Subgroups — H(3,𝔽₃)⋊#10620321 #31801251

Order 17739.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#10620321	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊#10620321	C₃	1	2	yes	maximal, complemented
3	C₃×H(3,𝔽₃)×C₇₃	C₃	1	1	yes	maximal, characteristic
4	C₃×#10620321	—	3	1	no	—
5	C₃×#10620321	—	3	3	no	—
6	C₃×#10620321	—	3	2	no	—
7	C₃³×C₇₃	—	3	1	no	—
8	C₃³×C₇₃	C₃²	1	1	yes	characteristic
9	H(3,𝔽₃)×C₇₃	—	3	2	no	—
10	H(3,𝔽₃)×C₇₃	C₉	1	3	yes	complemented
11	C₃²×C₇₃	—	3	1	no	—
12	C₃²×C₇₃	—	3	1	no	—
13	C₃²×C₇₃	—	3	1	no	—
14	C₃²×C₇₃	—	3	3	no	—
15	C₃²×C₇₃	—	9	1	no	—
16	C₃²×C₇₃	—	9	1	no	—
17	C₃²×C₇₃	—	9	1	no	—
18	C₃²×C₇₃	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₇₃	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₇₃	3₋¹⁺²	1	2	yes	—
21	#10620321	—	3	3	no	—
22	#10620321	—	9	2	no	—
23	H(3,𝔽₃)⋊C₉	—	73	1	no	maximal, Sylow
24	C₃×C₇₃	—	3	1	no	—
25	C₃×C₇₃	—	3	2	no	—
26	C₃×C₇₃	—	9	3	no	—
27	C₃×C₇₃	C₃²⋊C₉	1	1	yes	characteristic
28	C₃×C₇₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
29	C₃×C₇₃	#731	1	1	yes	characteristic
30	C₃×C₇₃	#732	1	1	yes	characteristic
31	C₃×H(3,𝔽₃)	C₇₃⋊C₃	1	1	yes	characteristic
32	C₃²×C₉	—	73	1	no	—
33	C₃²⋊C₉	—	73	2	no	—
34	C₇₃	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, complemented, Sylow
35	C₃³	—	3	1	no	—
36	C₃³	C₃×C₇₃⋊C₃	1	1	yes	characteristic
37	H(3,𝔽₃)	—	3	2	no	—
38	H(3,𝔽₃)	#10620321	1	3	yes	complemented
39	C₃×C₉	—	219	2	no	—
40	C₃×C₉	—	219	1	no	—
41	C₃×C₉	—	219	3	no	—
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×#10620321	1	1	yes	characteristic
50	C₃²	C₃²⋊(C₇₃⋊C₃)	1	1	yes	characteristic
51	C₃²	#14520297	1	2	yes	—
52	C₉	—	657	2	no	—
53	C₉	—	219	3	no	—
54	C₃	—	3	1	no	—
55	C₃	—	3	2	no	—
56	C₃	—	9	3	no	—
57	C₃	C₃²⋊#10620321	1	1	yes	characteristic
58	C₃	#20238267	1	1	yes	characteristic
59	C₃	#20238270	1	1	yes	characteristic
60	C₃	#20238273	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₇₃⋊C₃)
commutator	C₃²×C₇₃	C₃×C₉
Frattini	C₃³	C₃×C₇₃⋊C₃
Fitting, prime radical	C₃×H(3,𝔽₃)×C₇₃	C₃
socle	C₃²×C₇₃	H(3,𝔽₃)
cosocle	C₃³×C₇₃	C₃²
hypercenter	C₃×H(3,𝔽₃)	C₇₃⋊C₃
hypocenter	C₇₃	H(3,𝔽₃)⋊C₉
LCS	G ▷ C₃²×C₇₃ ▷ C₃×C₇₃ ▷ C₇₃
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×#10620321	C₃	semidirect
2. C₃²⋊#10620321	C₃	semidirect
10. H(3,𝔽₃)×C₇₃	C₉	semidirect
34. C₇₃	H(3,𝔽₃)⋊C₉	semidirect
38. H(3,𝔽₃)	#10620321	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#10620321	3
C₃²⋊#10620321	3
C₃×H(3,𝔽₃)×C₇₃	3
H(3,𝔽₃)⋊C₉	73

Subgroups — H(3,𝔽3)⋊#10620321 #31801251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)⋊#10620321 #31801251