Subgroups — #11757810

Order 972.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66440	C₃	1	3	yes	maximal, complemented
2	C₃×C₉×A₄	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×#733	C₃	1	1	yes	maximal, characteristic, complemented
4	#66366	C₃	1	3	yes	maximal, complemented
5	#66365	C₃	1	3	yes	maximal, complemented
6	#66359	C₃	1	1	yes	maximal, characteristic, complemented
7	#66358	C₃	1	1	yes	maximal, characteristic, complemented
8	#6908	—	4	1	no	maximal, Sylow
9	C₂×#733	—	3	1	no	—
10	C₃²×A₄	C₃²	1	1	yes	characteristic
11	C₂²×3₋¹⁺²	C₃²	1	3	yes	—
12	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
13	C₂²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
14	#1094	C₃²	1	3	yes	—
15	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
16	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
17	C₉⋊A₄	—	3	6	no	—
18	C₉×A₄	—	3	1	no	—
19	C₉×A₄	—	3	2	no	—
20	C₉×A₄	—	3	3	no	—
21	#737	—	4	3	no	—
22	C₃²×C₉	—	4	1	no	—
23	#733	—	4	1	no	—
24	#733	A₄	1	1	yes	characteristic, complemented
25	#732	—	4	1	no	—
26	#731	—	4	3	no	—
27	AΣL(1,𝔽₂₇)	—	4	3	no	—
28	C₂×3₋¹⁺²	—	3	3	no	—
29	C₂×C₃×C₉	—	3	1	no	—
30	C₂²×C₃²	C₃³	1	1	yes	characteristic
31	C₃×A₄	—	3	1	no	—
32	C₃×A₄	—	3	3	no	—
33	C₃×A₄	—	3	3	no	—
34	C₂²×C₉	—	3	1	no	—
35	C₂²×C₉	—	3	3	no	—
36	C₂²⋊C₉	—	3	1	no	—
37	C₂²⋊C₉	—	3	3	no	—
38	C₂²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
39	C₂²⋊C₉	H(3,𝔽₃)	1	2	yes	—
40	C₃³	—	4	1	no	—
41	3₋¹⁺²	—	4	3	no	—
42	3₋¹⁺²	—	12	6	no	—
43	3₋¹⁺²	C₃×A₄	1	3	yes	—
44	H(3,𝔽₃)	—	4	3	no	—
45	C₃×C₉	—	4	1	no	—
46	C₃×C₉	—	4	1	no	—
47	C₃×C₉	—	12	1	no	—
48	C₃×C₉	—	12	2	no	—
49	C₃×C₉	—	12	3	no	—
50	C₃×C₉	C₃×A₄	1	1	yes	characteristic
51	C₂×C₃²	—	3	1	no	—
52	C₂×C₉	—	9	1	no	—
53	C₂×C₉	—	9	3	no	—
54	C₂²×C₃	—	3	1	no	—
55	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	A₄	—	3	3	no	—
57	A₄	—	9	3	no	—
58	C₃²	—	12	3	no	—
59	C₃²	—	12	1	no	—
60	C₃²	—	12	3	no	—
61	C₃²	C₃²×A₄	1	1	yes	characteristic
62	C₉	—	3	1	no	—
63	C₉	—	3	3	no	—
64	C₉	—	4	1	no	—
65	C₉	—	4	2	no	—
66	C₉	—	12	1	no	—
67	C₉	—	12	3	no	—
68	C₂×C₃	—	3	1	no	—
69	C₂×C₃	—	9	1	no	—
70	C₂²	#6908	1	1	yes	characteristic, complemented, Sylow
71	C₃	—	3	1	no	—
72	C₃	—	12	3	no	—
73	C₃	—	36	3	no	—
74	C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
75	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	A₄×H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×#733	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
hypercenter	#733	A₄
hypocenter	C₂²	#6908
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #733

Complements

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

N	H ≅ G/N	type
1. #66440	C₃	semidirect
2. C₃×C₉×A₄	C₃	semidirect
3. C₂²×#733	C₃	semidirect
4. #66366	C₃	semidirect
5. #66365	C₃	semidirect
6. #66359	C₃	semidirect
7. #66358	C₃	semidirect
24. #733	A₄	semidirect
70. C₂²	#6908	semidirect

Maximal subgroups

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

subgroup	index
#66440	3
C₃×C₉×A₄	3
C₂²×#733	3
#66366	3
#66365	3
#66359	3
#66358	3
#6908	4