Subgroups — #11757459

Order 972.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#737	C₃	1	1	yes	maximal, characteristic
2	#66353	C₃	1	2	yes	maximal, complemented
3	C₃×C₂²⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	#6872	—	4	1	no	maximal, Sylow
5	C₂×#737	—	3	1	no	—
6	C₂²×3₋¹⁺²	—	3	2	no	—
7	C₂²×3₋¹⁺²	C₉	1	2	yes	—
8	C₂²×H(3,𝔽₃)	C₉	1	1	yes	characteristic
9	C₂²×C₃×C₉	—	3	1	no	—
10	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
11	C₂²⋊C₂₇	—	3	1	no	—
12	C₂²⋊C₂₇	—	3	2	no	—
13	#737	A₄	1	1	yes	characteristic
14	C₂₇⋊C₃	—	4	2	no	—
15	C₃×C₂₇	—	4	1	no	—
16	C₂×3₋¹⁺²	—	3	6	no	—
17	C₂×3₋¹⁺²	—	3	2	no	—
18	C₂×H(3,𝔽₃)	—	3	1	no	—
19	C₂×C₃×C₉	—	3	3	no	—
20	C₂×C₃×C₉	—	3	1	no	—
21	C₂²×C₃²	—	3	1	no	—
22	C₂²×C₃²	C₃×C₉	1	1	yes	characteristic
23	C₂²×C₉	—	3	2	no	—
24	C₂²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
25	C₂²×C₉	3₋¹⁺²	1	2	yes	—
26	3₋¹⁺²	—	3	2	no	—
27	3₋¹⁺²	C₂²⋊C₉	1	2	yes	—
28	H(3,𝔽₃)	C₂²⋊C₉	1	1	yes	characteristic
29	C₃×C₉	—	3	1	no	—
30	C₃×C₉	C₃×A₄	1	1	yes	characteristic
31	C₂₇	—	12	1	no	—
32	C₂₇	—	12	2	no	—
33	C₂×C₃²	—	3	3	no	—
34	C₂×C₃²	—	3	1	no	—
35	C₂×C₉	—	3	6	no	—
36	C₂×C₉	—	3	2	no	—
37	C₂×C₉	—	3	1	no	—
38	C₂²×C₃	—	3	1	no	—
39	C₂²×C₃	—	9	1	no	—
40	C₂²×C₃	C₃²⋊C₉	1	1	yes	characteristic
41	C₃²	—	3	1	no	—
42	C₃²	C₃×C₂²⋊C₉	1	1	yes	characteristic
43	C₉	—	3	2	no	—
44	C₉	#1094	1	2	yes	—
45	C₉	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
46	C₂×C₃	—	3	1	no	—
47	C₂×C₃	—	9	1	no	—
48	C₂×C₃	—	9	3	no	—
49	C₂²	#6872	1	1	yes	characteristic, complemented, Sylow
50	C₃	—	3	1	no	—
51	C₃	—	9	1	no	—
52	C₃	C₃²⋊(C₂²⋊C₉)	1	1	yes	characteristic
53	C₂	—	3	1	no	—

	subgroup	G/H
center	C₉	C₂²⋊H(3,𝔽₃)
commutator	C₂²×C₃²	C₃×C₉
Frattini	C₃×C₉	C₃×A₄
Fitting	C₂²×#737	C₃
socle	C₂²×C₃	C₃²⋊C₉
prime radical	C₂²×H(3,𝔽₃)	C₉
cosocle	C₂²×C₃×C₉	C₃²
hypercenter	#737	A₄
hypocenter	C₂²	#6872
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ #737

Complements

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

N	H ≅ G/N	type
2. #66353	C₃	semidirect
3. C₃×C₂²⋊C₂₇	C₃	semidirect
49. C₂²	#6872	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#737	3
#66353	3
C₃×C₂²⋊C₂₇	3
#6872	4