Subgroups — #28355751

Order 13851.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉⋊C₃×#732	C₃	1	3	yes	maximal, complemented
2	#17987403	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987374	C₃	1	2	yes	maximal, complemented
4	#17987373	C₃	1	6	yes	maximal, complemented
5	C₁₉×#6879	C₃	1	1	yes	maximal, characteristic, complemented
6	H(3,𝔽₃)×C₁₉⋊C₃	—	3	3	no	—
7	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
8	#13095320	C₃²	1	6	yes	complemented
9	#13095319	—	3	6	no	—
10	#13095305	—	3	2	no	—
11	#13095303	—	3	1	no	—
12	C₉×#4294	C₃²	1	2	yes	complemented
13	C₁₉×#732	C₃²	1	3	yes	complemented
14	C₉²×C₁₉	C₃²	1	1	yes	characteristic, complemented
15	#10624839	—	19	1	no	maximal, Sylow
16	C₃²×C₁₉⋊C₃	—	9	3	no	—
17	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
18	C₁₉×H(3,𝔽₃)	—	3	3	no	—
19	C₃×C₉×C₁₉	—	3	1	no	—
20	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
21	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
22	C₃²⋊(C₁₉⋊C₃)	—	9	6	no	—
23	C₃×#4294	—	3	2	no	—
24	C₃×#4294	H(3,𝔽₃)	1	2	yes	—
25	C₉×C₁₉⋊C₃	—	3	3	no	—
26	C₃×#732	—	19	3	no	—
27	#6887	—	19	1	no	—
28	#6879	—	19	6	no	—
29	#6879	—	19	2	no	—
30	#6879	C₁₉⋊C₃	1	1	yes	characteristic, complemented
31	C₃²×C₁₉	—	9	3	no	—
32	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₁₉⋊C₃	—	9	1	no	—
34	C₃×C₁₉⋊C₃	—	9	6	no	—
35	C₃×C₁₉⋊C₃	—	27	9	no	—
36	C₃×C₁₉⋊C₃	#732	1	3	yes	—
37	C₉×C₁₉	—	3	1	no	—
38	C₉×C₁₉	—	9	1	no	—
39	#4294	—	3	2	no	—
40	#4294	—	9	2	no	—
41	C₃×H(3,𝔽₃)	—	57	3	no	—
42	C₃²×C₉	—	19	1	no	—
43	#732	—	19	6	no	—
44	#732	—	57	6	no	—
45	#732	C₃×C₁₉⋊C₃	1	3	yes	complemented
46	C₉⋊C₉	—	57	2	no	—
47	C₃²⋊C₉	—	57	1	no	—
48	C₉²	—	19	2	no	—
49	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
50	C₃×C₁₉	—	3	1	no	—
51	C₃×C₁₉	—	27	3	no	—
52	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
53	C₁₉⋊C₃	—	3	3	no	—
54	C₁₉⋊C₃	—	27	6	no	—
55	C₃³	—	19	1	no	—
56	C₃³	—	171	3	no	—
57	H(3,𝔽₃)	—	3	3	no	—
58	H(3,𝔽₃)	—	57	6	no	—
59	H(3,𝔽₃)	—	171	6	no	—
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	19	2	no	—
62	C₃×C₉	—	57	3	no	—
63	C₃×C₉	—	57	2	no	—
64	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
65	C₁₉	#10624839	1	1	yes	characteristic, complemented, Sylow
66	C₃²	—	513	9	no	—
67	C₃²	—	9	3	no	—
68	C₃²	—	19	3	no	—
69	C₃²	—	171	6	no	—
70	C₃²	—	171	1	no	—
71	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
72	C₉	—	3	1	no	—
73	C₉	—	9	1	no	—
74	C₉	—	57	2	no	—
75	C₉	—	171	2	no	—
76	C₃	—	513	6	no	—
77	C₃	—	3	1	no	—
78	C₃	—	27	3	no	—
79	C₃	—	57	3	no	—
80	C₃	C₁₉⋊C₃×#732	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₁₉⋊C₃×#732
commutator, cosocle	C₃×C₉×C₁₉	C₃³
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6879	C₃
socle	C₃×C₁₉	C₃×#732
hypercenter	#6879	C₁₉⋊C₃
hypocenter	C₁₉	#10624839
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6879

Complements

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

N	H ≅ G/N	type
1. C₁₉⋊C₃×#732	C₃	semidirect
2. #17987403	C₃	semidirect
3. #17987374	C₃	semidirect
4. #17987373	C₃	semidirect
5. C₁₉×#6879	C₃	semidirect
8. #13095320	C₃²	semidirect
12. C₉×#4294	C₃²	semidirect
13. C₁₉×#732	C₃²	semidirect
14. C₉²×C₁₉	C₃²	semidirect
30. #6879	C₁₉⋊C₃	semidirect
45. #732	C₃×C₁₉⋊C₃	semidirect
49. C₉²	C₃×C₁₉⋊C₃	semidirect
65. C₁₉	#10624839	semidirect

Maximal subgroups

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

subgroup	index
C₁₉⋊C₃×#732	3
#17987403	3
#17987374	3
#17987373	3
C₁₉×#6879	3
#10624839	19