Subgroups — #41005225

Order 26973.

716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758608	C₃	1	3	yes	maximal
2	#22582441	C₃	1	9	yes	maximal, complemented
3	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	—	3	9	no	—
5	C₃×#11758608	—	3	27	no	—
6	C₃²×#66524	C₃²	1	9	yes	—
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	—	3	9	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	3	no	—
17	C₃×C₉×C₃₇	—	3	6	no	—
18	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
19	#11758608	—	3	81	no	—
20	C₃×#66524	—	3	9	no	—
21	C₃×#66524	—	3	18	no	—
22	C₃×#66524	3₋¹⁺²	1	27	yes	complemented
23	#10624571	—	37	1	no	maximal, Sylow
24	C₃²×C₃₇	—	3	9	no	—
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	9	9	no	—
27	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₃²×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
29	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
30	C₉×C₃₇	—	3	9	no	—
31	#66524	—	3	27	no	—
32	C₃²×3₋¹⁺²	—	37	3	no	—
33	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
34	#6855	—	37	9	no	—
35	C₃×C₃₇	—	9	3	no	—
36	C₃×C₃₇	#6891	1	3	yes	—
37	C₃×C₃₇	#6891	1	1	yes	characteristic
38	C₃×C₃₇	#6900	1	9	yes	—
39	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
40	C₃×3₋¹⁺²	—	3	3	no	—
41	C₃×3₋¹⁺²	—	3	9	no	—
42	C₃×3₋¹⁺²	—	111	27	no	—
43	C₃×3₋¹⁺²	—	111	9	no	—
44	C₃²×C₉	—	37	9	no	—
45	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
46	C₃₇	#10624571	1	1	yes	characteristic, complemented, Sylow
47	C₃³	—	3	9	no	—
48	C₃³	—	3	3	no	—
49	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
50	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
51	3₋¹⁺²	—	3	27	no	—
52	3₋¹⁺²	—	111	81	no	—
53	C₃×C₉	—	3	3	no	—
54	C₃×C₉	—	3	6	no	—
55	C₃×C₉	—	37	27	no	—
56	C₃×C₉	—	111	9	no	—
57	C₃×C₉	—	111	18	no	—
58	C₃×C₉	#11758608	1	9	yes	complemented
59	C₃²	—	3	9	no	—
60	C₃²	—	3	3	no	—
61	C₃²	—	9	9	no	—
62	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
63	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	9	yes	—
64	C₃²	C₃×#11758608	1	3	yes	—
65	C₉	—	3	9	no	—
66	C₉	—	111	27	no	—
67	C₃	—	9	3	no	—
68	C₃	#22582551	1	3	yes	—
69	C₃	#22582554	1	1	yes	characteristic
70	C₃	#22582601	1	9	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624571
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #22582441	C₃	semidirect
18. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
22. C₃×#66524	3₋¹⁺²	semidirect
46. C₃₇	#10624571	semidirect
58. C₃×C₉	#11758608	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#11758608	3
#22582441	3
C₃²×3₋¹⁺²×C₃₇	3
#10624571	37