Subgroups — #28355811

Order 13851.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987500	C₃	1	3	yes	maximal, complemented
2	#17987390	C₃	1	3	yes	maximal, complemented
3	#17987385	C₃	1	3	yes	maximal, complemented
4	#17987380	C₃	1	3	yes	maximal, complemented
5	C₁₉×#6887	C₃	1	1	yes	maximal, characteristic, complemented
6	#13095332	—	3	3	no	—
7	#13095322	C₃²	1	3	yes	—
8	#13095322	C₃²	1	3	yes	—
9	#13095320	C₃²	1	3	yes	—
10	#13095317	—	3	3	no	—
11	#13095314	—	3	3	no	—
12	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
13	C₁₉×C₉⋊C₉	—	3	1	no	—
14	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
15	C₁₉×C₃²⋊C₉	—	3	1	no	—
16	C₉²×C₁₉	—	3	1	no	—
17	#10624849	—	19	1	no	maximal, Sylow
18	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×C₉×C₁₉	—	3	3	no	—
20	C₃×C₉×C₁₉	—	3	3	no	—
21	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
24	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
25	#10587025	—	3	3	no	—
26	#10587025	—	3	3	no	—
27	#10587023	—	9	3	no	—
28	#10587022	—	9	3	no	—
29	C₉×C₁₉⋊C₃	—	9	3	no	—
30	#6908	—	19	3	no	—
31	#6887	C₁₉⋊C₃	1	1	yes	characteristic, complemented
32	#6883	—	19	3	no	—
33	#6882	—	19	3	no	—
34	#6881	—	19	3	no	—
35	C₃²×C₁₉	—	3	1	no	—
36	C₃²×C₁₉	—	9	1	no	—
37	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₁₉⋊C₃	—	9	3	no	—
39	C₉×C₁₉	—	3	1	no	—
40	C₉×C₁₉	—	3	1	no	—
41	C₉×C₁₉	—	9	3	no	—
42	C₉×C₁₉	#732	1	3	yes	—
43	#4294	—	9	3	no	—
44	#4294	—	9	3	no	—
45	#737	—	57	3	no	—
46	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
47	#733	—	19	3	no	—
48	#733	—	19	3	no	—
49	#732	—	19	3	no	—
50	#731	—	57	3	no	—
51	AΣL(1,𝔽₂₇)	—	57	3	no	—
52	C₉⋊C₉	—	3	1	no	—
53	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
54	C₃²⋊C₉	—	3	1	no	—
55	C₉²	—	3	1	no	—
56	C₃×C₁₉	—	3	1	no	—
57	C₃×C₁₉	—	9	1	no	—
58	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
59	C₁₉⋊C₃	—	27	3	no	—
60	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
61	3₋¹⁺²	—	57	3	no	—
62	3₋¹⁺²	—	57	3	no	—
63	3₋¹⁺²	—	171	3	no	—
64	3₋¹⁺²	—	171	3	no	—
65	H(3,𝔽₃)	—	57	3	no	—
66	C₃×C₉	—	3	3	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	171	3	no	—
69	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
70	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
71	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
72	C₁₉	#10624849	1	1	yes	characteristic, complemented, Sylow
73	C₃²	—	3	1	no	—
74	C₃²	—	9	1	no	—
75	C₃²	—	171	3	no	—
76	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
77	C₉	—	3	1	no	—
78	C₉	—	3	1	no	—
79	C₉	—	9	3	no	—
80	C₉	—	171	3	no	—
81	C₉	—	171	3	no	—
82	C₉	#13095320	1	3	yes	—
83	C₃	—	513	3	no	—
84	C₃	—	3	1	no	—
85	C₃	—	9	1	no	—
86	C₃	C₃×#13095320	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×#13095320
commutator, cosocle	C₃×C₉×C₁₉	C₃³
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6887	C₃
socle	C₃×C₁₉	C₃×#732
hypercenter	#6887	C₁₉⋊C₃
hypocenter	C₁₉	#10624849
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. #17987500	C₃	semidirect
2. #17987390	C₃	semidirect
3. #17987385	C₃	semidirect
4. #17987380	C₃	semidirect
5. C₁₉×#6887	C₃	semidirect
31. #6887	C₁₉⋊C₃	semidirect
72. C₁₉	#10624849	semidirect

Maximal subgroups

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

subgroup	index
#17987500	3
#17987390	3
#17987385	3
#17987380	3
C₁₉×#6887	3
#10624849	19