Subgroups — #41004791

Order 26973.

174 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582538	C₃	1	3	yes	maximal
2	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic
3	#15688506	—	3	3	no	—
4	#15688505	—	3	3	no	—
5	C₃²⋊#66524	—	3	3	no	—
6	C₉×#66524	C₉	1	9	yes	complemented
7	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
8	C₃₇×C₉⋊C₉	—	3	2	no	—
9	C₃₇×C₃²⋊C₉	—	3	1	no	—
10	C₉²×C₃₇	C₉	1	3	yes	complemented
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃×C₉×C₃₇	—	3	3	no	—
13	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
14	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
16	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
17	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
18	C₃×#66524	—	3	9	no	—
19	H(3,ℤ/9)	—	37	1	no	maximal, Sylow
20	C₃²×C₃₇	—	9	1	no	—
21	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
22	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
23	C₉×C₃₇	—	9	3	no	—
24	C₉×C₃₇	C₉²	1	1	yes	characteristic
25	C₉×C₃₇	C₉⋊C₉	1	6	yes	—
26	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
27	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
28	#66524	—	9	9	no	—
29	#6887	—	37	3	no	—
30	#6887	C₃₇⋊C₃	1	1	yes	characteristic
31	C₃×C₃₇	—	3	3	no	—
32	C₃×C₃₇	—	3	1	no	—
33	C₃×C₃₇	#6855	1	1	yes	characteristic
34	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
35	C₉⋊C₉	—	3	2	no	—
36	C₉⋊C₉	—	111	3	no	—
37	C₉⋊C₉	—	111	3	no	—
38	C₃²⋊C₉	—	3	1	no	—
39	C₃²⋊C₉	—	111	3	no	—
40	C₉²	—	37	9	no	—
41	C₉²	#66524	1	3	yes	complemented
42	C₃₇	H(3,ℤ/9)	1	1	yes	characteristic, complemented, Sylow
43	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
44	C₃×C₉	—	3	3	no	—
45	C₃×C₉	—	111	9	no	—
46	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
47	C₃×C₉	#11758605	1	3	yes	—
48	C₃×C₉	#11758606	1	3	yes	—
49	C₃×C₉	C₃×#66524	1	1	yes	characteristic
50	C₃×C₉	#11758608	1	2	yes	—
51	C₃²	—	9	1	no	—
52	C₃²	#15688502	1	3	yes	—
53	C₃²	C₃²⋊#66524	1	1	yes	characteristic
54	C₉	—	9	3	no	—
55	C₉	—	333	9	no	—
56	C₉	C₉×#66524	1	1	yes	characteristic
57	C₉	#15688504	1	6	yes	—
58	C₉	#15688505	1	1	yes	characteristic
59	C₉	#15688506	1	1	yes	characteristic
60	C₃	—	3	1	no	—
61	C₃	—	3	3	no	—
62	C₃	#22582441	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₉×#66524
commutator	C₉×C₃₇	C₉²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃×C₃₇	#6855
prime radical	C₃³×C₃₇	H(3,𝔽₃)
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	H(3,ℤ/9)
LCS	G ▷ C₉×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
6. C₉×#66524	C₉	semidirect
10. C₉²×C₃₇	C₉	semidirect
41. C₉²	#66524	semidirect
42. C₃₇	H(3,ℤ/9)	semidirect

Maximal subgroups

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

subgroup	index
#22582538	3
C₃₇×#6887	3
H(3,ℤ/9)	37