Subgroups — #41008209

Order 26973.

936 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582680	C₃	1	3	yes	maximal
2	#22582652	C₃	1	24	yes	maximal, complemented
3	C₃×#15688521	C₃	1	12	yes	maximal, complemented
4	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
5	#15688530	—	3	27	no	—
6	C₃×#11758608	C₃²	1	36	yes	—
7	#15688521	C₃²	1	9	yes	—
8	#15688521	C₃²	1	72	yes	—
9	C₃₇×#737	—	3	9	no	—
10	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
13	C₃³×C₃₇	C₃³	1	4	yes	—
14	3₋¹⁺²×C₃₇	—	3	24	no	—
15	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
16	C₃×C₉×C₃₇	—	3	12	no	—
17	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
18	C₃×C₉×C₃₇	C₃³	1	8	yes	—
19	#11758608	—	3	72	no	—
20	#11758608	C₃³	1	27	yes	—
21	C₃×#66524	—	3	36	no	—
22	#10624933	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	9	4	no	—
24	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
25	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
26	C₉×C₃₇	—	3	1	no	—
27	C₉×C₃₇	—	3	8	no	—
28	#66524	—	3	27	no	—
29	3₋¹⁺⁴	—	37	3	no	—
30	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic
31	#6912	—	37	24	no	—
32	C₃×#733	—	37	12	no	—
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	—	3	12	no	—
35	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
36	#737	—	3	9	no	—
37	#737	—	111	27	no	—
38	C₃×3₋¹⁺²	—	37	36	no	—
39	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	8	yes	—
40	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
41	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	—
42	#733	—	37	72	no	—
43	#733	—	37	9	no	—
44	C₃₇	#10624933	1	1	yes	characteristic, complemented, Sylow
45	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
46	3₋¹⁺²	—	3	24	no	—
47	3₋¹⁺²	—	37	27	no	—
48	3₋¹⁺²	—	111	72	no	—
49	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
50	C₃×C₉	—	3	12	no	—
51	C₃×C₉	—	111	36	no	—
52	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃×C₉	C₃²×C₃₇⋊C₃	1	8	yes	—
54	C₃²	—	9	4	no	—
55	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
56	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
57	C₉	—	3	1	no	—
58	C₉	—	3	8	no	—
59	C₉	—	111	27	no	—
60	C₃	—	3	1	no	—
61	C₃	—	3	12	no	—
62	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃×C₃₇×#737	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624933
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
2. #22582652	C₃	semidirect
3. C₃×#15688521	C₃	semidirect
44. C₃₇	#10624933	semidirect

Maximal subgroups

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

subgroup	index
#22582680	3
#22582652	3
C₃×#15688521	3
C₃×C₃₇×#737	3
#10624933	37