Subgroups — #17987432

Order 4617.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#10587025	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²×#4294	C₃	1	1	yes	maximal, characteristic, complemented
4	#13095303	C₃	1	9	yes	maximal, complemented
5	C₃×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₁₉	C₃²	1	1	yes	characteristic
7	C₃²×C₁₉⋊C₃	C₃²	1	3	yes	—
8	C₁₉×3₋¹⁺²	—	3	3	no	—
9	C₃×C₉×C₁₉	C₃²	1	3	yes	—
10	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
11	#10587025	—	3	3	no	—
12	C₃×#4294	—	3	3	no	—
13	C₃×#4294	C₃²	1	3	yes	—
14	C₃×#4294	C₃²	1	3	yes	—
15	#6893	—	19	1	no	maximal, Sylow
16	C₃²×C₁₉	—	3	3	no	—
17	C₃²×C₁₉	H(3,𝔽₃)	1	3	yes	—
18	C₃²×C₁₉	C₃³	1	1	yes	characteristic
19	C₃×C₁₉⋊C₃	—	3	3	no	—
20	C₃×C₁₉⋊C₃	—	3	9	no	—
21	C₉×C₁₉	—	3	3	no	—
22	#4294	—	3	3	no	—
23	#4294	—	3	3	no	—
24	C₃×3₋¹⁺²	—	19	1	no	—
25	C₃×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
26	C₃×H(3,𝔽₃)	—	19	1	no	—
27	C₃²×C₉	—	19	1	no	—
28	C₃²⋊C₉	—	19	9	no	—
29	C₃×C₁₉	—	3	3	no	—
30	C₃×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×C₁₉	#737	1	3	yes	—
32	C₁₉⋊C₃	—	9	3	no	—
33	C₃³	—	19	3	no	—
34	C₃³	C₃×C₁₉⋊C₃	1	1	yes	characteristic
35	3₋¹⁺²	—	3	3	no	—
36	3₋¹⁺²	—	57	3	no	—
37	H(3,𝔽₃)	—	57	3	no	—
38	C₃×C₉	—	19	3	no	—
39	C₃×C₉	—	19	3	no	—
40	C₃×C₉	—	57	3	no	—
41	C₃×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
42	C₁₉	#6893	1	1	yes	characteristic, complemented, Sylow
43	C₃²	—	3	3	no	—
44	C₃²	—	57	3	no	—
45	C₃²	—	57	9	no	—
46	C₃²	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
47	C₃²	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
48	C₉	—	3	3	no	—
49	C₉	—	57	3	no	—
50	C₉	—	57	3	no	—
51	C₃	—	3	3	no	—
52	C₃	—	171	3	no	—
53	C₃	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
54	C₃	#13095333	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃³
Fitting	C₃×C₁₉×3₋¹⁺²	C₃
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₃
hypercenter	C₃×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#6893
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×#10587025	C₃	semidirect
3. C₃²×#4294	C₃	semidirect
4. #13095303	C₃	semidirect
5. C₃×C₁₉×3₋¹⁺²	C₃	semidirect
25. C₃×3₋¹⁺²	C₁₉⋊C₃	semidirect
42. C₁₉	#6893	semidirect

Maximal subgroups

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

subgroup	index
C₃×#10587025	3
C₃×C₃²⋊(C₁₉⋊C₃)	3
C₃²×#4294	3
#13095303	3
C₃×C₁₉×3₋¹⁺²	3
#6893	19