Subgroups — C₃×C₃₇⋊H(5,𝔽₃) #41008253

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×C₃²⋊(C₃₇⋊C₃)	C₃	1	12	yes	maximal, complemented
3	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	27	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	C₃²	1	36	yes	complemented
6	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	72	yes	complemented
7	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	9	yes	complemented
8	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	324	yes	complemented
9	H(3,𝔽₃)×C₃₇⋊C₃	C₃²	1	729	yes	complemented
10	C₃⁴×C₃₇	C₃²	1	4	yes	complemented
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	complemented
12	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	27	yes	complemented
13	C₃³×C₃₇	—	9	12	no	—
14	C₃³×C₃₇	C₃³	1	12	yes	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃³×C₃₇	C₃³	1	36	yes	complemented
17	C₃²×C₃₇⋊C₃	—	9	108	no	—
18	C₃²×C₃₇⋊C₃	C₃³	1	27	yes	—
19	C₃²×C₃₇⋊C₃	C₃³	1	324	yes	complemented
20	H(3,𝔽₃)×C₃₇	C₃³	1	81	yes	complemented
21	C₃²⋊(C₃₇⋊C₃)	C₃³	1	648	yes	complemented
22	C₃²⋊(C₃₇⋊C₃)	C₃³	1	81	yes	complemented
23	C₃×H(5,𝔽₃)	—	37	1	no	maximal, Sylow
24	C₃²×C₃₇	—	3	36	no	—
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	9	36	no	—
27	C₃²×C₃₇	C₃⁴	1	3	yes	—
28	C₃²×C₃₇	C₃⁴	1	36	yes	—
29	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
30	C₃×C₃₇⋊C₃	—	3	81	no	—
31	C₃×C₃₇⋊C₃	—	9	324	no	—
32	C₃×C₃₇⋊C₃	C₃⁴	1	81	yes	—
33	H(5,𝔽₃)	—	37	81	no	—
34	C₃²×H(3,𝔽₃)	—	37	12	no	—
35	C₃²×H(3,𝔽₃)	—	37	27	no	—
36	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
37	C₃×C₃₇	—	3	36	no	—
38	C₃×C₃₇	—	3	3	no	—
39	C₃×C₃₇	H(5,𝔽₃)	1	3	yes	complemented
40	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
41	C₃₇⋊C₃	—	3	81	no	—
42	C₃⁴	—	37	36	no	—
43	C₃⁴	C₃×C₃₇⋊C₃	1	4	yes	complemented
44	C₃×H(3,𝔽₃)	—	37	9	no	—
45	C₃×H(3,𝔽₃)	—	37	72	no	—
46	C₃×H(3,𝔽₃)	—	37	324	no	—
47	C₃×H(3,𝔽₃)	—	37	729	no	—
48	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	9	yes	complemented
49	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	27	yes	complemented
50	C₃₇	C₃×H(5,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
51	C₃³	—	9	12	no	—
52	C₃³	—	37	324	no	—
53	C₃³	—	37	27	no	—
54	C₃³	—	333	108	no	—
55	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
56	C₃³	C₃²×C₃₇⋊C₃	1	12	yes	—
57	C₃³	C₃²×C₃₇⋊C₃	1	36	yes	complemented
58	H(3,𝔽₃)	—	37	81	no	—
59	H(3,𝔽₃)	—	37	648	no	—
60	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	81	yes	complemented
61	C₃²	—	3	3	no	—
62	C₃²	—	3	36	no	—
63	C₃²	—	9	36	no	—
64	C₃²	—	37	81	no	—
65	C₃²	—	333	324	no	—
66	C₃²	—	111	81	no	—
67	C₃²	C₃³×C₃₇⋊C₃	1	3	yes	—
68	C₃²	C₃³×C₃₇⋊C₃	1	36	yes	—
69	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
70	C₃	—	3	3	no	—
71	C₃	—	3	36	no	—
72	C₃	—	111	81	no	—
73	C₃	C₃₇⋊H(5,𝔽₃)	1	3	yes	direct product factor
74	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃³×C₃₇⋊C₃
commutator, cosocle	C₃×C₃₇	C₃⁵
Frattini	C₃	C₃⁴×C₃₇⋊C₃
Fitting	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃²×C₃₇	C₃⁴
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	C₃×H(5,𝔽₃)
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃)

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×C₃²⋊(C₃₇⋊C₃)	C₃	semidirect
3. C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	semidirect
4. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
5. C₃³×C₃₇⋊C₃	C₃²	semidirect
6. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
7. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
8. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
9. H(3,𝔽₃)×C₃₇⋊C₃	C₃²	semidirect
10. C₃⁴×C₃₇	C₃²	semidirect
11. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
12. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
16. C₃³×C₃₇	C₃³	semidirect
19. C₃²×C₃₇⋊C₃	C₃³	semidirect
20. H(3,𝔽₃)×C₃₇	C₃³	semidirect
21. C₃²⋊(C₃₇⋊C₃)	C₃³	semidirect
22. C₃²⋊(C₃₇⋊C₃)	C₃³	semidirect
36. C₃²×H(3,𝔽₃)	C₃₇⋊C₃	semidirect
39. C₃×C₃₇	H(5,𝔽₃)	semidirect
43. C₃⁴	C₃×C₃₇⋊C₃	semidirect
48. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	semidirect
49. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	semidirect
50. C₃₇	C₃×H(5,𝔽₃)	semidirect
57. C₃³	C₃²×C₃₇⋊C₃	semidirect
60. H(3,𝔽₃)	C₃²×C₃₇⋊C₃	semidirect
73. 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×C₃²⋊(C₃₇⋊C₃)	3
C₃×H(3,𝔽₃)×C₃₇⋊C₃	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃×H(5,𝔽₃)	37

Subgroups — C3×C37⋊H(5,𝔽3) #41008253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇⋊H(5,𝔽₃) #41008253