Subgroups — #40996751

Order 26487.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15668883	C₃	1	1	yes	maximal, characteristic
2	C₃²×#11758279	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#11758279	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊C₉×C₁₀₉	C₃	1	1	yes	maximal, characteristic
5	#15668883	—	3	1	no	—
6	#15668883	—	3	1	no	—
7	#15668883	C₉	1	3	yes	complemented
8	C₃×#11758279	—	3	1	no	—
9	C₃×#11758279	—	3	1	no	—
10	C₃×#11758279	—	3	3	no	—
11	C₃×#11758279	—	3	1	no	—
12	C₃³×C₁₀₉	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₁₀₉	—	3	1	no	—
14	C₃²×C₁₀₉	—	3	1	no	—
15	C₃²×C₁₀₉	—	3	1	no	—
16	C₃²×C₁₀₉	—	3	1	no	—
17	C₃²×C₁₀₉	C₃×C₉	1	1	yes	characteristic
18	C₃²×C₁₀₉	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₁₀₉	3₋¹⁺²	1	1	yes	characteristic
20	C₃²×C₁₀₉	3₋¹⁺²	1	1	yes	characteristic
21	C₉×C₁₀₉	—	9	1	no	—
22	#11758279	—	3	3	no	—
23	#11758279	—	3	3	no	—
24	#11758279	—	9	1	no	—
25	C₃×C₁₀₉	—	3	1	no	—
26	C₃×C₁₀₉	—	3	1	no	—
27	C₃×C₁₀₉	—	3	1	no	—
28	C₃×C₁₀₉	C₃²⋊C₉	1	1	yes	characteristic
29	C₃×C₁₀₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
30	C₃×C₁₀₉	#731	1	1	yes	characteristic
31	C₃×C₁₀₉	#733	1	1	yes	characteristic
32	3₋¹⁺²⋊C₉	—	109	1	no	maximal, Sylow
33	C₁₀₉	3₋¹⁺²⋊C₉	1	1	yes	characteristic, complemented, Sylow
34	C₃×3₋¹⁺²	—	109	1	no	—
35	C₃²×C₉	—	109	1	no	—
36	C₃²⋊C₉	—	109	1	no	—
37	C₃²⋊C₉	C₁₀₉⋊C₃	1	1	yes	characteristic
38	C₃³	C₃×C₁₀₉⋊C₃	1	1	yes	characteristic
39	3₋¹⁺²	—	327	1	no	—
40	3₋¹⁺²	—	327	1	no	—
41	3₋¹⁺²	—	109	3	no	—
42	C₃×C₉	—	3	1	no	—
43	C₃×C₉	—	327	1	no	—
44	C₃×C₉	—	327	1	no	—
45	C₃×C₉	—	327	3	no	—
46	C₃×C₉	—	327	1	no	—
47	C₃²	—	3	1	no	—
48	C₃²	—	3	1	no	—
49	C₃²	—	3	1	no	—
50	C₃²	C₉×C₁₀₉⋊C₃	1	1	yes	characteristic
51	C₃²	C₃²⋊(C₁₀₉⋊C₃)	1	1	yes	characteristic
52	C₃²	#15668881	1	1	yes	characteristic
53	C₃²	#15668882	1	1	yes	characteristic
54	C₉	—	9	1	no	—
55	C₉	—	327	3	no	—
56	C₉	—	327	3	no	—
57	C₉	—	981	1	no	—
58	C₃	—	3	1	no	—
59	C₃	—	3	1	no	—
60	C₃	—	3	1	no	—
61	C₃	#22579486	1	1	yes	characteristic
62	C₃	#22579498	1	1	yes	characteristic
63	C₃	#22579499	1	1	yes	characteristic
64	C₃	#22579504	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₁₀₉⋊C₃)
commutator	C₃²×C₁₀₉	C₃×C₉
Frattini	C₃³	C₃×C₁₀₉⋊C₃
Fitting	C₃²⋊C₉×C₁₀₉	C₃
socle	C₃²×C₁₀₉	H(3,𝔽₃)
prime radical, cosocle	C₃³×C₁₀₉	C₃²
hypercenter	C₃²⋊C₉	C₁₀₉⋊C₃
hypocenter	C₁₀₉	3₋¹⁺²⋊C₉
LCS	G ▷ C₃²×C₁₀₉ ▷ C₃×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
7. #15668883	C₉	semidirect
33. C₁₀₉	3₋¹⁺²⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15668883	3
C₃²×#11758279	3
C₃²⋊#11758279	3
C₃²⋊C₉×C₁₀₉	3
3₋¹⁺²⋊C₉	109