Subgroups — #41006742

Order 26973.

356 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582554	C₃	1	3	yes	maximal
2	#22582472	C₃	1	9	yes	maximal, complemented
3	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	—	3	3	no	—
5	C₃²⋊#66524	—	3	6	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	—	9	2	no	—
13	C₃³×C₃₇	—	9	1	no	—
14	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	3₋¹⁺²×C₃₇	—	3	27	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	—	3	6	no	—
19	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
20	#11758608	—	9	9	no	—
21	C₃×#66524	—	3	9	no	—
22	#10624772	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	2	no	—
24	C₃²×C₃₇	—	3	1	no	—
25	C₃²×C₃₇	—	3	6	no	—
26	C₃²×C₃₇	—	3	3	no	—
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	—	9	9	no	—
29	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
31	C₉×C₃₇	—	3	9	no	—
32	#66524	—	9	9	no	—
33	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
34	#6891	—	37	3	no	—
35	#6868	—	37	9	no	—
36	C₃×C₃₇	—	3	3	no	—
37	C₃×C₃₇	—	9	3	no	—
38	C₃×C₃₇	#6891	1	1	yes	characteristic
39	C₃×C₃₇	#6911	1	1	yes	characteristic
40	C₃×C₃₇	#6912	1	2	yes	—
41	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
42	C₃×3₋¹⁺²	—	3	3	no	—
43	C₃×3₋¹⁺²	—	3	9	no	—
44	C₃×3₋¹⁺²	—	111	3	no	—
45	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
46	C₃²⋊C₉	—	37	9	no	—
47	C₃²⋊C₉	—	111	6	no	—
48	C₃₇	#10624772	1	1	yes	characteristic, complemented, Sylow
49	C₃³	—	3	3	no	—
50	C₃³	—	9	1	no	—
51	C₃³	—	9	2	no	—
52	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
53	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
54	3₋¹⁺²	—	3	27	no	—
55	3₋¹⁺²	—	333	9	no	—
56	C₃×C₉	—	3	3	no	—
57	C₃×C₉	—	3	6	no	—
58	C₃×C₉	—	111	9	no	—
59	C₃×C₉	#11758608	1	9	yes	complemented
60	C₃²	—	3	1	no	—
61	C₃²	—	3	2	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	6	no	—
65	C₃²	—	9	9	no	—
66	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
67	C₃²	C₃×#11758608	1	3	yes	—
68	C₉	—	3	9	no	—
69	C₉	—	333	9	no	—
70	C₃	—	3	3	no	—
71	C₃	—	9	3	no	—
72	C₃	#22582554	1	1	yes	characteristic
73	C₃	#22582648	1	1	yes	characteristic
74	C₃	#22582652	1	2	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	#10624772
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
2. #22582472	C₃	semidirect
19. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
48. C₃₇	#10624772	semidirect
59. 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
#22582554	3
#22582472	3
C₃²×3₋¹⁺²×C₃₇	3
#10624772	37