Subgroups — 3₋¹⁺²⋊#10620321 #31801254

Order 17739.

82 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
2	C₃²⋊#10620321	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#10620321	C₃	1	1	yes	maximal, characteristic
4	C₃×3₋¹⁺²×C₇₃	C₃	1	1	yes	maximal, characteristic
5	C₃×#10620321	—	3	1	no	—
6	C₃×#10620321	—	3	3	no	—
7	C₃×#10620321	—	3	1	no	—
8	C₃×#10620321	—	3	1	no	—
9	C₃³×C₇₃	C₃²	1	1	yes	characteristic
10	3₋¹⁺²×C₇₃	—	3	2	no	—
11	3₋¹⁺²×C₇₃	C₉	1	3	yes	complemented
12	C₃×C₉×C₇₃	—	3	1	no	—
13	C₃²×C₇₃	—	3	1	no	—
14	C₃²×C₇₃	—	3	1	no	—
15	C₃²×C₇₃	—	3	1	no	—
16	C₃²×C₇₃	C₃×C₉	1	1	yes	characteristic
17	C₃²×C₇₃	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃²×C₇₃	3₋¹⁺²	1	2	yes	—
19	C₉×C₇₃	—	3	3	no	—
20	#10620321	—	3	3	no	—
21	#10620321	—	9	1	no	—
22	#10620321	—	9	1	no	—
23	3₋¹⁺²⋊C₉	—	73	1	no	maximal, Sylow
24	C₃×C₇₃	—	3	1	no	—
25	C₃×C₇₃	—	3	2	no	—
26	C₃×C₇₃	C₃²⋊C₉	1	1	yes	characteristic
27	C₃×C₇₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
28	C₃×C₇₃	#731	1	1	yes	characteristic
29	C₃×C₇₃	#733	1	1	yes	characteristic
30	C₃×3₋¹⁺²	C₇₃⋊C₃	1	1	yes	characteristic
31	C₃²×C₉	—	73	1	no	—
32	C₃²⋊C₉	—	73	1	no	—
33	C₃²⋊C₉	—	73	1	no	—
34	C₇₃	3₋¹⁺²⋊C₉	1	1	yes	characteristic, complemented, Sylow
35	C₃³	C₃×C₇₃⋊C₃	1	1	yes	characteristic
36	3₋¹⁺²	—	3	2	no	—
37	3₋¹⁺²	#10620321	1	3	yes	complemented
38	C₃×C₉	—	3	1	no	—
39	C₃×C₉	—	219	1	no	—
40	C₃×C₉	—	219	1	no	—
41	C₃×C₉	—	219	1	no	—
42	C₃×C₉	—	219	3	no	—
43	C₃²	—	3	1	no	—
44	C₃²	—	3	1	no	—
45	C₃²	—	3	1	no	—
46	C₃²	C₃×#10620321	1	1	yes	characteristic
47	C₃²	C₃²⋊(C₇₃⋊C₃)	1	1	yes	characteristic
48	C₃²	#14520297	1	2	yes	—
49	C₉	—	3	3	no	—
50	C₉	—	657	1	no	—
51	C₉	—	657	1	no	—
52	C₉	—	219	3	no	—
53	C₃	—	3	1	no	—
54	C₃	—	3	2	no	—
55	C₃	C₃²⋊#10620321	1	1	yes	characteristic
56	C₃	#20238269	1	1	yes	characteristic
57	C₃	#20238272	1	1	yes	characteristic
58	C₃	#20238275	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₇₃⋊C₃)
commutator	C₃²×C₇₃	C₃×C₉
Frattini	C₃³	C₃×C₇₃⋊C₃
Fitting	C₃×3₋¹⁺²×C₇₃	C₃
socle	C₃²×C₇₃	H(3,𝔽₃)
prime radical, cosocle	C₃³×C₇₃	C₃²
hypercenter	C₃×3₋¹⁺²	C₇₃⋊C₃
hypocenter	C₇₃	3₋¹⁺²⋊C₉
LCS	G ▷ C₃²×C₇₃ ▷ C₃×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
11. 3₋¹⁺²×C₇₃	C₉	semidirect
34. C₇₃	3₋¹⁺²⋊C₉	semidirect
37. 3₋¹⁺²	#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
C₃×3₋¹⁺²×C₇₃	3
3₋¹⁺²⋊C₉	73

Subgroups — 3−1+2⋊#10620321 #31801254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²⋊#10620321 #31801254