Subgroups — #41006499

Order 26973.

518 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688521	C₃	1	9	yes	maximal, complemented
2	#22582554	C₃	1	3	yes	maximal
3	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	C₃²	1	9	yes	—
5	#15688521	—	3	27	no	—
6	C₃²⋊#66524	—	3	9	no	—
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃²×C₉×C₃₇	—	3	3	no	—
9	C₃²×C₉×C₃₇	—	3	9	no	—
10	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
11	C₃³×C₃₇	—	3	9	no	—
12	C₃³×C₃₇	—	3	3	no	—
13	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	C₃×C₉×C₃₇	—	3	27	no	—
16	C₃×C₉×C₃₇	—	3	6	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	H(3,𝔽₃)	1	9	yes	—
19	#11758608	—	3	27	no	—
20	C₃×#66524	—	3	9	no	—
21	#10624733	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	3	27	no	—
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	6	no	—
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	#733	1	9	yes	complemented
27	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
28	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₉×C₃₇	—	3	9	no	—
30	#66524	—	9	9	no	—
31	C₃³×C₉	C₃₇⋊C₃	1	1	yes	characteristic
32	C₃×#733	—	37	9	no	—
33	#6891	—	37	3	no	—
34	C₃×C₃₇	—	3	9	no	—
35	C₃×C₃₇	—	3	3	no	—
36	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₃₇	C₃×#733	1	3	yes	—
38	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	37	9	no	—
40	C₃²×C₉	—	3	3	no	—
41	C₃²×C₉	—	3	9	no	—
42	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
43	#733	—	111	27	no	—
44	C₃²⋊C₉	—	111	9	no	—
45	C₃₇	#10624733	1	1	yes	characteristic, complemented, Sylow
46	C₃³	—	3	9	no	—
47	C₃³	—	3	3	no	—
48	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
49	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
50	3₋¹⁺²	—	111	27	no	—
51	C₃×C₉	—	3	27	no	—
52	C₃×C₉	—	3	6	no	—
53	C₃×C₉	—	3	3	no	—
54	C₃×C₉	—	111	9	no	—
55	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
56	C₃²	—	3	27	no	—
57	C₃²	—	3	3	no	—
58	C₃²	—	3	6	no	—
59	C₃²	—	3	3	no	—
60	C₃²	#15688521	1	9	yes	complemented
61	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
62	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
63	C₉	—	3	9	no	—
64	C₉	—	333	9	no	—
65	C₃	—	3	9	no	—
66	C₃	—	3	3	no	—
67	C₃	#22582550	1	1	yes	characteristic
68	C₃	C₃×#15688521	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃³×C₉×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃³×C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624733
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
1. C₃×#15688521	C₃	semidirect
26. C₃²×C₃₇	#733	semidirect
45. C₃₇	#10624733	semidirect
60. C₃²	#15688521	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688521	3
#22582554	3
C₃³×C₉×C₃₇	3
#10624733	37