Subgroups — #41006319

Order 26973.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688511	C₃	1	3	yes	maximal
2	#22582489	C₃	1	9	yes	maximal, complemented
3	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic
4	#15688511	—	3	9	no	—
5	C₃×#11758601	C₃²	1	9	yes	—
6	C₃²×C₉×C₃₇	C₉	1	3	yes	—
7	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
8	C₉²×C₃₇	C₉	1	2	yes	—
9	C₉²×C₃₇	C₉	1	3	yes	—
10	C₉²×C₃₇	C₉	1	1	yes	characteristic
11	C₉²×C₃₇	C₃²	1	3	yes	—
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃×C₉×C₃₇	—	3	9	no	—
14	C₃×C₉×C₃₇	C₃×C₉	1	6	yes	—
15	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
17	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
18	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	—
19	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
20	#11758601	—	3	9	no	—
21	#10624706	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	3	3	no	—
23	C₃²×C₃₇	C₉⋊C₉	1	3	yes	—
24	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
25	C₉×C₃₇	—	3	6	no	—
26	C₉×C₃₇	C₉⋊C₉	1	3	yes	—
27	C₉×C₃₇	C₉⋊C₉	1	3	yes	—
28	C₉×C₃₇	C₂₇⋊C₃	1	9	yes	complemented
29	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
30	C₃×C₂₇⋊C₃	—	37	3	no	—
31	C₃×C₉²	C₃₇⋊C₃	1	1	yes	characteristic
32	C₉⋊C₂₇	—	37	9	no	—
33	C₃×C₃₇	—	3	3	no	—
34	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
35	C₃×C₃₇	C₃×C₂₇⋊C₃	1	3	yes	—
36	C₃²×C₉	#66524	1	3	yes	—
37	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
38	C₂₇⋊C₃	—	111	9	no	—
39	C₃×C₂₇	—	37	9	no	—
40	C₉²	#66524	1	2	yes	—
41	C₉²	#66524	1	3	yes	—
42	C₉²	#66524	1	1	yes	characteristic
43	C₉²	C₃×C₃₇⋊C₃	1	3	yes	—
44	C₃₇	#10624706	1	1	yes	characteristic, complemented, Sylow
45	C₃³	C₃×#66524	1	1	yes	characteristic
46	C₃×C₉	—	3	9	no	—
47	C₃×C₉	#11758605	1	9	yes	—
48	C₃×C₉	C₃×#66524	1	6	yes	—
49	C₃×C₉	C₃×#66524	1	1	yes	characteristic
50	C₃×C₉	C₃×#66524	1	1	yes	characteristic
51	C₃×C₉	C₃×#66524	1	3	yes	—
52	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
53	C₂₇	—	111	9	no	—
54	C₃²	—	3	3	no	—
55	C₃²	#15688505	1	3	yes	—
56	C₃²	C₃²×#66524	1	1	yes	characteristic
57	C₉	—	3	6	no	—
58	C₉	#15688505	1	3	yes	—
59	C₉	#15688505	1	3	yes	—
60	C₉	#15688511	1	9	yes	complemented
61	C₉	C₃×#11758605	1	3	yes	—
62	C₃	—	3	3	no	—
63	C₃	C₃×#15688505	1	1	yes	characteristic
64	C₃	C₃×#15688511	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃×C₉²×C₃₇	C₃
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₃×C₉²	C₃₇⋊C₃
hypocenter	C₃₇	#10624706
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₉²

Complements

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

N	H ≅ G/N	type
2. #22582489	C₃	semidirect
28. C₉×C₃₇	C₂₇⋊C₃	semidirect
44. C₃₇	#10624706	semidirect
60. C₉	#15688511	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688511	3
#22582489	3
C₃×C₉²×C₃₇	3
#10624706	37