Subgroups — #11757456

Order 972.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal
4	#6858	—	4	1	no	maximal, Sylow
5	C₂×C₃×3₋¹⁺²	—	3	1	no	—
6	C₂²×C₃³	C₃²	1	1	yes	characteristic
7	C₂²×3₋¹⁺²	—	3	3	no	—
8	C₂²×C₃×C₉	—	3	1	no	—
9	C₃×C₂²⋊C₉	—	3	1	no	—
10	C₃×C₂²⋊C₉	—	3	2	no	—
11	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
12	C₃²⋊C₉	—	4	2	no	—
13	C₃²⋊C₉	—	4	1	no	—
14	C₂×C₃³	—	3	1	no	—
15	C₂×3₋¹⁺²	—	3	9	no	—
16	C₂×C₃×C₉	—	3	3	no	—
17	C₂²×C₃²	—	3	1	no	—
18	C₂²×C₃²	—	3	2	no	—
19	C₂²×C₃²	—	3	1	no	—
20	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
21	C₂²×C₉	—	3	3	no	—
22	C₂²⋊C₉	—	9	1	no	—
23	C₂²⋊C₉	—	9	2	no	—
24	C₃³	C₃×A₄	1	1	yes	characteristic
25	3₋¹⁺²	—	3	3	no	—
26	C₃×C₉	—	3	1	no	—
27	C₃×C₉	—	12	2	no	—
28	C₃×C₉	—	12	1	no	—
29	C₂×C₃²	—	3	1	no	—
30	C₂×C₃²	—	3	3	no	—
31	C₂×C₃²	—	9	1	no	—
32	C₂×C₃²	—	9	2	no	—
33	C₂×C₉	—	3	9	no	—
34	C₂²×C₃	—	9	1	no	—
35	C₂²×C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
36	C₂²×C₃	#731	1	2	yes	—
37	C₂²×C₃	#733	1	1	yes	characteristic
38	C₃²	—	3	1	no	—
39	C₃²	—	3	2	no	—
40	C₃²	—	3	1	no	—
41	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
42	C₉	—	3	3	no	—
43	C₉	—	36	2	no	—
44	C₉	—	36	1	no	—
45	C₂×C₃	—	3	1	no	—
46	C₂×C₃	—	3	2	no	—
47	C₂×C₃	—	3	1	no	—
48	C₂×C₃	—	9	3	no	—
49	C₂²	#6858	1	1	yes	characteristic, complemented, Sylow
50	C₃	—	9	1	no	—
51	C₃	#66364	1	1	yes	characteristic
52	C₃	#66365	1	2	yes	—
53	C₃	#66368	1	1	yes	characteristic
54	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator, prime radical, cosocle	C₂²×C₃³	C₃²
Frattini	C₃³	C₃×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	#6858
LCS	G ▷ C₂²×C₃³ ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
49. C₂²	#6858	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×3₋¹⁺²	3
C₃²⋊(C₂²⋊C₉)	3
#6858	4