Subgroups — #17987419

Order 4617.

252 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
2	C₃×#10587023	C₃	1	3	yes	maximal, complemented
3	C₃²⋊#4294	C₃	1	2	yes	maximal, complemented
4	#13095303	C₃	1	6	yes	maximal, complemented
5	C₁₉×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₁₉	C₃²	1	1	yes	characteristic
7	C₃²×C₁₉⋊C₃	—	3	3	no	—
8	C₃²×C₁₉⋊C₃	—	3	3	no	—
9	C₃²×C₁₉⋊C₃	—	3	6	no	—
10	C₃²×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
11	C₃²×C₁₉⋊C₃	C₃²	1	2	yes	—
12	C₃×C₉×C₁₉	C₃²	1	3	yes	complemented
13	C₃×#4294	C₃²	1	6	yes	complemented
14	#10587023	—	3	9	no	—
15	#6891	—	19	1	no	maximal, Sylow
16	C₃²×C₁₉	—	3	1	no	—
17	C₃²×C₁₉	—	3	2	no	—
18	C₃²×C₁₉	3₋¹⁺²	1	3	yes	complemented
19	C₃²×C₁₉	C₃³	1	1	yes	characteristic
20	C₃×C₁₉⋊C₃	—	3	27	no	—
21	C₃×C₁₉⋊C₃	—	3	2	no	—
22	C₃×C₁₉⋊C₃	—	3	2	no	—
23	C₃×C₁₉⋊C₃	—	3	2	no	—
24	C₃×C₁₉⋊C₃	—	3	1	no	—
25	C₃×C₁₉⋊C₃	—	3	2	no	—
26	C₃×C₁₉⋊C₃	H(3,𝔽₃)	1	3	yes	—
27	C₃×C₁₉⋊C₃	3₋¹⁺²	1	6	yes	complemented
28	C₉×C₁₉	—	3	3	no	—
29	#4294	—	3	6	no	—
30	C₃⁴	—	19	1	no	—
31	C₃×3₋¹⁺²	—	19	3	no	—
32	C₃²⋊C₉	—	19	2	no	—
33	C₃²⋊C₉	—	19	6	no	—
34	C₃²⋊C₉	C₁₉⋊C₃	1	1	yes	characteristic, complemented
35	C₃×C₁₉	—	3	3	no	—
36	C₃×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₁₉	C₃×3₋¹⁺²	1	2	yes	—
38	C₃×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
39	C₁₉⋊C₃	—	3	3	no	—
40	C₁₉⋊C₃	—	3	6	no	—
41	C₃³	—	19	2	no	—
42	C₃³	—	19	1	no	—
43	C₃³	—	57	3	no	—
44	C₃³	—	57	6	no	—
45	C₃³	—	57	3	no	—
46	C₃³	C₃×C₁₉⋊C₃	1	1	yes	characteristic
47	3₋¹⁺²	—	57	9	no	—
48	C₃×C₉	—	19	6	no	—
49	C₃×C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
50	C₁₉	#6891	1	1	yes	characteristic, complemented, Sylow
51	C₃²	—	3	1	no	—
52	C₃²	—	3	2	no	—
53	C₃²	—	19	3	no	—
54	C₃²	—	19	6	no	—
55	C₃²	—	57	27	no	—
56	C₃²	—	57	2	no	—
57	C₃²	—	57	2	no	—
58	C₃²	—	57	2	no	—
59	C₃²	—	57	2	no	—
60	C₃²	—	57	1	no	—
61	C₃²	#10587023	1	3	yes	complemented
62	C₃²	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
63	C₉	—	3	3	no	—
64	C₉	—	57	6	no	—
65	C₃	—	3	3	no	—
66	C₃	—	57	3	no	—
67	C₃	—	57	6	no	—
68	C₃	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
69	C₃	3₋¹⁺²×C₁₉⋊C₃	1	2	yes	—
70	C₃	C₃×#10587023	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃³
Fitting	C₁₉×C₃²⋊C₉	C₃
prime radical	C₃³×C₁₉⋊C₃	C₃
hypercenter	C₃²⋊C₉	C₁₉⋊C₃
hypocenter	C₁₉	#6891
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃×#10587023	C₃	semidirect
3. C₃²⋊#4294	C₃	semidirect
4. #13095303	C₃	semidirect
5. C₁₉×C₃²⋊C₉	C₃	semidirect
12. C₃×C₉×C₁₉	C₃²	semidirect
13. C₃×#4294	C₃²	semidirect
18. C₃²×C₁₉	3₋¹⁺²	semidirect
27. C₃×C₁₉⋊C₃	3₋¹⁺²	semidirect
34. C₃²⋊C₉	C₁₉⋊C₃	semidirect
49. C₃×C₉	C₃×C₁₉⋊C₃	semidirect
50. C₁₉	#6891	semidirect
61. C₃²	#10587023	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×#10587023	3
C₃²⋊#4294	3
#13095303	3
C₁₉×C₃²⋊C₉	3
#6891	19