Subgroups — #41004794

Order 26973.

156 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	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic
3	#15688511	—	3	3	no	—
4	#15688511	—	3	3	no	—
5	#15688511	—	3	3	no	—
6	C₃×#11758601	C₉	1	9	yes	complemented
7	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
8	C₃₇×C₉⋊C₉	—	3	1	no	—
9	C₃₇×C₉⋊C₉	—	3	1	no	—
10	C₃₇×C₃²⋊C₉	—	3	1	no	—
11	C₉²×C₃₇	C₉	1	3	yes	—
12	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
13	C₃×C₉×C₃₇	—	3	3	no	—
14	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
15	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
18	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
19	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
20	#11758601	—	3	9	no	—
21	#10624465	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	9	1	no	—
23	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
24	C₃²×C₃₇	C₉⋊C₉	1	3	yes	—
25	C₉×C₃₇	—	9	3	no	—
26	C₉×C₃₇	C₉²	1	1	yes	characteristic
27	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
28	C₉×C₃₇	C₉⋊C₉	1	3	yes	—
29	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
30	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
31	C₃×C₂₇⋊C₃	—	37	3	no	—
32	#6887	C₃₇⋊C₃	1	1	yes	characteristic
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	—	3	3	no	—
35	C₃×C₃₇	#6855	1	1	yes	characteristic
36	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
37	C₂₇⋊C₃	—	111	3	no	—
38	C₂₇⋊C₃	—	111	3	no	—
39	C₂₇⋊C₃	—	111	3	no	—
40	C₃×C₂₇	—	37	9	no	—
41	C₉⋊C₉	—	3	1	no	—
42	C₉⋊C₉	—	3	1	no	—
43	C₃²⋊C₉	—	3	1	no	—
44	C₉²	#66524	1	3	yes	—
45	C₃₇	#10624465	1	1	yes	characteristic, complemented, Sylow
46	C₃³	#11758608	1	1	yes	characteristic
47	C₃×C₉	—	3	3	no	—
48	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
49	C₃×C₉	#11758605	1	3	yes	—
50	C₃×C₉	#11758606	1	3	yes	—
51	C₃×C₉	C₃×#66524	1	1	yes	characteristic
52	C₃×C₉	#11758608	1	1	yes	characteristic
53	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
54	C₂₇	—	111	9	no	—
55	C₃²	—	9	1	no	—
56	C₃²	C₃²⋊#66524	1	1	yes	characteristic
57	C₃²	#15688504	1	3	yes	—
58	C₉	—	9	3	no	—
59	C₉	C₉×#66524	1	1	yes	characteristic
60	C₉	#15688502	1	3	yes	—
61	C₉	#15688504	1	3	yes	—
62	C₉	#15688505	1	1	yes	characteristic
63	C₉	#15688506	1	1	yes	characteristic
64	C₃	—	3	1	no	—
65	C₃	—	3	3	no	—
66	C₃	#22582441	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₉×#66524
commutator	C₉×C₃₇	C₉²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃×C₃₇	#6855
prime radical	C₃³×C₃₇	3₋¹⁺²
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624465
LCS	G ▷ C₉×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
6. C₃×#11758601	C₉	semidirect
45. C₃₇	#10624465	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
C₃₇×#6887	3
#10624465	37