Subgroups — #11757853

Order 972.

164 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	#66436	C₃	1	3	yes	maximal, complemented
3	#66368	C₃	1	3	yes	maximal, complemented
4	#66359	C₃	1	2	yes	maximal, complemented
5	#66358	C₃	1	1	yes	maximal, characteristic, complemented
6	#66357	C₃	1	3	yes	maximal, complemented
7	#6913	—	4	1	no	maximal, Sylow
8	C₂×C₃×3₋¹⁺²	—	3	1	no	—
9	C₂²×C₃³	C₃²	1	1	yes	characteristic
10	C₂²×3₋¹⁺²	—	3	3	no	—
11	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
12	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
13	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
14	C₂²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
15	#1094	C₃²	1	6	yes	—
16	C₉⋊A₄	—	3	6	no	—
17	C₉×A₄	—	3	3	no	—
18	#737	—	4	3	no	—
19	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
20	#733	—	4	2	no	—
21	#732	—	4	1	no	—
22	#731	—	4	3	no	—
23	AΣL(1,𝔽₂₇)	—	4	3	no	—
24	C₂×C₃³	—	3	1	no	—
25	C₂×3₋¹⁺²	—	3	9	no	—
26	C₂×C₃×C₉	—	3	1	no	—
27	C₂×C₃×C₉	—	3	1	no	—
28	C₂×C₃×C₉	—	3	1	no	—
29	C₂²×C₃²	—	3	1	no	—
30	C₂²×C₃²	—	9	1	no	—
31	C₂²×C₃²	C₃³	1	1	yes	characteristic
32	C₃×A₄	—	3	3	no	—
33	C₂²×C₉	—	3	1	no	—
34	C₂²×C₉	—	3	1	no	—
35	C₂²×C₉	H(3,𝔽₃)	1	3	yes	—
36	C₂²⋊C₉	—	3	6	no	—
37	C₃³	C₃×A₄	1	1	yes	characteristic
38	3₋¹⁺²	—	3	3	no	—
39	3₋¹⁺²	—	4	6	no	—
40	3₋¹⁺²	—	12	6	no	—
41	H(3,𝔽₃)	—	4	3	no	—
42	C₃×C₉	—	12	3	no	—
43	C₃×C₉	C₃×A₄	1	1	yes	characteristic
44	C₃×C₉	C₃×A₄	1	1	yes	characteristic
45	C₃×C₉	C₃×A₄	1	1	yes	characteristic
46	C₂×C₃²	—	3	1	no	—
47	C₂×C₃²	—	3	3	no	—
48	C₂×C₃²	—	9	3	no	—
49	C₂×C₉	—	3	3	no	—
50	C₂×C₉	—	3	3	no	—
51	C₂×C₉	—	3	3	no	—
52	C₂²×C₃	—	3	1	no	—
53	C₂²×C₃	—	9	1	no	—
54	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	A₄	—	9	3	no	—
56	C₃²	—	3	1	no	—
57	C₃²	—	9	1	no	—
58	C₃²	—	12	3	no	—
59	C₃²	C₃²×A₄	1	1	yes	characteristic
60	C₉	—	3	1	no	—
61	C₉	—	3	1	no	—
62	C₉	—	12	6	no	—
63	C₉	C₂²⋊H(3,𝔽₃)	1	3	yes	—
64	C₂×C₃	—	3	3	no	—
65	C₂×C₃	—	3	1	no	—
66	C₂×C₃	—	9	3	no	—
67	C₂²	#6913	1	1	yes	characteristic, complemented, Sylow
68	C₃	—	3	1	no	—
69	C₃	—	9	1	no	—
70	C₃	—	36	3	no	—
71	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
72	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)
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	#6913
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ 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
2. #66436	C₃	semidirect
3. #66368	C₃	semidirect
4. #66359	C₃	semidirect
5. #66358	C₃	semidirect
6. #66357	C₃	semidirect
19. C₃×3₋¹⁺²	A₄	semidirect
67. C₂²	#6913	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
#66436	3
#66368	3
#66359	3
#66358	3
#66357	3
#6913	4