Subgroups — H(3,𝔽₃)⋊#4294 #17987343

Order 4617.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#4294	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊#4294	C₃	1	2	yes	maximal, complemented
3	C₃×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃³×C₁₉	—	3	1	no	—
5	C₃³×C₁₉	C₃²	1	1	yes	characteristic
6	C₁₉×H(3,𝔽₃)	—	3	2	no	—
7	C₁₉×H(3,𝔽₃)	C₉	1	3	yes	complemented
8	C₃×#4294	—	3	1	no	—
9	C₃×#4294	—	3	3	no	—
10	C₃×#4294	—	3	2	no	—
11	H(3,𝔽₃)⋊C₉	—	19	1	no	maximal, Sylow
12	C₃²×C₁₉	—	3	1	no	—
13	C₃²×C₁₉	—	3	1	no	—
14	C₃²×C₁₉	—	3	1	no	—
15	C₃²×C₁₉	—	3	3	no	—
16	C₃²×C₁₉	—	9	1	no	—
17	C₃²×C₁₉	—	9	1	no	—
18	C₃²×C₁₉	—	9	1	no	—
19	C₃²×C₁₉	C₃×C₉	1	1	yes	characteristic
20	C₃²×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃²×C₁₉	3₋¹⁺²	1	2	yes	—
22	#4294	—	3	3	no	—
23	#4294	—	9	2	no	—
24	C₃×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
25	C₃²×C₉	—	19	1	no	—
26	C₃²⋊C₉	—	19	2	no	—
27	C₃×C₁₉	—	3	1	no	—
28	C₃×C₁₉	—	3	2	no	—
29	C₃×C₁₉	—	9	3	no	—
30	C₃×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
31	C₃×C₁₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
32	C₃×C₁₉	#731	1	1	yes	characteristic
33	C₃×C₁₉	#732	1	1	yes	characteristic
34	C₃³	—	3	1	no	—
35	C₃³	C₃×C₁₉⋊C₃	1	1	yes	characteristic
36	H(3,𝔽₃)	—	3	2	no	—
37	H(3,𝔽₃)	#4294	1	3	yes	complemented
38	C₃×C₉	—	57	1	no	—
39	C₃×C₉	—	57	3	no	—
40	C₃×C₉	—	57	2	no	—
41	C₁₉	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, complemented, Sylow
42	C₃²	—	3	1	no	—
43	C₃²	—	3	1	no	—
44	C₃²	—	3	1	no	—
45	C₃²	—	3	3	no	—
46	C₃²	—	9	1	no	—
47	C₃²	—	9	1	no	—
48	C₃²	—	9	1	no	—
49	C₃²	C₃×#4294	1	1	yes	characteristic
50	C₃²	#10587025	1	2	yes	—
51	C₃²	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
52	C₉	—	57	3	no	—
53	C₉	—	171	2	no	—
54	C₃	—	3	1	no	—
55	C₃	—	3	2	no	—
56	C₃	—	9	3	no	—
57	C₃	C₃²⋊#4294	1	1	yes	characteristic
58	C₃	#13095313	1	1	yes	characteristic
59	C₃	#13095316	1	1	yes	characteristic
60	C₃	#13095319	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₁₉⋊C₃)
commutator	C₃²×C₁₉	C₃×C₉
Frattini	C₃³	C₃×C₁₉⋊C₃
Fitting, prime radical	C₃×C₁₉×H(3,𝔽₃)	C₃
socle	C₃²×C₁₉	H(3,𝔽₃)
cosocle	C₃³×C₁₉	C₃²
hypercenter	C₃×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	H(3,𝔽₃)⋊C₉
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×#4294	C₃	semidirect
2. C₃²⋊#4294	C₃	semidirect
7. C₁₉×H(3,𝔽₃)	C₉	semidirect
37. H(3,𝔽₃)	#4294	semidirect
41. C₁₉	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
C₃²×#4294	3
C₃²⋊#4294	3
C₃×C₁₉×H(3,𝔽₃)	3
H(3,𝔽₃)⋊C₉	19

Subgroups — H(3,𝔽3)⋊#4294 #17987343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)⋊#4294 #17987343