Subgroups — C₃×D₉×H(3,𝔽₃) #13091358

Order 1458.

674 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×D₃×H(3,𝔽₃)	—	3	1	no	maximal
3	C₃³×D₉	C₃	1	4	yes	maximal, complemented
4	D₉×H(3,𝔽₃)	C₃	1	9	yes	maximal, direct product factor
5	C₃²×H(3,𝔽₃)	D₃	1	1	yes	characteristic
6	C₃³×C₉	C₂×C₃	1	4	yes	complemented
7	C₉×H(3,𝔽₃)	—	2	9	no	—
8	C₉×H(3,𝔽₃)	C₂×C₃	1	9	yes	complemented
9	C₃×C₃²⋊C₉	—	2	4	no	—
10	C₃³×D₃	—	3	4	no	—
11	C₂×C₃×H(3,𝔽₃)	—	9	1	no	—
12	D₃×H(3,𝔽₃)	—	3	9	no	—
13	C₃²×D₉	—	3	12	no	—
14	C₃²×D₉	C₃²	1	1	yes	characteristic
15	C₃²×D₉	C₃²	1	12	yes	complemented
16	C₃⁴	C₃×D₃	1	4	yes	—
17	C₃×H(3,𝔽₃)	—	2	4	no	—
18	C₃×H(3,𝔽₃)	—	2	9	no	—
19	C₃×H(3,𝔽₃)	D₉	1	1	yes	characteristic, direct product factor
20	C₃×H(3,𝔽₃)	C₃×D₃	1	9	yes	—
21	C₃²×C₉	—	2	4	no	—
22	C₃²×C₉	—	2	12	no	—
23	C₃²×C₉	—	3	12	no	—
24	C₃²×C₉	—	6	12	no	—
25	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
26	C₃²×C₉	C₂×C₃²	1	12	yes	complemented
27	C₃²⋊C₉	—	2	36	no	—
28	C₂×C₃³	—	9	4	no	—
29	C₃²×D₃	—	3	1	no	—
30	C₃²×D₃	—	3	12	no	—
31	C₃²×D₃	—	9	12	no	—
32	C₂×H(3,𝔽₃)	—	9	9	no	—
33	C₃×D₉	—	3	12	no	—
34	C₃×D₉	H(3,𝔽₃)	1	3	yes	direct product factor
35	C₃×D₉	C₃³	1	1	yes	characteristic
36	C₃³	—	2	4	no	—
37	C₃³	—	2	12	no	—
38	C₃³	—	3	12	no	—
39	C₃³	—	6	12	no	—
40	C₃³	C₃×D₉	1	4	yes	complemented
41	C₃³	C₃²×D₃	1	1	yes	characteristic
42	C₃³	C₃²×D₃	1	12	yes	—
43	H(3,𝔽₃)	—	2	36	no	—
44	H(3,𝔽₃)	C₃×D₉	1	9	yes	direct product factor
45	C₃×C₉	—	2	3	no	—
46	C₃×C₉	—	2	1	no	—
47	C₃×C₉	—	2	12	no	—
48	C₃×C₉	—	3	12	no	—
49	C₃×C₉	—	6	36	no	—
50	C₃×C₉	—	6	12	no	—
51	C₃×C₉	—	6	12	no	—
52	C₃×C₉	C₂×H(3,𝔽₃)	1	3	yes	complemented
53	C₃×C₉	C₂×C₃³	1	1	yes	characteristic
54	C₂×C₃²	—	9	1	no	—
55	C₂×C₃²	—	9	12	no	—
56	C₂×C₃²	—	27	12	no	—
57	C₃×D₃	—	3	3	no	—
58	C₃×D₃	—	3	1	no	—
59	C₃×D₃	—	9	12	no	—
60	D₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor
61	C₃²	—	2	3	no	—
62	C₃²	—	2	1	no	—
63	C₃²	—	2	12	no	—
64	C₃²	—	3	12	no	—
65	C₃²	—	3	12	no	—
66	C₃²	—	6	12	no	—
67	C₃²	—	6	36	no	—
68	C₃²	—	6	12	no	—
69	C₃²	C₃²×D₉	1	1	yes	characteristic
70	C₃²	C₃²×D₉	1	12	yes	complemented
71	C₃²	D₃×H(3,𝔽₃)	1	3	yes	—
72	C₃²	C₃³×D₃	1	1	yes	characteristic
73	C₉	—	2	3	no	—
74	C₉	—	2	1	no	—
75	C₉	—	6	12	no	—
76	C₉	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic, complemented
77	C₂×C₃	—	9	3	no	—
78	C₂×C₃	—	9	1	no	—
79	C₂×C₃	—	27	12	no	—
80	D₃	—	3	1	no	—
81	C₃	—	2	3	no	—
82	C₃	—	2	1	no	—
83	C₃	—	3	12	no	—
84	C₃	—	6	12	no	—
85	C₃	D₉×H(3,𝔽₃)	1	3	yes	direct product factor
86	C₃	C₃³×D₉	1	1	yes	characteristic
87	C₃	C₃×D₃×H(3,𝔽₃)	1	1	yes	characteristic
88	C₂	—	9	1	no	Sylow

	subgroup	G/H
center	C₃²	C₃²×D₉
commutator, cosocle	C₃×C₉	C₂×C₃³
Frattini	C₃²	C₃³×D₃
Fitting	C₃×C₉×H(3,𝔽₃)	C₂
socle	C₃³	C₃²×D₃
hypercenter	C₃×H(3,𝔽₃)	D₉
hypocenter	C₉	C₂×C₃×H(3,𝔽₃)
LCS	G ▷ C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₃² ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×H(3,𝔽₃)	C₂	semidirect
3. C₃³×D₉	C₃	semidirect
4. D₉×H(3,𝔽₃)	C₃	direct (N × H)
6. C₃³×C₉	C₂×C₃	semidirect
8. C₉×H(3,𝔽₃)	C₂×C₃	semidirect
15. C₃²×D₉	C₃²	semidirect
19. C₃×H(3,𝔽₃)	D₉	direct (N × H)
26. C₃²×C₉	C₂×C₃²	semidirect
34. C₃×D₉	H(3,𝔽₃)	direct (N × H)
40. C₃³	C₃×D₉	semidirect
44. H(3,𝔽₃)	C₃×D₉	direct (N × H)
52. C₃×C₉	C₂×H(3,𝔽₃)	semidirect
60. D₉	C₃×H(3,𝔽₃)	direct (N × H)
70. C₃²	C₃²×D₉	semidirect
76. C₉	C₂×C₃×H(3,𝔽₃)	semidirect
85. C₃	D₉×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
C₃×C₉×H(3,𝔽₃)	2
C₃×D₃×H(3,𝔽₃)	3
C₃³×D₉	3
D₉×H(3,𝔽₃)	3

Subgroups — C3×D9×H(3,𝔽3) #13091358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₉×H(3,𝔽₃) #13091358