Subgroups — #41006523

Order 26973.

446 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688521	C₃	1	9	yes	maximal, complemented
2	#22582554	C₃	1	3	yes	maximal
3	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	C₃²	1	9	yes	—
5	#15688521	—	3	27	no	—
6	C₃²⋊#66524	—	3	9	no	—
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₇	—	3	3	no	—
9	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
10	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
11	C₃³×C₃₇	—	3	3	no	—
12	C₃³×C₃₇	—	9	3	no	—
13	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	3₋¹⁺²×C₃₇	—	3	27	no	—
16	C₃×C₉×C₃₇	—	3	6	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	H(3,𝔽₃)	1	9	yes	—
19	#11758608	—	3	27	no	—
20	C₃×#66524	—	3	9	no	—
21	#10624738	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	3	3	no	—
23	C₃²×C₃₇	—	3	9	no	—
24	C₃²×C₃₇	—	3	3	no	—
25	C₃²×C₃₇	—	9	9	no	—
26	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
27	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₉×C₃₇	—	3	9	no	—
29	#66524	—	9	9	no	—
30	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
31	C₃×#733	—	37	9	no	—
32	#6891	—	37	3	no	—
33	C₃×C₃₇	—	3	3	no	—
34	C₃×C₃₇	—	9	3	no	—
35	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
36	C₃×C₃₇	#6912	1	3	yes	—
37	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
38	C₃×3₋¹⁺²	—	3	3	no	—
39	C₃×3₋¹⁺²	—	3	9	no	—
40	C₃×3₋¹⁺²	—	37	9	no	—
41	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
42	#733	—	111	27	no	—
43	C₃²⋊C₉	—	111	9	no	—
44	C₃₇	#10624738	1	1	yes	characteristic, complemented, Sylow
45	C₃³	—	3	3	no	—
46	C₃³	—	9	3	no	—
47	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
48	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
49	3₋¹⁺²	—	3	27	no	—
50	3₋¹⁺²	—	111	27	no	—
51	C₃×C₉	—	3	6	no	—
52	C₃×C₉	—	3	3	no	—
53	C₃×C₉	—	111	9	no	—
54	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
55	C₃²	—	3	3	no	—
56	C₃²	—	3	3	no	—
57	C₃²	—	3	9	no	—
58	C₃²	—	9	9	no	—
59	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
60	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
61	C₉	—	3	9	no	—
62	C₉	—	333	9	no	—
63	C₃	—	3	3	no	—
64	C₃	—	9	3	no	—
65	C₃	#22582550	1	1	yes	characteristic
66	C₃	#22582652	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624738
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×#15688521	C₃	semidirect
44. C₃₇	#10624738	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688521	3
#22582554	3
C₃²×3₋¹⁺²×C₃₇	3
#10624738	37