Subgroups — #28356616

Order 13851.

936 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987547	C₃	1	3	yes	maximal
2	#17987519	C₃	1	24	yes	maximal, complemented
3	C₃×#13095322	C₃	1	12	yes	maximal, complemented
4	C₃×C₁₉×#737	C₃	1	1	yes	maximal, characteristic
5	#13095331	—	3	27	no	—
6	C₃×#10587025	C₃²	1	36	yes	—
7	#13095322	C₃²	1	9	yes	—
8	#13095322	C₃²	1	72	yes	—
9	C₁₉×#737	—	3	9	no	—
10	C₃×C₁₉×3₋¹⁺²	C₃²	1	8	yes	—
11	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₁₉	C₃²	1	4	yes	—
13	#10624933	—	19	1	no	maximal, Sylow
14	C₃³×C₁₉	C₃³	1	4	yes	—
15	C₁₉×3₋¹⁺²	—	3	24	no	—
16	C₁₉×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
17	C₃×C₉×C₁₉	—	3	12	no	—
18	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
19	C₃×C₉×C₁₉	C₃³	1	8	yes	—
20	#10587025	—	3	72	no	—
21	#10587025	C₃³	1	27	yes	—
22	C₃×#4294	—	3	36	no	—
23	3₋¹⁺⁴	—	19	3	no	—
24	C₃×#737	C₁₉⋊C₃	1	1	yes	characteristic
25	#6912	—	19	24	no	—
26	C₃×#733	—	19	12	no	—
27	C₃²×C₁₉	—	9	4	no	—
28	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	12	yes	—
29	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
30	C₉×C₁₉	—	3	1	no	—
31	C₉×C₁₉	—	3	8	no	—
32	#4294	—	3	27	no	—
33	#737	—	3	9	no	—
34	#737	—	57	27	no	—
35	C₃×3₋¹⁺²	—	19	36	no	—
36	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	8	yes	—
37	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
38	C₃²×C₉	C₃×C₁₉⋊C₃	1	4	yes	—
39	#733	—	19	72	no	—
40	#733	—	19	9	no	—
41	C₃×C₁₉	—	3	1	no	—
42	C₃×C₁₉	—	3	12	no	—
43	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
44	C₃³	C₃²×C₁₉⋊C₃	1	4	yes	—
45	3₋¹⁺²	—	3	24	no	—
46	3₋¹⁺²	—	19	27	no	—
47	3₋¹⁺²	—	57	72	no	—
48	H(3,𝔽₃)	C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
49	C₃×C₉	—	3	12	no	—
50	C₃×C₉	—	57	36	no	—
51	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
52	C₃×C₉	C₃²×C₁₉⋊C₃	1	8	yes	—
53	C₁₉	#10624933	1	1	yes	characteristic, complemented, Sylow
54	C₃²	—	9	4	no	—
55	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	12	yes	—
56	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
57	C₉	—	3	1	no	—
58	C₉	—	3	8	no	—
59	C₉	—	57	27	no	—
60	C₃	—	3	1	no	—
61	C₃	—	3	12	no	—
62	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₃×C₁₉×#737	C₃
socle	C₃×C₁₉	C₃²×H(3,𝔽₃)
prime radical	C₃×C₁₉×H(3,𝔽₃)	C₃²
hypercenter	C₃×#737	C₁₉⋊C₃
hypocenter	C₁₉	#10624933
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
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. #17987519	C₃	semidirect
3. C₃×#13095322	C₃	semidirect
53. C₁₉	#10624933	semidirect

Maximal subgroups

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

subgroup	index
#17987547	3
#17987519	3
C₃×#13095322	3
C₃×C₁₉×#737	3
#10624933	19