Subgroups — #41008421

Order 26973.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688534	C₃	1	3	yes	maximal
2	C₂₇×#66524	C₃	1	1	yes	maximal, characteristic
3	#15688534	—	3	3	no	—
4	C₃×#11758601	C₉	1	1	yes	characteristic
5	C₃×#11758601	C₃²	1	1	yes	characteristic
6	C₉×#66524	C₉	1	1	yes	characteristic
7	C₃×C₂₇×C₃₇	C₉	1	1	yes	characteristic
8	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
9	C₃×#66524	C₂₇	1	1	yes	characteristic
10	C₃×#66524	C₂₇	1	1	yes	characteristic
11	C₃×#66524	C₂₇	1	1	yes	characteristic
12	C₂₇×C₃₇	—	3	1	no	—
13	#11758601	—	3	1	no	—
14	#11758601	3₋¹⁺²	1	3	yes	—
15	C₉⋊C₈₁	—	37	1	no	maximal, Sylow
16	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
17	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
18	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
19	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
20	#66524	—	3	1	no	—
21	#66524	—	3	1	no	—
22	#66524	C₈₁	1	3	yes	complemented
23	C₃×C₈₁	—	37	3	no	—
24	C₉×C₂₇	—	37	1	no	—
25	C₃×C₃₇	C₉⋊C₂₇	1	1	yes	characteristic
26	C₃×C₃₇	C₃×C₈₁	1	1	yes	characteristic
27	C₃×C₃₇	C₈₁⋊C₃	1	1	yes	characteristic
28	C₃×C₃₇	C₈₁⋊C₃	1	1	yes	characteristic
29	C₃×C₂₇	—	37	1	no	—
30	C₃×C₂₇	—	37	1	no	—
31	C₃×C₂₇	#66526	1	1	yes	characteristic
32	C₉²	—	37	1	no	—
33	C₈₁	—	111	3	no	—
34	C₃₇	C₉⋊C₈₁	1	1	yes	characteristic, complemented, Sylow
35	C₃×C₉	—	37	1	no	—
36	C₃×C₉	—	37	1	no	—
37	C₃×C₉	—	37	1	no	—
38	C₃×C₉	C₃×#66526	1	1	yes	characteristic
39	C₂₇	—	3	1	no	—
40	C₂₇	—	37	3	no	—
41	C₂₇	—	111	1	no	—
42	C₃²	(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
43	C₉	—	37	3	no	—
44	C₉	—	111	1	no	—
45	C₉	—	111	1	no	—
46	C₉	#15688538	1	1	yes	characteristic
47	C₉	#15688541	1	1	yes	characteristic
48	C₉	#15688542	1	1	yes	characteristic
49	C₃	#22582706	1	1	yes	characteristic
50	C₃	(C₃₇⋊C₃)⋊C₈₁	1	1	yes	characteristic
51	C₃	#22582716	1	1	yes	characteristic
52	C₃	#22582717	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃×C₉	C₃×#66526
commutator	C₃×C₃₇	C₃×C₈₁
Fitting	C₃×C₂₇×C₃₇	C₉
socle, prime radical	C₃²×C₃₇	C₃×C₂₇
cosocle	C₃×#11758601	C₃²
hypercenter	C₃×C₂₇	#66526
hypocenter	C₃₇	C₉⋊C₈₁
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₂₇

Complements

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

N	H ≅ G/N	type
22. #66524	C₈₁	semidirect
34. C₃₇	C₉⋊C₈₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688534	3
C₂₇×#66524	3
C₉⋊C₈₁	37