Subgroups — C₂³×C₉×H(3,𝔽₃) #14518556

Order 1944.

1616 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×H(3,𝔽₃)	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₂³×C₃²×C₉	C₃	1	4	yes	maximal, complemented
4	C₂³×C₃²⋊C₉	C₃	1	8	yes	maximal, complemented
5	C₂×C₉×H(3,𝔽₃)	C₂²	1	7	yes	direct product factor
6	C₂²×C₃×H(3,𝔽₃)	C₂×C₃	1	7	yes	—
7	C₂²×C₃²×C₉	C₂×C₃	1	28	yes	complemented
8	C₂²×C₃²⋊C₉	C₂×C₃	1	56	yes	complemented
9	C₉×H(3,𝔽₃)	C₂³	1	1	yes	characteristic, direct product factor, Sylow
10	C₂³×C₃³	C₃²	1	4	yes	—
11	C₂³×H(3,𝔽₃)	C₉	1	9	yes	direct product factor
12	C₂³×C₃×C₉	—	3	12	no	—
13	C₂³×C₃×C₉	C₃²	1	8	yes	—
14	C₂³×C₃×C₉	C₃²	1	1	yes	characteristic
15	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	7	yes	—
16	C₂×C₃²×C₉	C₂²×C₃	1	28	yes	complemented
17	C₂×C₃²⋊C₉	C₂²×C₃	1	56	yes	complemented
18	C₂²×C₃³	C₂×C₃²	1	28	yes	—
19	C₂²×H(3,𝔽₃)	C₂×C₉	1	63	yes	direct product factor
20	C₂²×C₃×C₉	—	3	84	no	—
21	C₂²×C₃×C₉	C₂×C₃²	1	56	yes	—
22	C₂²×C₃×C₉	C₂×C₃²	1	7	yes	—
23	C₃×H(3,𝔽₃)	C₂³×C₃	1	1	yes	characteristic
24	C₃²×C₉	C₂³×C₃	1	4	yes	complemented
25	C₃²⋊C₉	C₂³×C₃	1	8	yes	complemented
26	C₂³×C₃²	—	3	4	no	—
27	C₂³×C₃²	—	3	8	no	—
28	C₂³×C₃²	C₃×C₉	1	12	yes	complemented
29	C₂³×C₃²	C₃³	1	1	yes	characteristic
30	C₂³×C₉	—	3	8	no	—
31	C₂³×C₉	H(3,𝔽₃)	1	3	yes	direct product factor
32	C₂×C₃³	C₂²×C₃²	1	28	yes	—
33	C₂×H(3,𝔽₃)	C₂²×C₉	1	63	yes	direct product factor
34	C₂×C₃×C₉	—	3	84	no	—
35	C₂×C₃×C₉	C₂²×C₃²	1	56	yes	—
36	C₂×C₃×C₉	C₂²×C₃²	1	7	yes	—
37	C₂²×C₃²	—	3	28	no	—
38	C₂²×C₃²	—	3	56	no	—
39	C₂²×C₃²	C₂×C₃×C₉	1	84	yes	complemented
40	C₂²×C₃²	C₂×C₃³	1	7	yes	—
41	C₂²×C₉	—	3	56	no	—
42	C₂²×C₉	C₂×H(3,𝔽₃)	1	21	yes	direct product factor
43	C₃³	C₂³×C₃²	1	4	yes	—
44	H(3,𝔽₃)	C₂³×C₉	1	9	yes	direct product factor
45	C₃×C₉	—	3	12	no	—
46	C₃×C₉	C₂³×C₃²	1	8	yes	—
47	C₃×C₉	C₂³×C₃²	1	1	yes	characteristic
48	C₂³×C₃	—	3	12	no	—
49	C₂³×C₃	C₃²×C₉	1	1	yes	characteristic
50	C₂³×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₂³×C₃	#737	1	2	yes	—
52	C₂×C₃²	—	3	28	no	—
53	C₂×C₃²	—	3	56	no	—
54	C₂×C₃²	C₂²×C₃×C₉	1	84	yes	complemented
55	C₂×C₃²	C₂²×C₃³	1	7	yes	—
56	C₂×C₉	—	3	56	no	—
57	C₂×C₉	C₂²×H(3,𝔽₃)	1	21	yes	direct product factor
58	C₂²×C₃	—	3	84	no	—
59	C₂²×C₃	C₂×C₃²×C₉	1	7	yes	—
60	C₂²×C₃	C₂×C₃×H(3,𝔽₃)	1	7	yes	—
61	C₂²×C₃	C₂×#737	1	14	yes	—
62	C₃²	—	3	4	no	—
63	C₃²	—	3	8	no	—
64	C₃²	C₂³×C₃×C₉	1	12	yes	complemented
65	C₃²	C₂³×C₃³	1	1	yes	characteristic
66	C₉	—	3	8	no	—
67	C₉	C₂³×H(3,𝔽₃)	1	3	yes	direct product factor
68	C₂³	C₉×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃	—	3	84	no	—
70	C₂×C₃	C₂²×C₃²×C₉	1	7	yes	—
71	C₂×C₃	C₂²×C₃×H(3,𝔽₃)	1	7	yes	—
72	C₂×C₃	C₂²×#737	1	14	yes	—
73	C₂²	C₂×C₉×H(3,𝔽₃)	1	7	yes	direct product factor
74	C₃	—	3	12	no	—
75	C₃	C₂³×C₃²×C₉	1	1	yes	characteristic
76	C₃	C₂³×C₃×H(3,𝔽₃)	1	1	yes	characteristic
77	C₃	C₂³×#737	1	2	yes	—
78	C₂	C₂²×C₉×H(3,𝔽₃)	1	7	yes	direct product factor

	subgroup	G/H
center	C₂³×C₃×C₉	C₃²
commutator	C₃	C₂³×C₃²×C₉
Frattini, cosocle	C₃²	C₂³×C₃³
socle	C₂³×C₃²	C₃³
prime radical	C₂³×C₃×H(3,𝔽₃)	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×H(3,𝔽₃)	C₂	direct (N × H)
3. C₂³×C₃²×C₉	C₃	semidirect
4. C₂³×C₃²⋊C₉	C₃	semidirect
5. C₂×C₉×H(3,𝔽₃)	C₂²	direct (N × H)
7. C₂²×C₃²×C₉	C₂×C₃	semidirect
8. C₂²×C₃²⋊C₉	C₂×C₃	semidirect
9. C₉×H(3,𝔽₃)	C₂³	direct (N × H)
11. C₂³×H(3,𝔽₃)	C₉	direct (N × H)
16. C₂×C₃²×C₉	C₂²×C₃	semidirect
17. C₂×C₃²⋊C₉	C₂²×C₃	semidirect
19. C₂²×H(3,𝔽₃)	C₂×C₉	direct (N × H)
24. C₃²×C₉	C₂³×C₃	semidirect
25. C₃²⋊C₉	C₂³×C₃	semidirect
28. C₂³×C₃²	C₃×C₉	semidirect
31. C₂³×C₉	H(3,𝔽₃)	direct (N × H)
33. C₂×H(3,𝔽₃)	C₂²×C₉	direct (N × H)
39. C₂²×C₃²	C₂×C₃×C₉	semidirect
42. C₂²×C₉	C₂×H(3,𝔽₃)	direct (N × H)
44. H(3,𝔽₃)	C₂³×C₉	direct (N × H)
54. C₂×C₃²	C₂²×C₃×C₉	semidirect
57. C₂×C₉	C₂²×H(3,𝔽₃)	direct (N × H)
64. C₃²	C₂³×C₃×C₉	semidirect
67. C₉	C₂³×H(3,𝔽₃)	direct (N × H)
68. C₂³	C₉×H(3,𝔽₃)	direct (N × H)
73. C₂²	C₂×C₉×H(3,𝔽₃)	direct (N × H)
78. C₂	C₂²×C₉×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
C₂²×C₉×H(3,𝔽₃)	2
C₂³×C₃×H(3,𝔽₃)	3
C₂³×C₃²×C₉	3
C₂³×C₃²⋊C₉	3

Subgroups — C23×C9×H(3,𝔽3) #14518556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₉×H(3,𝔽₃) #14518556