Subgroups — #28355805

Order 13851.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#13095322	C₃	1	2	yes	maximal, complemented
2	C₃×#13095320	C₃	1	1	yes	maximal, characteristic
3	#17987390	C₃	1	9	yes	maximal, complemented
4	C₁₉×#6887	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#10587025	—	3	2	no	—
6	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
7	#13095322	—	3	2	no	—
8	#13095322	—	3	2	no	—
9	#13095322	C₃²	1	6	yes	complemented
10	#13095320	—	3	1	no	—
11	#13095320	—	3	1	no	—
12	#13095320	C₃²	1	3	yes	—
13	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
14	C₁₉×C₉⋊C₉	—	3	1	no	—
15	C₁₉×C₉⋊C₉	C₃²	1	3	yes	complemented
16	C₁₉×C₃²⋊C₉	—	3	1	no	—
17	C₉²×C₁₉	—	3	1	no	—
18	#10624848	—	19	1	no	maximal, Sylow
19	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₁₉⋊C₃	—	9	1	no	—
21	C₃×C₉×C₁₉	—	3	2	no	—
22	C₃×C₉×C₁₉	—	3	1	no	—
23	C₃×C₉×C₁₉	—	3	3	no	—
24	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
27	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
28	C₃²⋊(C₁₉⋊C₃)	—	9	2	no	—
29	#10587025	—	3	6	no	—
30	#10587025	—	9	2	no	—
31	#10587025	—	9	2	no	—
32	C₃×#4294	—	9	2	no	—
33	C₃×#733	—	19	2	no	—
34	C₃×#732	—	19	1	no	—
35	#6887	C₁₉⋊C₃	1	1	yes	characteristic, complemented
36	#6883	—	19	9	no	—
37	C₃²×C₁₉	—	9	1	no	—
38	C₃²×C₁₉	#732	1	2	yes	—
39	C₃²×C₁₉	#732	1	1	yes	characteristic
40	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃×C₁₉⋊C₃	—	9	3	no	—
42	C₃×C₁₉⋊C₃	—	27	3	no	—
43	C₉×C₁₉	—	3	1	no	—
44	C₉×C₁₉	—	3	1	no	—
45	C₉×C₁₉	—	3	1	no	—
46	C₉×C₁₉	—	9	3	no	—
47	#4294	—	9	6	no	—
48	C₃×3₋¹⁺²	—	57	2	no	—
49	C₃×H(3,𝔽₃)	—	57	1	no	—
50	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
51	#733	—	19	6	no	—
52	#733	—	57	2	no	—
53	#733	—	57	2	no	—
54	#732	—	19	3	no	—
55	#732	—	57	1	no	—
56	#732	—	57	1	no	—
57	C₉⋊C₉	—	3	1	no	—
58	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
59	C₃²⋊C₉	—	3	1	no	—
60	C₉²	—	3	1	no	—
61	C₃×C₁₉	—	3	1	no	—
62	C₃×C₁₉	—	3	1	no	—
63	C₃×C₁₉	—	3	2	no	—
64	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
65	C₁₉⋊C₃	—	27	3	no	—
66	C₃³	—	171	1	no	—
67	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
68	3₋¹⁺²	—	57	6	no	—
69	3₋¹⁺²	—	171	2	no	—
70	3₋¹⁺²	—	171	2	no	—
71	H(3,𝔽₃)	—	57	3	no	—
72	H(3,𝔽₃)	—	171	2	no	—
73	C₃×C₉	—	3	2	no	—
74	C₃×C₉	—	3	1	no	—
75	C₃×C₉	—	3	3	no	—
76	C₃×C₉	—	171	2	no	—
77	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
78	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
79	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
80	C₁₉	#10624848	1	1	yes	characteristic, complemented, Sylow
81	C₃²	—	513	3	no	—
82	C₃²	—	9	1	no	—
83	C₃²	—	171	3	no	—
84	C₃²	#13095320	1	2	yes	—
85	C₃²	#13095320	1	1	yes	characteristic
86	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
87	C₉	—	3	1	no	—
88	C₉	—	3	1	no	—
89	C₉	—	3	1	no	—
90	C₉	—	9	3	no	—
91	C₉	—	171	6	no	—
92	C₃	—	513	3	no	—
93	C₃	—	3	1	no	—
94	C₃	—	3	1	no	—
95	C₃	—	3	2	no	—
96	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
prime radical	C₃×#13095320	C₃
hypercenter	#6887	C₁₉⋊C₃
hypocenter	C₁₉	#10624848
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. C₃×#13095322	C₃	semidirect
3. #17987390	C₃	semidirect
4. C₁₉×#6887	C₃	semidirect
9. #13095322	C₃²	semidirect
15. C₁₉×C₉⋊C₉	C₃²	semidirect
35. #6887	C₁₉⋊C₃	semidirect
58. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
80. C₁₉	#10624848	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095322	3
C₃×#13095320	3
#17987390	3
C₁₉×#6887	3
#10624848	19