Subgroups — #41008388

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	#22582537	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66526	—	3	3	no	—
4	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
5	#15688506	C₉	1	1	yes	characteristic, complemented
6	#15688505	C₉	1	1	yes	characteristic, complemented
7	#15688504	—	3	1	no	—
8	#15688502	—	3	1	no	—
9	C₉×#66524	—	3	1	no	—
10	C₃₇×C₉⋊C₉	C₉	1	1	yes	characteristic, complemented
11	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃×#66526	—	9	3	no	—
13	C₃×C₉×C₃₇	—	3	1	no	—
14	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃×#66524	—	3	1	no	—
16	C₃×#66524	—	3	1	no	—
17	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
18	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
19	C₉×C₃₇⋊C₃	—	3	3	no	—
20	#10624515	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
22	C₃×C₃₇⋊C₃	—	9	1	no	—
23	C₃×C₃₇⋊C₃	#731	1	3	yes	—
24	#66526	—	27	3	no	—
25	C₉×C₃₇	—	3	1	no	—
26	C₉×C₃₇	—	9	1	no	—
27	#66524	—	3	1	no	—
28	#66524	—	3	1	no	—
29	#66524	—	9	1	no	—
30	#66524	—	9	1	no	—
31	#6887	—	37	1	no	—
32	#6867	—	37	3	no	—
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	#6867	1	1	yes	characteristic
35	C₃₇⋊C₃	—	3	3	no	—
36	C₃²×C₉	—	37	1	no	—
37	C₉⋊C₉	—	37	1	no	—
38	C₉⋊C₉	—	37	1	no	—
39	C₉⋊C₉	—	111	1	no	—
40	C₉⋊C₉	#66526	1	1	yes	characteristic, complemented
41	C₃²⋊C₉	—	111	1	no	—
42	C₃²⋊C₉	—	111	3	no	—
43	C₉²	—	111	1	no	—
44	C₃₇	#10624515	1	1	yes	characteristic, complemented, Sylow
45	C₃³	—	37	1	no	—
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	37	1	no	—
48	C₃×C₉	—	37	1	no	—
49	C₃×C₉	—	333	3	no	—
50	C₃×C₉	—	111	3	no	—
51	C₃×C₉	—	111	1	no	—
52	C₃×C₉	—	111	1	no	—
53	C₃×C₉	C₃×#66526	1	1	yes	characteristic
54	C₃²	—	37	3	no	—
55	C₃²	—	333	1	no	—
56	C₃²	C₃²⋊#66526	1	1	yes	characteristic
57	C₉	—	3	1	no	—
58	C₉	—	9	1	no	—
59	C₉	—	333	1	no	—
60	C₉	—	333	1	no	—
61	C₉	—	111	1	no	—
62	C₉	—	111	1	no	—
63	C₉	—	999	3	no	—
64	C₃	—	3	1	no	—
65	C₃	—	111	3	no	—
66	C₃	#22582692	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582692
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃×C₉	C₃×#66526
Fitting	C₃₇×C₉⋊C₉	C₉
socle	C₃×C₃₇	#6867
prime radical	C₃²×C₃₇⋊C₃	H(3,𝔽₃)
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	C₉⋊C₉	#66526
hypocenter	C₃₇	#10624515
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ C₉⋊C₉

Complements

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

N	H ≅ G/N	type
5. #15688506	C₉	semidirect
6. #15688505	C₉	semidirect
10. C₃₇×C₉⋊C₉	C₉	semidirect
40. C₉⋊C₉	#66526	semidirect
44. C₃₇	#10624515	semidirect

Maximal subgroups

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

subgroup	index
#22582692	3
#22582537	3
#10624515	37