Subgroups — #11757806

Order 972.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66437	C₃	1	3	yes	maximal, complemented
2	C₃×C₉×A₄	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×#732	C₃	1	1	yes	maximal, characteristic, complemented
4	#66363	C₃	1	3	yes	maximal, complemented
5	#66361	C₃	1	3	yes	maximal, complemented
6	#66359	C₃	1	2	yes	maximal, complemented
7	#6908	—	4	1	no	maximal, Sylow
8	C₂×#732	—	3	1	no	—
9	C₃²×A₄	C₃²	1	1	yes	characteristic
10	C₂²×H(3,𝔽₃)	C₃²	1	3	yes	—
11	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
12	#1094	—	3	6	no	—
13	#1094	C₃²	1	6	yes	—
14	C₃×C₂²⋊C₉	—	3	3	no	—
15	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
16	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
17	C₉×A₄	—	3	1	no	—
18	C₉×A₄	—	3	2	no	—
19	#737	—	4	3	no	—
20	C₃²×C₉	—	4	1	no	—
21	#733	—	4	2	no	—
22	#732	A₄	1	1	yes	characteristic, complemented
23	#731	—	4	3	no	—
24	AΣL(1,𝔽₂₇)	—	4	3	no	—
25	C₂×H(3,𝔽₃)	—	3	3	no	—
26	C₂×C₃×C₉	—	3	1	no	—
27	C₂²×C₃²	—	3	3	no	—
28	C₂²×C₃²	C₃³	1	1	yes	characteristic
29	C₃×A₄	—	3	1	no	—
30	C₃×A₄	—	3	3	no	—
31	C₂²×C₉	—	3	1	no	—
32	C₂²⋊C₉	—	3	1	no	—
33	C₂²⋊C₉	—	3	6	no	—
34	C₂²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
35	C₂²⋊C₉	H(3,𝔽₃)	1	2	yes	—
36	C₃³	—	4	1	no	—
37	3₋¹⁺²	—	4	6	no	—
38	3₋¹⁺²	—	12	6	no	—
39	H(3,𝔽₃)	C₃×A₄	1	3	yes	—
40	C₃×C₉	—	4	1	no	—
41	C₃×C₉	—	4	1	no	—
42	C₃×C₉	—	12	1	no	—
43	C₃×C₉	—	12	2	no	—
44	C₃×C₉	—	12	3	no	—
45	C₃×C₉	C₃×A₄	1	1	yes	characteristic
46	C₂×C₃²	—	3	1	no	—
47	C₂×C₃²	—	9	3	no	—
48	C₂×C₉	—	9	1	no	—
49	C₂²×C₃	—	3	1	no	—
50	C₂²×C₃	—	9	3	no	—
51	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	A₄	—	3	3	no	—
53	C₃²	—	3	3	no	—
54	C₃²	—	12	3	no	—
55	C₃²	—	12	1	no	—
56	C₃²	C₃²×A₄	1	1	yes	characteristic
57	C₉	—	3	1	no	—
58	C₉	—	4	1	no	—
59	C₉	—	4	2	no	—
60	C₉	—	12	1	no	—
61	C₉	—	12	6	no	—
62	C₂×C₃	—	3	1	no	—
63	C₂×C₃	—	9	1	no	—
64	C₂×C₃	—	27	3	no	—
65	C₂²	#6908	1	1	yes	characteristic, complemented, Sylow
66	C₃	—	3	1	no	—
67	C₃	—	9	3	no	—
68	C₃	—	12	3	no	—
69	C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
70	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	A₄×H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×#732	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
hypercenter	#732	A₄
hypocenter	C₂²	#6908
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #732

Complements

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

N	H ≅ G/N	type
1. #66437	C₃	semidirect
2. C₃×C₉×A₄	C₃	semidirect
3. C₂²×#732	C₃	semidirect
4. #66363	C₃	semidirect
5. #66361	C₃	semidirect
6. #66359	C₃	semidirect
22. #732	A₄	semidirect
65. C₂²	#6908	semidirect

Maximal subgroups

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

subgroup	index
#66437	3
C₃×C₉×A₄	3
C₂²×#732	3
#66363	3
#66361	3
#66359	3
#6908	4