Subgroups — #28355754

Order 13851.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉⋊C₃×#733	C₃	1	3	yes	maximal
2	#17987403	C₃	1	1	yes	maximal, characteristic
3	#17987377	C₃	1	2	yes	maximal, complemented
4	#17987376	C₃	1	3	yes	maximal, complemented
5	#17987375	C₃	1	3	yes	maximal, complemented
6	C₁₉×#6880	C₃	1	1	yes	maximal, characteristic, complemented
7	3₋¹⁺²×C₁₉⋊C₃	—	3	3	no	—
8	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	#13095322	C₃²	1	6	yes	—
10	#13095321	—	3	6	no	—
11	#13095305	—	3	2	no	—
12	#13095303	—	3	1	no	—
13	C₉×#4294	C₃²	1	1	yes	characteristic
14	C₉×#4294	C₃²	1	1	yes	characteristic
15	C₁₉×#733	C₃²	1	3	yes	—
16	C₉²×C₁₉	C₃²	1	1	yes	characteristic
17	#10624840	—	19	1	no	maximal, Sylow
18	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
19	C₁₉×3₋¹⁺²	—	3	3	no	—
20	C₃×C₉×C₁₉	—	3	1	no	—
21	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
22	#10587025	—	3	6	no	—
23	C₃×#4294	—	3	1	no	—
24	C₃×#4294	—	3	1	no	—
25	C₃×#4294	H(3,𝔽₃)	1	2	yes	—
26	#10587023	—	9	3	no	—
27	#10587022	—	9	3	no	—
28	C₉×C₁₉⋊C₃	—	3	3	no	—
29	C₉×C₁₉⋊C₃	—	9	3	no	—
30	C₃×#733	—	19	3	no	—
31	#6887	—	19	1	no	—
32	#6880	—	19	3	no	—
33	#6880	—	19	3	no	—
34	#6880	—	19	2	no	—
35	#6880	C₁₉⋊C₃	1	1	yes	characteristic, complemented
36	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₁₉⋊C₃	—	9	1	no	—
38	C₃×C₁₉⋊C₃	#732	1	3	yes	—
39	C₉×C₁₉	—	3	1	no	—
40	C₉×C₁₉	—	9	1	no	—
41	C₉×C₁₉	—	9	3	no	—
42	#4294	—	3	2	no	—
43	#4294	—	9	1	no	—
44	#4294	—	9	1	no	—
45	#4294	—	9	6	no	—
46	C₃×3₋¹⁺²	—	57	3	no	—
47	C₃²×C₉	—	19	1	no	—
48	#733	—	19	6	no	—
49	#733	—	57	6	no	—
50	#733	C₃×C₁₉⋊C₃	1	3	yes	—
51	C₉⋊C₉	—	57	2	no	—
52	C₃²⋊C₉	—	57	1	no	—
53	C₉²	—	19	1	no	—
54	C₉²	—	19	1	no	—
55	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
56	C₃×C₁₉	—	3	1	no	—
57	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
58	C₁₉⋊C₃	—	3	3	no	—
59	C₃³	—	19	1	no	—
60	3₋¹⁺²	—	3	3	no	—
61	3₋¹⁺²	—	57	6	no	—
62	3₋¹⁺²	—	171	3	no	—
63	3₋¹⁺²	—	171	3	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	19	2	no	—
66	C₃×C₉	—	57	3	no	—
67	C₃×C₉	—	57	1	no	—
68	C₃×C₉	—	57	1	no	—
69	C₃×C₉	—	171	3	no	—
70	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
71	C₁₉	#10624840	1	1	yes	characteristic, complemented, Sylow
72	C₃²	—	19	3	no	—
73	C₃²	—	171	1	no	—
74	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
75	C₉	—	3	1	no	—
76	C₉	—	9	1	no	—
77	C₉	—	9	3	no	—
78	C₉	—	57	2	no	—
79	C₉	—	171	1	no	—
80	C₉	—	171	1	no	—
81	C₉	—	171	6	no	—
82	C₃	—	3	1	no	—
83	C₃	—	57	3	no	—
84	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×#6880	C₃
socle	C₃×C₁₉	C₃×#732
prime radical	C₃²×C₁₉⋊C₃	H(3,𝔽₃)
hypercenter	#6880	C₁₉⋊C₃
hypocenter	C₁₉	#10624840
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6880

Complements

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

N	H ≅ G/N	type
3. #17987377	C₃	semidirect
4. #17987376	C₃	semidirect
5. #17987375	C₃	semidirect
6. C₁₉×#6880	C₃	semidirect
35. #6880	C₁₉⋊C₃	semidirect
71. C₁₉	#10624840	semidirect

Maximal subgroups

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

subgroup	index
C₁₉⋊C₃×#733	3
#17987403	3
#17987377	3
#17987376	3
#17987375	3
C₁₉×#6880	3
#10624840	19