Subgroups — #41005049

Order 26973.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688500	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688500	C₃	1	1	yes	maximal, characteristic
3	C₂₇×#66524	C₃	1	1	yes	maximal, characteristic
4	C₃×C₃₇×C₈₁	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758601	C₉	1	1	yes	characteristic
6	C₃×#11758601	C₉	1	1	yes	characteristic
7	C₉×#66524	C₉	1	1	yes	characteristic
8	#15688500	—	3	1	no	—
9	#15688500	—	3	1	no	—
10	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
11	C₃₇×C₈₁	—	3	1	no	—
12	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃×#66524	C₂₇	1	1	yes	characteristic
14	C₃×#66524	C₂₇	1	1	yes	characteristic
15	C₃×#66524	C₂₇	1	1	yes	characteristic
16	C₂₇×C₃₇	3₋¹⁺²	1	1	yes	characteristic
17	C₂₇×C₃₇	3₋¹⁺²	1	1	yes	characteristic
18	C₂₇×C₃₇	3₋¹⁺²	1	1	yes	characteristic
19	#11758601	—	3	1	no	—
20	#11758601	—	3	1	no	—
21	C₉⋊C₈₁	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
23	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
24	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
25	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
26	#66524	—	3	1	no	—
27	#66524	—	3	1	no	—
28	#66524	C₈₁	1	3	yes	complemented
29	C₃×C₈₁	—	37	1	no	—
30	C₃×C₈₁	—	37	1	no	—
31	C₃×C₈₁	C₃₇⋊C₃	1	1	yes	characteristic
32	C₉×C₂₇	—	37	1	no	—
33	C₃×C₃₇	C₉⋊C₂₇	1	1	yes	characteristic
34	C₃×C₃₇	C₃×C₈₁	1	1	yes	characteristic
35	C₃×C₃₇	C₈₁⋊C₃	1	1	yes	characteristic
36	C₃×C₃₇	C₈₁⋊C₃	1	1	yes	characteristic
37	C₃×C₂₇	—	37	1	no	—
38	C₃×C₂₇	—	37	1	no	—
39	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
40	C₉²	—	37	1	no	—
41	C₈₁	—	3	1	no	—
42	C₈₁	—	111	1	no	—
43	C₈₁	—	111	1	no	—
44	C₃₇	C₉⋊C₈₁	1	1	yes	characteristic, complemented, Sylow
45	C₃×C₉	—	37	1	no	—
46	C₃×C₉	—	37	1	no	—
47	C₃×C₉	—	37	1	no	—
48	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
49	C₂₇	—	111	1	no	—
50	C₂₇	—	111	1	no	—
51	C₂₇	#11758605	1	1	yes	characteristic
52	C₂₇	#11758606	1	1	yes	characteristic
53	C₂₇	#11758608	1	1	yes	characteristic
54	C₃²	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
55	C₉	—	37	3	no	—
56	C₉	—	111	1	no	—
57	C₉	—	111	1	no	—
58	C₉	#15688504	1	1	yes	characteristic
59	C₉	#15688509	1	1	yes	characteristic
60	C₉	#15688510	1	1	yes	characteristic
61	C₃	#22582488	1	1	yes	characteristic
62	C₃	C₈₁×C₃₇⋊C₃	1	1	yes	characteristic
63	C₃	#22582498	1	1	yes	characteristic
64	C₃	#22582499	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
28. #66524	C₈₁	semidirect
44. 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×#15688500	3
C₂₇×#66524	3
C₃×C₃₇×C₈₁	3
C₉⋊C₈₁	37