Subgroups — H(3,𝔽₃)⋊(C₂²⋊C₉) #11757453

Order 972.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
4	H(3,𝔽₃)⋊C₉	—	4	1	no	maximal, Sylow
5	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
6	C₂²×C₃³	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₂²×H(3,𝔽₃)	—	3	2	no	—
9	C₂²×H(3,𝔽₃)	C₉	1	3	yes	complemented
10	C₃×C₂²⋊C₉	—	3	1	no	—
11	C₃×C₂²⋊C₉	—	3	3	no	—
12	C₃×C₂²⋊C₉	—	3	2	no	—
13	C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
14	C₃²×C₉	—	4	1	no	—
15	C₃²⋊C₉	—	4	2	no	—
16	C₂×C₃³	—	3	3	no	—
17	C₂×C₃³	—	3	1	no	—
18	C₂×H(3,𝔽₃)	—	3	6	no	—
19	C₂×H(3,𝔽₃)	—	3	3	no	—
20	C₂²×C₃²	—	3	1	no	—
21	C₂²×C₃²	—	3	1	no	—
22	C₂²×C₃²	—	3	1	no	—
23	C₂²×C₃²	—	3	3	no	—
24	C₂²×C₃²	—	9	1	no	—
25	C₂²×C₃²	—	9	1	no	—
26	C₂²×C₃²	—	9	1	no	—
27	C₂²×C₃²	C₃×C₉	1	1	yes	characteristic
28	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
29	C₂²×C₃²	3₋¹⁺²	1	2	yes	—
30	C₂²⋊C₉	—	3	3	no	—
31	C₂²⋊C₉	—	9	2	no	—
32	C₃³	—	3	1	no	—
33	C₃³	C₃×A₄	1	1	yes	characteristic
34	H(3,𝔽₃)	—	3	2	no	—
35	H(3,𝔽₃)	C₂²⋊C₉	1	3	yes	complemented
36	C₃×C₉	—	12	1	no	—
37	C₃×C₉	—	12	3	no	—
38	C₃×C₉	—	12	2	no	—
39	C₂×C₃²	—	3	1	no	—
40	C₂×C₃²	—	3	9	no	—
41	C₂×C₃²	—	3	1	no	—
42	C₂×C₃²	—	3	2	no	—
43	C₂×C₃²	—	9	1	no	—
44	C₂×C₃²	—	9	1	no	—
45	C₂×C₃²	—	9	1	no	—
46	C₂×C₃²	—	9	3	no	—
47	C₂×C₃²	—	9	3	no	—
48	C₂×C₃²	—	9	3	no	—
49	C₂²×C₃	—	3	1	no	—
50	C₂²×C₃	—	3	2	no	—
51	C₂²×C₃	—	9	3	no	—
52	C₂²×C₃	C₃²⋊C₉	1	1	yes	characteristic
53	C₂²×C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
54	C₂²×C₃	#731	1	1	yes	characteristic
55	C₂²×C₃	#732	1	1	yes	characteristic
56	C₃²	—	3	1	no	—
57	C₃²	—	3	1	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	3	3	no	—
60	C₃²	—	9	1	no	—
61	C₃²	—	9	1	no	—
62	C₃²	—	9	1	no	—
63	C₃²	C₃×C₂²⋊C₉	1	1	yes	characteristic
64	C₃²	#1094	1	2	yes	—
65	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
66	C₉	—	12	3	no	—
67	C₉	—	36	2	no	—
68	C₂×C₃	—	3	1	no	—
69	C₂×C₃	—	3	1	no	—
70	C₂×C₃	—	3	1	no	—
71	C₂×C₃	—	3	1	no	—
72	C₂×C₃	—	9	1	no	—
73	C₂×C₃	—	9	2	no	—
74	C₂×C₃	—	9	9	no	—
75	C₂²	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, complemented, Sylow
76	C₃	—	3	1	no	—
77	C₃	—	3	2	no	—
78	C₃	—	9	3	no	—
79	C₃	#66361	1	1	yes	characteristic
80	C₃	#66362	1	1	yes	characteristic
81	C₃	#66363	1	1	yes	characteristic
82	C₃	C₃²⋊(C₂²⋊C₉)	1	1	yes	characteristic
83	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator	C₂²×C₃²	C₃×C₉
Frattini	C₃³	C₃×A₄
Fitting, prime radical	C₂²×C₃×H(3,𝔽₃)	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
cosocle	C₂²×C₃³	C₃²
hypercenter	C₃×H(3,𝔽₃)	A₄
hypocenter	C₂²	H(3,𝔽₃)⋊C₉
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. C₃²×C₂²⋊C₉	C₃	semidirect
3. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
9. C₂²×H(3,𝔽₃)	C₉	semidirect
35. H(3,𝔽₃)	C₂²⋊C₉	semidirect
75. C₂²	H(3,𝔽₃)⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×H(3,𝔽₃)	3
C₃²×C₂²⋊C₉	3
C₃²⋊(C₂²⋊C₉)	3
H(3,𝔽₃)⋊C₉	4

Subgroups — H(3,𝔽3)⋊(C22⋊C9) #11757453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)⋊(C₂²⋊C₉) #11757453