Subgroups — #11757849

Order 972.

174 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, complemented
2	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#1094	C₃	1	2	yes	maximal, complemented
4	#66357	C₃	1	9	yes	maximal, complemented
5	#6910	—	4	1	no	maximal, Sylow
6	C₂×C₃×3₋¹⁺²	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₃²×A₄	—	3	1	no	—
9	C₂²×3₋¹⁺²	—	3	2	no	—
10	C₂²×3₋¹⁺²	—	3	1	no	—
11	C₂²×C₃×C₉	C₃²	1	3	yes	complemented
12	C₂²⋊H(3,𝔽₃)	—	3	2	no	—
13	C₂²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
14	#1094	—	3	2	no	—
15	#1094	—	3	2	no	—
16	#1094	C₃²	1	6	yes	complemented
17	C₃×C₂²⋊C₉	—	3	2	no	—
18	C₃×3₋¹⁺²	—	4	2	no	—
19	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
20	C₃×H(3,𝔽₃)	—	4	1	no	—
21	#731	—	4	9	no	—
22	C₂×C₃³	—	3	1	no	—
23	C₂×3₋¹⁺²	—	3	6	no	—
24	C₂×3₋¹⁺²	—	3	3	no	—
25	C₂×C₃×C₉	—	3	3	no	—
26	C₂²×C₃²	—	9	1	no	—
27	C₂²×C₃²	H(3,𝔽₃)	1	2	yes	—
28	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
29	C₂²×C₃²	C₃³	1	1	yes	characteristic
30	C₃×A₄	—	3	3	no	—
31	C₃×A₄	—	9	3	no	—
32	C₂²×C₉	—	3	3	no	—
33	C₂²⋊C₉	—	3	6	no	—
34	C₃³	—	12	1	no	—
35	C₃³	C₃×A₄	1	1	yes	characteristic
36	3₋¹⁺²	—	3	2	no	—
37	3₋¹⁺²	—	3	1	no	—
38	3₋¹⁺²	—	4	6	no	—
39	3₋¹⁺²	—	12	2	no	—
40	3₋¹⁺²	—	12	2	no	—
41	H(3,𝔽₃)	—	4	3	no	—
42	H(3,𝔽₃)	—	12	2	no	—
43	C₃×C₉	—	12	2	no	—
44	C₃×C₉	C₃×A₄	1	3	yes	complemented
45	C₂×C₃²	—	3	1	no	—
46	C₂×C₃²	—	3	2	no	—
47	C₂×C₃²	—	3	1	no	—
48	C₂×C₃²	—	9	3	no	—
49	C₂×C₉	—	3	9	no	—
50	C₂²×C₃	—	3	1	no	—
51	C₂²×C₃	—	3	1	no	—
52	C₂²×C₃	—	3	2	no	—
53	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	A₄	—	9	3	no	—
55	C₃²	—	9	1	no	—
56	C₃²	—	12	3	no	—
57	C₃²	—	36	3	no	—
58	C₃²	C₂²⋊H(3,𝔽₃)	1	2	yes	—
59	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
60	C₃²	C₃²×A₄	1	1	yes	characteristic
61	C₉	—	3	3	no	—
62	C₉	—	12	6	no	—
63	C₂×C₃	—	3	3	no	—
64	C₂×C₃	—	3	1	no	—
65	C₂×C₃	—	9	1	no	—
66	C₂×C₃	—	9	2	no	—
67	C₂²	#6910	1	1	yes	characteristic, complemented, Sylow
68	C₃	—	3	1	no	—
69	C₃	—	3	1	no	—
70	C₃	—	3	2	no	—
71	C₃	—	36	3	no	—
72	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
73	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	C₃×C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
prime radical	C₃×C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	#6910
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×3₋¹⁺²	C₃	semidirect
3. C₃×#1094	C₃	semidirect
4. #66357	C₃	semidirect
11. C₂²×C₃×C₉	C₃²	semidirect
16. #1094	C₃²	semidirect
19. C₃×3₋¹⁺²	A₄	semidirect
44. C₃×C₉	C₃×A₄	semidirect
67. C₂²	#6910	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
C₃×C₂²⋊H(3,𝔽₃)	3
C₃×#1094	3
#66357	3
#6910	4