Subgroups — #28355929

Order 13851.

944 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×#4294	C₃	1	27	yes	maximal, complemented
2	#17987405	C₃	1	9	yes	maximal, complemented
3	C₃×C₃²⋊#4294	C₃	1	3	yes	maximal
4	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
5	C₃²×#4294	C₃²	1	9	yes	—
6	#13095307	C₃²	1	27	yes	complemented
7	#13095306	C₃²	1	27	yes	complemented
8	C₃²⋊#4294	C₃²	1	9	yes	—
9	C₃²⋊#4294	C₃²	1	18	yes	—
10	C₉×#4294	C₃²	1	27	yes	complemented
11	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
12	C₃×C₁₉×3₋¹⁺²	C₉	1	27	yes	complemented
13	C₃×C₁₉×3₋¹⁺²	C₃²	1	9	yes	—
14	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
15	#10624863	—	19	1	no	maximal, Sylow
16	C₃³×C₁₉	—	9	3	no	—
17	C₃³×C₁₉	C₃×C₉	1	9	yes	—
18	C₃³×C₁₉	C₃³	1	1	yes	characteristic
19	C₃³×C₁₉	C₃³	1	3	yes	—
20	C₁₉×3₋¹⁺²	C₃×C₉	1	81	yes	complemented
21	C₃×C₉×C₁₉	C₃×C₉	1	27	yes	complemented
22	C₃×C₉×C₁₉	C₃³	1	9	yes	—
23	C₃×#4294	—	9	9	no	—
24	C₃×#4294	C₃³	1	27	yes	—
25	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic
26	C₉×3₋¹⁺²	—	19	27	no	—
27	#6887	—	19	9	no	—
28	C₃×C₃²⋊C₉	—	19	3	no	—
29	C₃²×C₁₉	—	3	3	no	—
30	C₃²×C₁₉	—	3	3	no	—
31	C₃²×C₁₉	—	3	6	no	—
32	C₃²×C₁₉	—	9	9	no	—
33	C₃²×C₁₉	C₃²×C₉	1	3	yes	—
34	C₃²×C₁₉	C₃²×C₉	1	9	yes	—
35	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
36	C₉×C₁₉	C₃²×C₉	1	27	yes	—
37	#4294	—	3	27	no	—
38	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
39	C₃×3₋¹⁺²	#4294	1	27	yes	complemented
40	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	9	yes	—
41	C₃²×C₉	—	19	9	no	—
42	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
43	C₉⋊C₉	—	19	27	no	—
44	C₉⋊C₉	—	19	27	no	—
45	C₃²⋊C₉	—	19	9	no	—
46	C₃²⋊C₉	—	19	18	no	—
47	C₉²	—	19	27	no	—
48	C₃×C₁₉	—	3	9	no	—
49	C₃×C₁₉	—	3	3	no	—
50	C₃×C₁₉	C₃³×C₉	1	1	yes	characteristic
51	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
52	C₃³	—	9	3	no	—
53	C₃³	C₃×#4294	1	9	yes	—
54	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
55	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
56	3₋¹⁺²	C₃×#4294	1	81	yes	complemented
57	C₃×C₉	—	19	27	no	—
58	C₃×C₉	—	171	9	no	—
59	C₃×C₉	C₃×#4294	1	27	yes	complemented
60	C₃×C₉	C₃²×C₁₉⋊C₃	1	9	yes	—
61	C₁₉	#10624863	1	1	yes	characteristic, complemented, Sylow
62	C₃²	—	3	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	6	no	—
65	C₃²	—	9	9	no	—
66	C₃²	C₃²×#4294	1	3	yes	—
67	C₃²	C₃²×#4294	1	9	yes	—
68	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
69	C₉	—	57	27	no	—
70	C₉	C₃²×#4294	1	27	yes	—
71	C₃	—	3	9	no	—
72	C₃	—	3	3	no	—
73	C₃	C₃³×#4294	1	1	yes	characteristic
74	C₃	#17987546	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator	C₃×C₁₉	C₃³×C₉
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
socle, cosocle	C₃²×C₁₉	C₃⁴
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624863
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×#4294	C₃	semidirect
2. #17987405	C₃	semidirect
6. #13095307	C₃²	semidirect
7. #13095306	C₃²	semidirect
10. C₉×#4294	C₃²	semidirect
12. C₃×C₁₉×3₋¹⁺²	C₉	semidirect
20. C₁₉×3₋¹⁺²	C₃×C₉	semidirect
21. C₃×C₉×C₁₉	C₃×C₉	semidirect
39. C₃×3₋¹⁺²	#4294	semidirect
56. 3₋¹⁺²	C₃×#4294	semidirect
59. C₃×C₉	C₃×#4294	semidirect
61. C₁₉	#10624863	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
#17987405	3
C₃×C₃²⋊#4294	3
C₃²×C₁₉×3₋¹⁺²	3
#10624863	19