Subgroups — (C₃×C₁₉⋊C₃)⋊H(3,𝔽₃) #17987416

Order 4617.

378 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃²⋊(C₁₉⋊C₃)	C₃	1	9	yes	maximal, complemented
3	C₃×C₃²⋊(C₁₉⋊C₃)	C₃	1	2	yes	maximal, complemented
4	C₃×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₁₉	C₃²	1	1	yes	characteristic
6	C₃³×C₁₉	C₃²	1	3	yes	complemented
7	C₃²×C₁₉⋊C₃	—	3	3	no	—
8	C₃²×C₁₉⋊C₃	—	3	9	no	—
9	C₃²×C₁₉⋊C₃	C₃²	1	3	yes	—
10	C₃²×C₁₉⋊C₃	C₃²	1	6	yes	complemented
11	C₁₉×H(3,𝔽₃)	—	3	3	no	—
12	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
13	C₃²⋊(C₁₉⋊C₃)	—	3	27	no	—
14	C₃²⋊H(3,𝔽₃)	—	19	1	no	maximal, Sylow
15	C₃²×C₁₉	—	3	3	no	—
16	C₃²×C₁₉	—	3	9	no	—
17	C₃²×C₁₉	—	3	3	no	—
18	C₃²×C₁₉	H(3,𝔽₃)	1	3	yes	complemented
19	C₃²×C₁₉	C₃³	1	1	yes	characteristic
20	C₃×C₁₉⋊C₃	—	3	27	no	—
21	C₃×C₁₉⋊C₃	—	3	6	no	—
22	C₃×C₁₉⋊C₃	—	3	3	no	—
23	C₃×C₁₉⋊C₃	—	3	18	no	—
24	C₃×C₁₉⋊C₃	—	3	6	no	—
25	C₃×C₁₉⋊C₃	H(3,𝔽₃)	1	9	yes	complemented
26	C₃⁴	—	19	1	no	—
27	C₃×H(3,𝔽₃)	—	19	2	no	—
28	C₃×H(3,𝔽₃)	—	19	9	no	—
29	C₃×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic, complemented
30	C₃×C₁₉	—	3	3	no	—
31	C₃×C₁₉	—	9	3	no	—
32	C₃×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
33	C₃×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₁₉⋊C₃	—	3	9	no	—
35	C₁₉⋊C₃	—	9	6	no	—
36	C₃³	—	19	3	no	—
37	C₃³	—	19	6	no	—
38	C₃³	—	57	9	no	—
39	C₃³	—	57	3	no	—
40	C₃³	C₃×C₁₉⋊C₃	1	1	yes	characteristic
41	C₃³	C₃×C₁₉⋊C₃	1	3	yes	complemented
42	H(3,𝔽₃)	—	3	3	no	—
43	H(3,𝔽₃)	—	57	27	no	—
44	H(3,𝔽₃)	—	57	6	no	—
45	C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
46	C₃²	—	3	3	no	—
47	C₃²	—	3	9	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	19	9	no	—
50	C₃²	—	57	27	no	—
51	C₃²	—	57	6	no	—
52	C₃²	—	57	3	no	—
53	C₃²	—	57	18	no	—
54	C₃²	—	57	6	no	—
55	C₃²	C₃²⋊(C₁₉⋊C₃)	1	3	yes	complemented
56	C₃²	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
57	C₃	—	3	3	no	—
58	C₃	—	9	3	no	—
59	C₃	—	57	9	no	—
60	C₃	—	171	6	no	—
61	C₃	H(3,𝔽₃)×C₁₉⋊C₃	1	3	yes	—
62	C₃	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃³×C₁₉⋊C₃	C₃	semidirect
2. C₃×C₃²⋊(C₁₉⋊C₃)	C₃	semidirect
3. C₃×C₃²⋊(C₁₉⋊C₃)	C₃	semidirect
4. C₃×C₁₉×H(3,𝔽₃)	C₃	semidirect
6. C₃³×C₁₉	C₃²	semidirect
10. C₃²×C₁₉⋊C₃	C₃²	semidirect
18. C₃²×C₁₉	H(3,𝔽₃)	semidirect
25. C₃×C₁₉⋊C₃	H(3,𝔽₃)	semidirect
29. C₃×H(3,𝔽₃)	C₁₉⋊C₃	semidirect
41. C₃³	C₃×C₁₉⋊C₃	semidirect
45. C₁₉	C₃²⋊H(3,𝔽₃)	semidirect
55. C₃²	C₃²⋊(C₁₉⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₁₉⋊C₃	3
C₃×C₃²⋊(C₁₉⋊C₃)	3
C₃×C₁₉×H(3,𝔽₃)	3
C₃²⋊H(3,𝔽₃)	19

Subgroups — (C3×C19⋊C3)⋊H(3,𝔽3) #17987416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₃×C₁₉⋊C₃)⋊H(3,𝔽₃) #17987416