Subgroups — C₁₉×#10624638 #28352834

Order 13851.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×C₉⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
2	C₁₉×#6887	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#6861	C₃	1	3	yes	maximal, complemented
4	C₁₉×#6859	C₃	1	3	yes	maximal, complemented
5	C₁₉×#6856	C₃	1	3	yes	maximal, complemented
6	C₃×C₁₉×3₋¹⁺²	—	3	3	no	—
7	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	3	yes	—
8	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
9	C₁₉×C₉⋊C₉	—	3	2	no	—
10	C₁₉×C₃²⋊C₉	—	3	6	no	—
11	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
12	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
13	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
14	C₉²×C₁₉	—	3	1	no	—
15	#10624638	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₃³×C₁₉	—	3	3	no	—
17	C₃³×C₁₉	C₃³	1	1	yes	characteristic
18	C₁₉×3₋¹⁺²	—	9	9	no	—
19	C₁₉×H(3,𝔽₃)	—	9	3	no	—
20	C₃×C₉×C₁₉	—	3	3	no	—
21	C₃×C₉×C₁₉	—	3	3	no	—
22	C₃×C₉×C₁₉	—	3	3	no	—
23	C₃×C₉×C₁₉	—	3	3	no	—
24	C₃×C₉×C₁₉	H(3,𝔽₃)	1	2	yes	—
25	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
26	C₉⋊H(3,𝔽₃)	C₃×C₁₉	1	3	yes	complemented
27	#6887	C₃×C₁₉	1	1	yes	characteristic, complemented
28	#6861	C₃×C₁₉	1	3	yes	complemented
29	#6859	C₃×C₁₉	1	3	yes	complemented
30	#6856	C₃×C₁₉	1	3	yes	complemented
31	C₃²×C₁₉	—	3	3	no	—
32	C₃²×C₁₉	—	3	1	no	—
33	C₃²×C₁₉	—	9	3	no	—
34	C₃²×C₁₉	—	9	6	no	—
35	C₃²×C₁₉	—	9	3	no	—
36	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₉×C₁₉	—	3	2	no	—
38	C₉×C₁₉	—	9	3	no	—
39	C₉×C₁₉	—	9	3	no	—
40	C₉×C₁₉	—	9	3	no	—
41	C₉×C₁₉	#732	1	3	yes	—
42	C₃×3₋¹⁺²	—	3	3	no	—
43	C₃×H(3,𝔽₃)	C₃²×C₁₉	1	3	yes	—
44	C₃²×C₉	C₃²×C₁₉	1	1	yes	characteristic
45	C₉⋊C₉	—	3	2	no	—
46	C₃²⋊C₉	—	3	6	no	—
47	C₃²⋊C₉	C₃²×C₁₉	1	3	yes	—
48	C₃²⋊C₉	C₃²×C₁₉	1	3	yes	—
49	C₃²⋊C₉	C₃²×C₁₉	1	3	yes	—
50	C₉²	—	3	1	no	—
51	C₃×C₁₉	—	9	1	no	—
52	C₃×C₁₉	—	27	3	no	—
53	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
54	C₃×C₁₉	#6913	1	3	yes	—
55	C₃³	—	3	3	no	—
56	C₃³	C₃³×C₁₉	1	1	yes	characteristic
57	3₋¹⁺²	—	9	9	no	—
58	H(3,𝔽₃)	—	9	3	no	—
59	C₃×C₉	—	3	3	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	3	3	no	—
63	C₃×C₉	C₁₉×H(3,𝔽₃)	1	2	yes	—
64	C₃×C₉	C₁₉×H(3,𝔽₃)	1	1	yes	characteristic
65	C₁₉	#10624638	1	1	yes	characteristic, direct product factor, Sylow
66	C₃²	—	3	1	no	—
67	C₃²	—	3	3	no	—
68	C₃²	—	9	3	no	—
69	C₃²	—	9	3	no	—
70	C₃²	—	9	6	no	—
71	C₃²	C₃×C₁₉×H(3,𝔽₃)	1	1	yes	characteristic
72	C₉	—	3	2	no	—
73	C₉	—	9	3	no	—
74	C₉	—	9	3	no	—
75	C₉	—	9	3	no	—
76	C₉	C₁₉×#732	1	3	yes	—
77	C₃	—	9	1	no	—
78	C₃	—	27	3	no	—
79	C₃	C₃×C₁₉×#732	1	1	yes	characteristic
80	C₃	C₁₉×#6913	1	3	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 ◁ C₃²×C₉×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×C₉⋊H(3,𝔽₃)	C₃	semidirect
2. C₁₉×#6887	C₃	semidirect
3. C₁₉×#6861	C₃	semidirect
4. C₁₉×#6859	C₃	semidirect
5. C₁₉×#6856	C₃	semidirect
15. #10624638	C₁₉	direct (N × H)
26. C₉⋊H(3,𝔽₃)	C₃×C₁₉	semidirect
27. #6887	C₃×C₁₉	semidirect
28. #6861	C₃×C₁₉	semidirect
29. #6859	C₃×C₁₉	semidirect
30. #6856	C₃×C₁₉	semidirect
65. C₁₉	#10624638	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×C₉⋊H(3,𝔽₃)	3
C₁₉×#6887	3
C₁₉×#6861	3
C₁₉×#6859	3
C₁₉×#6856	3
#10624638	19

Subgroups — C19×#10624638 #28352834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10624638 #28352834