Subgroups — #28354646

Order 13851.

404 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊#4294	C₃	1	3	yes	maximal
2	#17987337	C₃	1	9	yes	maximal, complemented
3	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
4	C₃²×#4294	—	3	3	no	—
5	C₃²⋊#4294	—	3	6	no	—
6	C₃²⋊#4294	C₃²	1	9	yes	—
7	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
8	C₃×C₁₉×3₋¹⁺²	—	3	3	no	—
9	C₃×C₁₉×3₋¹⁺²	—	3	6	no	—
10	C₃×C₁₉×3₋¹⁺²	C₉	1	9	yes	complemented
11	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
12	#10624690	—	19	1	no	maximal, Sylow
13	C₃³×C₁₉	—	9	3	no	—
14	C₃³×C₁₉	C₃×C₉	1	3	yes	—
15	C₃³×C₁₉	H(3,𝔽₃)	1	3	yes	—
16	C₃³×C₁₉	3₋¹⁺²	1	6	yes	—
17	C₃³×C₁₉	C₃³	1	1	yes	characteristic
18	C₁₉×3₋¹⁺²	—	3	27	no	—
19	C₃×C₉×C₁₉	—	3	3	no	—
20	C₃×C₉×C₁₉	—	3	6	no	—
21	C₃×C₉×C₁₉	C₃×C₉	1	9	yes	complemented
22	C₃×#4294	—	3	9	no	—
23	C₃×#4294	—	9	9	no	—
24	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic
25	C₃×C₃²⋊C₉	—	19	3	no	—
26	#6867	—	19	9	no	—
27	C₃²×C₁₉	—	3	3	no	—
28	C₃²×C₁₉	—	3	6	no	—
29	C₃²×C₁₉	—	3	3	no	—
30	C₃²×C₁₉	—	9	3	no	—
31	C₃²×C₁₉	—	9	6	no	—
32	C₃²×C₁₉	C₃²⋊C₉	1	9	yes	—
33	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
34	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₃²×C₁₉	C₃×3₋¹⁺²	1	2	yes	—
36	C₉×C₁₉	—	3	9	no	—
37	#4294	—	9	9	no	—
38	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	3	6	no	—
40	C₃×3₋¹⁺²	—	3	3	no	—
41	C₃×3₋¹⁺²	#4294	1	9	yes	complemented
42	C₃²×C₉	—	57	3	no	—
43	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
44	C₃²⋊C₉	—	19	9	no	—
45	C₃²⋊C₉	—	57	6	no	—
46	C₃×C₁₉	—	3	9	no	—
47	C₃×C₁₉	—	3	1	no	—
48	C₃×C₁₉	—	3	2	no	—
49	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
50	C₃×C₁₉	#6910	1	3	yes	—
51	C₃³	—	9	3	no	—
52	C₃³	C₃×#4294	1	3	yes	—
53	C₃³	#10587025	1	6	yes	—
54	C₃³	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
55	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
56	3₋¹⁺²	—	3	27	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	—	3	6	no	—
59	C₃×C₉	—	57	9	no	—
60	C₃×C₉	—	171	9	no	—
61	C₃×C₉	C₃×#4294	1	9	yes	complemented
62	C₁₉	#10624690	1	1	yes	characteristic, complemented, Sylow
63	C₃²	—	3	3	no	—
64	C₃²	—	3	6	no	—
65	C₃²	—	3	3	no	—
66	C₃²	—	9	3	no	—
67	C₃²	—	9	6	no	—
68	C₃²	C₃²⋊#4294	1	9	yes	—
69	C₃²	C₃²×#4294	1	1	yes	characteristic
70	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
71	C₃²	C₃×#10587025	1	2	yes	—
72	C₉	—	3	9	no	—
73	C₉	—	171	9	no	—
74	C₃	—	3	9	no	—
75	C₃	—	3	1	no	—
76	C₃	—	3	2	no	—
77	C₃	C₃×C₃²⋊#4294	1	1	yes	characteristic
78	C₃	#17987510	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₁₉	C₃²
cosocle	C₃³×C₁₉	C₃³
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624690
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #17987337	C₃	semidirect
10. C₃×C₁₉×3₋¹⁺²	C₉	semidirect
21. C₃×C₉×C₁₉	C₃×C₉	semidirect
41. C₃×3₋¹⁺²	#4294	semidirect
61. C₃×C₉	C₃×#4294	semidirect
62. C₁₉	#10624690	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊#4294	3
#17987337	3
C₃²×C₁₉×3₋¹⁺²	3
#10624690	19