Subgroups — C₁₉×#10624665 #28352861

Order 13851.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#6891	C₃	1	2	yes	maximal, complemented
2	C₁₉×C₃²⋊H(3,𝔽₃)	C₃	1	2	yes	maximal, complemented
3	C₁₉×#6856	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
5	C₃×C₁₉×3₋¹⁺²	—	3	2	no	—
6	C₃×C₁₉×H(3,𝔽₃)	—	3	4	no	—
7	C₃×C₁₉×H(3,𝔽₃)	—	3	2	no	—
8	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	6	yes	—
9	C₁₉×C₃²⋊C₉	—	3	4	no	—
10	C₁₉×C₃²⋊C₉	C₃²	1	6	yes	—
11	#10624665	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₃³×C₁₉	—	3	6	no	—
13	C₃³×C₁₉	—	9	2	no	—
14	C₃³×C₁₉	—	9	2	no	—
15	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
16	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃³×C₁₉	C₃³	1	1	yes	characteristic
18	C₁₉×3₋¹⁺²	—	9	6	no	—
19	C₁₉×H(3,𝔽₃)	—	9	6	no	—
20	C₁₉×H(3,𝔽₃)	—	9	6	no	—
21	C₁₉×H(3,𝔽₃)	—	9	12	no	—
22	C₃×C₉×C₁₉	—	3	6	no	—
23	#6891	C₃×C₁₉	1	2	yes	complemented
24	C₃²⋊H(3,𝔽₃)	C₃×C₁₉	1	2	yes	complemented
25	#6856	C₃×C₁₉	1	9	yes	complemented
26	C₃²×C₁₉	—	3	2	no	—
27	C₃²×C₁₉	—	3	2	no	—
28	C₃²×C₁₉	—	3	4	no	—
29	C₃²×C₁₉	—	3	4	no	—
30	C₃²×C₁₉	—	3	2	no	—
31	C₃²×C₁₉	—	3	2	no	—
32	C₃²×C₁₉	—	9	9	no	—
33	C₃²×C₁₉	—	9	6	no	—
34	C₃²×C₁₉	—	9	6	no	—
35	C₃²×C₁₉	—	9	12	no	—
36	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₉×C₁₉	—	9	6	no	—
38	C₃⁴	C₃²×C₁₉	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃×H(3,𝔽₃)	—	3	4	no	—
41	C₃×H(3,𝔽₃)	—	3	2	no	—
42	C₃×H(3,𝔽₃)	C₃²×C₁₉	1	6	yes	—
43	C₃²⋊C₉	—	3	4	no	—
44	C₃²⋊C₉	C₃²×C₁₉	1	6	yes	—
45	C₃×C₁₉	—	9	1	no	—
46	C₃×C₁₉	—	9	1	no	—
47	C₃×C₁₉	—	9	2	no	—
48	C₃×C₁₉	—	27	6	no	—
49	C₃×C₁₉	#6909	1	2	yes	—
50	C₃×C₁₉	#6911	1	2	yes	—
51	C₃³	—	3	6	no	—
52	C₃³	—	9	2	no	—
53	C₃³	—	9	2	no	—
54	C₃³	C₁₉×H(3,𝔽₃)	1	2	yes	—
55	C₃³	C₁₉×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃³	C₃³×C₁₉	1	1	yes	characteristic
57	3₋¹⁺²	—	9	6	no	—
58	H(3,𝔽₃)	—	9	6	no	—
59	H(3,𝔽₃)	—	9	6	no	—
60	H(3,𝔽₃)	—	9	12	no	—
61	C₃×C₉	—	3	6	no	—
62	C₁₉	#10624665	1	1	yes	characteristic, direct product factor, Sylow
63	C₃²	—	3	2	no	—
64	C₃²	—	3	2	no	—
65	C₃²	—	3	4	no	—
66	C₃²	—	3	4	no	—
67	C₃²	—	3	2	no	—
68	C₃²	—	3	2	no	—
69	C₃²	—	9	9	no	—
70	C₃²	—	9	6	no	—
71	C₃²	—	9	6	no	—
72	C₃²	—	9	12	no	—
73	C₃²	C₃×C₁₉×H(3,𝔽₃)	1	1	yes	characteristic
74	C₉	—	9	6	no	—
75	C₃	—	9	1	no	—
76	C₃	—	9	1	no	—
77	C₃	—	9	2	no	—
78	C₃	—	27	6	no	—
79	C₃	C₁₉×#6909	1	2	yes	—
80	C₃	C₁₉×#6911	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₉
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₁₉ ◁ C₃⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#6891	C₃	semidirect
2. C₁₉×C₃²⋊H(3,𝔽₃)	C₃	semidirect
3. C₁₉×#6856	C₃	semidirect
11. #10624665	C₁₉	direct (N × H)
23. #6891	C₃×C₁₉	semidirect
24. C₃²⋊H(3,𝔽₃)	C₃×C₁₉	semidirect
25. #6856	C₃×C₁₉	semidirect
62. C₁₉	#10624665	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#6891	3
C₁₉×C₃²⋊H(3,𝔽₃)	3
C₁₉×#6856	3
#10624665	19

Subgroups — C19×#10624665 #28352861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10624665 #28352861