Subgroups — #28355817

Order 13851.

202 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	C₃×C₁₉×C₉⋊C₉	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	—	3	1	no	—
13	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
14	C₁₉×C₉⋊C₉	—	3	1	no	—
15	C₁₉×C₉⋊C₉	—	3	1	no	—
16	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
17	#10624850	—	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	—	9	3	no	—
22	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
25	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
26	#10587025	—	3	3	no	—
27	#10587025	—	3	3	no	—
28	#10587023	—	9	3	no	—
29	#10587022	—	9	3	no	—
30	C₉×C₁₉⋊C₃	—	9	3	no	—
31	#6908	—	19	3	no	—
32	C₃×C₉⋊C₉	C₁₉⋊C₃	1	1	yes	characteristic, complemented
33	#6883	—	19	3	no	—
34	#6882	—	19	3	no	—
35	#6881	—	19	3	no	—
36	C₃²×C₁₉	—	3	3	no	—
37	C₃²×C₁₉	—	3	1	no	—
38	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃×C₁₉⋊C₃	—	9	3	no	—
40	C₉×C₁₉	—	3	1	no	—
41	C₉×C₁₉	—	3	1	no	—
42	C₉×C₁₉	—	9	3	no	—
43	C₉×C₁₉	#732	1	3	yes	—
44	#4294	—	9	3	no	—
45	#4294	—	9	3	no	—
46	#737	—	57	3	no	—
47	C₃²×C₉	—	3	1	no	—
48	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
49	#733	—	19	3	no	—
50	#733	—	19	3	no	—
51	#732	—	19	3	no	—
52	#731	—	57	3	no	—
53	AΣL(1,𝔽₂₇)	—	57	3	no	—
54	C₉⋊C₉	—	3	1	no	—
55	C₉⋊C₉	—	3	1	no	—
56	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
57	C₃×C₁₉	—	3	1	no	—
58	C₃×C₁₉	—	3	3	no	—
59	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
60	C₁₉⋊C₃	—	27	3	no	—
61	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
62	3₋¹⁺²	—	57	3	no	—
63	3₋¹⁺²	—	57	3	no	—
64	3₋¹⁺²	—	171	3	no	—
65	3₋¹⁺²	—	171	3	no	—
66	H(3,𝔽₃)	—	57	3	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	9	3	no	—
70	C₃×C₉	—	171	3	no	—
71	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
72	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
73	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
74	C₁₉	#10624850	1	1	yes	characteristic, complemented, Sylow
75	C₃²	—	3	3	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	171	3	no	—
78	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
79	C₉	—	3	1	no	—
80	C₉	—	3	1	no	—
81	C₉	—	9	3	no	—
82	C₉	—	171	3	no	—
83	C₉	—	171	3	no	—
84	C₉	#13095320	1	3	yes	—
85	C₃	—	513	3	no	—
86	C₃	—	3	1	no	—
87	C₃	—	3	3	no	—
88	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	C₃×C₁₉×C₉⋊C₉	C₃
socle	C₃×C₁₉	C₃×#732
hypercenter	C₃×C₉⋊C₉	C₁₉⋊C₃
hypocenter	C₁₉	#10624850
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

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. C₃×C₁₉×C₉⋊C₉	C₃	semidirect
32. C₃×C₉⋊C₉	C₁₉⋊C₃	semidirect
74. C₁₉	#10624850	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
C₃×C₁₉×C₉⋊C₉	3
#10624850	19