Subgroups — C₉×(C₃⋊D₃)⋊C₃ #92253

Order 486.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₉×H(3,𝔽₃)	C₂	semidirect
2. C₉×C₃⋊D₃	C₃	semidirect
5. (C₃⋊D₃)⋊C₉	C₃	semidirect
8. C₃²×C₉	D₃	semidirect
9. C₃²×C₉	C₂×C₃	semidirect
12. C₃²⋊C₉	C₂×C₃	semidirect
16. (C₃⋊D₃)⋊C₃	C₉	direct (N × H)
24. H(3,𝔽₃)	C₂×C₉	semidirect
36. C₃⋊D₃	C₃×C₉	semidirect
50. C₃²	D₃×C₉	semidirect
51. C₃²	C₂×C₃×C₉	semidirect
57. C₉	(C₃⋊D₃)⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×H(3,𝔽₃)	2
C₉×C₃⋊D₃	3
C₃×(C₃⋊D₃)⋊C₃	3
C₃×D₃×C₉	3
(C₃⋊D₃)⋊C₉	3

Subgroups — C9×(C3⋊D3)⋊C3 #92253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×(C₃⋊D₃)⋊C₃ #92253