Subgroups — #41008384

Order 26973.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582692	C₃	1	3	yes	maximal
2	#22582536	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66526	—	3	3	no	—
4	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
5	#15688504	—	3	2	no	—
6	#15688502	—	3	1	no	—
7	C₉×#66524	C₉	1	2	yes	complemented
8	C₉²×C₃₇	C₉	1	1	yes	characteristic, complemented
9	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
10	C₃×#66526	—	9	3	no	—
11	C₃×C₉×C₃₇	—	3	1	no	—
12	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃×#66524	—	3	2	no	—
14	C₃×#66524	3₋¹⁺²	1	2	yes	—
15	C₉×C₃₇⋊C₃	—	3	3	no	—
16	#10624513	—	37	1	no	maximal, Sylow
17	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
18	C₃×C₃₇⋊C₃	—	9	1	no	—
19	C₃×C₃₇⋊C₃	#731	1	3	yes	—
20	#66526	—	27	3	no	—
21	C₉×C₃₇	—	3	1	no	—
22	C₉×C₃₇	—	9	1	no	—
23	#66524	—	3	2	no	—
24	#66524	—	9	2	no	—
25	#6887	—	37	1	no	—
26	#6867	—	37	3	no	—
27	C₃×C₃₇	—	3	1	no	—
28	C₃×C₃₇	#6867	1	1	yes	characteristic
29	C₃₇⋊C₃	—	3	3	no	—
30	C₃²×C₉	—	37	1	no	—
31	C₉⋊C₉	—	111	2	no	—
32	C₃²⋊C₉	—	111	1	no	—
33	C₃²⋊C₉	—	111	3	no	—
34	C₉²	—	37	2	no	—
35	C₉²	#66526	1	1	yes	characteristic, complemented
36	C₃₇	#10624513	1	1	yes	characteristic, complemented, Sylow
37	C₃³	—	37	1	no	—
38	C₃×C₉	—	3	1	no	—
39	C₃×C₉	—	37	2	no	—
40	C₃×C₉	—	333	3	no	—
41	C₃×C₉	—	111	3	no	—
42	C₃×C₉	—	111	2	no	—
43	C₃×C₉	C₃×#66526	1	1	yes	characteristic
44	C₃²	—	37	3	no	—
45	C₃²	—	333	1	no	—
46	C₃²	C₃²⋊#66526	1	1	yes	characteristic
47	C₉	—	3	1	no	—
48	C₉	—	9	1	no	—
49	C₉	—	333	2	no	—
50	C₉	—	111	2	no	—
51	C₉	—	999	3	no	—
52	C₃	—	3	1	no	—
53	C₃	—	111	3	no	—
54	C₃	#22582692	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582692
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃×C₉	C₃×#66526
Fitting	C₉²×C₃₇	C₉
socle	C₃×C₃₇	#6867
prime radical	C₃²×C₃₇⋊C₃	H(3,𝔽₃)
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	C₉²	#66526
hypocenter	C₃₇	#10624513
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
7. C₉×#66524	C₉	semidirect
8. C₉²×C₃₇	C₉	semidirect
35. C₉²	#66526	semidirect
36. C₃₇	#10624513	semidirect

Maximal subgroups

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

subgroup	index
#22582692	3
#22582536	3
#10624513	37