—	3	243	no	—
52	C₃²	—	3	108	no	—
53	C₃²	—	3	36	no	—
54	C₃²	—	3	108	no	—
55	C₃²	—	9	81	no	—
56	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	12	yes	—
57	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	36	yes	complemented
58	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
59	C₃	—	3	36	no	—
60	C₃	—	9	27	no	—
61	C₃	C₃₇×C₃²⋊H(3,𝔽₃)	1	9	yes	direct product factor
62	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	4	yes	—

	subgroup	G/H
center, socle	C₃³×C₃₇	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×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	semidirect
3. C₃₇×C₃²⋊H(3,𝔽₃)	C₃	direct (N × H)
6. C₃⁴×C₃₇	C₃²	semidirect
7. C₃⁴×C₃₇	C₃²	semidirect
9. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
14. C₃³×C₃₇	H(3,𝔽₃)	semidirect
17. C₃³×C₃₇	C₃³	semidirect
19. C₃×C₃²⋊H(3,𝔽₃)	C₃₇	direct (N × H)
26. C₃²×C₃₇	C₃×H(3,𝔽₃)	semidirect
28. C₃⁵	C₃×C₃₇	semidirect
29. C₃²×H(3,𝔽₃)	C₃×C₃₇	semidirect
30. C₃²⋊H(3,𝔽₃)	C₃×C₃₇	direct (N × H)
33. C₃×C₃₇	C₃²⋊H(3,𝔽₃)	direct (N × H)
37. C₃⁴	C₃²×C₃₇	semidirect
38. C₃⁴	C₃²×C₃₇	semidirect
40. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
41. C₃₇	C₃×C₃²⋊H(3,𝔽₃)	direct (N × H)
46. C₃³	H(3,𝔽₃)×C₃₇	semidirect
49. C₃³	C₃³×C₃₇	semidirect
57. C₃²	C₃×H(3,𝔽₃)×C₃₇	semidirect
61. C₃	C₃₇×C₃²⋊H(3,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
C₃⁵×C₃₇	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
C₃×C₃²⋊H(3,𝔽₃)	37

Subgroups — C3×C37×C32⋊H(3,𝔽3) #41004654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×C₃²⋊H(3,𝔽₃) #41004654