Subgroups — #737×#4294 #28355920

Order 13851.

986 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×#4294	C₃	1	24	yes	maximal, complemented
2	H(3,𝔽₃)×#4294	C₃	1	3	yes	maximal
3	#17987405	C₃	1	8	yes	maximal, complemented
4	C₃×C₉×#4294	C₃	1	4	yes	maximal, complemented
5	C₃×C₁₉×#737	C₃	1	1	yes	maximal, characteristic
6	C₃²×#4294	C₃²	1	12	yes	—
7	#13095307	C₃²	1	24	yes	—
8	#13095306	C₃²	1	24	yes	—
9	C₃²⋊#4294	C₃²	1	24	yes	—
10	C₉×#4294	C₃²	1	1	yes	characteristic
11	C₉×#4294	C₃²	1	8	yes	—
12	C₉×#4294	C₃²	1	24	yes	—
13	C₁₉×#737	C₉	1	27	yes	complemented
14	C₃×C₁₉×3₋¹⁺²	C₃²	1	8	yes	—
15	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₁₉	C₃²	1	4	yes	—
17	C₉×#737	—	19	1	no	maximal, Sylow
18	C₃³×C₁₉	C₃³	1	4	yes	—
19	C₁₉×3₋¹⁺²	C₃×C₉	1	72	yes	complemented
20	C₁₉×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
21	C₃×C₉×C₁₉	C₃×C₉	1	36	yes	complemented
22	C₃×C₉×C₁₉	C₃³	1	8	yes	—
23	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
24	C₃×#4294	—	3	36	no	—
25	C₃×#4294	C₃³	1	24	yes	—
26	C₃×#4294	C₃³	1	3	yes	—
27	C₃×#737	C₁₉⋊C₃	1	1	yes	characteristic
28	C₉×3₋¹⁺²	—	19	24	no	—
29	C₉×H(3,𝔽₃)	—	19	3	no	—
30	#6887	—	19	8	no	—
31	C₃×C₉²	—	19	4	no	—
32	C₃²×C₁₉	—	3	12	no	—
33	C₃²×C₁₉	C₃²×C₉	1	12	yes	—
34	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
35	C₉×C₁₉	C₃²×C₉	1	3	yes	—
36	C₉×C₁₉	C₃²×C₉	1	24	yes	—
37	#4294	—	3	24	no	—
38	#4294	#737	1	9	yes	direct product factor
39	#737	#4294	1	27	yes	direct product factor
40	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	8	yes	—
41	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
42	C₃²×C₉	—	19	12	no	—
43	C₃²×C₉	C₃×C₁₉⋊C₃	1	4	yes	—
44	C₉⋊C₉	—	19	24	no	—
45	C₉⋊C₉	—	19	24	no	—
46	C₃²⋊C₉	—	19	24	no	—
47	C₉²	—	19	24	no	—
48	C₉²	—	19	1	no	—
49	C₉²	—	19	8	no	—
50	C₃×C₁₉	—	3	12	no	—
51	C₃×C₁₉	C₃³×C₉	1	1	yes	characteristic
52	C₃×C₁₉	C₃×#737	1	3	yes	—
53	C₃³	C₃²×C₁₉⋊C₃	1	4	yes	—
54	3₋¹⁺²	C₃×#4294	1	72	yes	complemented
55	H(3,𝔽₃)	C₃×#4294	1	9	yes	—
56	C₃×C₉	—	19	3	no	—
57	C₃×C₉	—	19	24	no	—
58	C₃×C₉	—	57	36	no	—
59	C₃×C₉	C₃×#4294	1	36	yes	complemented
60	C₃×C₉	C₃²×C₁₉⋊C₃	1	8	yes	—
61	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
62	C₁₉	C₉×#737	1	1	yes	characteristic, complemented, Sylow
63	C₃²	—	3	12	no	—
64	C₃²	C₃²×#4294	1	12	yes	—
65	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
66	C₉	—	19	9	no	—
67	C₉	—	57	24	no	—
68	C₉	C₃²×#4294	1	3	yes	—
69	C₉	C₃²×#4294	1	24	yes	—
70	C₃	—	3	12	no	—
71	C₃	C₃³×#4294	1	1	yes	characteristic
72	C₃	C₁₉⋊C₃×#737	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²×C₁₉⋊C₃
commutator	C₃×C₁₉	C₃³×C₉
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₃×C₁₉×#737	C₃
socle, cosocle	C₃²×C₁₉	C₃⁴
prime radical	C₃×C₁₉×H(3,𝔽₃)	C₃²
hypercenter	C₃×#737	C₁₉⋊C₃
hypocenter	C₁₉	C₉×#737
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃×C₉ ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×#4294	C₃	semidirect
3. #17987405	C₃	semidirect
4. C₃×C₉×#4294	C₃	semidirect
13. C₁₉×#737	C₉	semidirect
19. C₁₉×3₋¹⁺²	C₃×C₉	semidirect
21. C₃×C₉×C₁₉	C₃×C₉	semidirect
38. #4294	#737	direct (N × H)
39. #737	#4294	direct (N × H)
54. 3₋¹⁺²	C₃×#4294	semidirect
59. C₃×C₉	C₃×#4294	semidirect
62. C₁₉	C₉×#737	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×#4294	3
H(3,𝔽₃)×#4294	3
#17987405	3
C₃×C₉×#4294	3
C₃×C₁₉×#737	3
C₉×#737	19