Subgroups — #28355723

Order 13851.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987479	C₃	1	1	yes	maximal, characteristic
2	#17987367	C₃	1	2	yes	maximal, complemented
3	#17987366	C₃	1	3	yes	maximal, complemented
4	#17987365	C₃	1	3	yes	maximal, complemented
5	C₁₉⋊C₃×C₈₁	C₃	1	3	yes	maximal, complemented
6	C₃×C₁₉×C₈₁	C₃	1	1	yes	maximal, characteristic, complemented
7	#13095332	C₉	1	1	yes	characteristic
8	#13095312	C₉	1	2	yes	—
9	#13095311	C₉	1	3	yes	—
10	#13095310	C₉	1	3	yes	—
11	C₂₇×C₁₉⋊C₃	C₃²	1	3	yes	—
12	#13095301	C₃²	1	6	yes	—
13	C₃×C₁₉×C₂₇	C₃²	1	1	yes	characteristic
14	C₁₉×C₈₁	C₃²	1	1	yes	characteristic
15	C₁₉×C₈₁	C₃²	1	1	yes	characteristic
16	C₁₉×C₈₁	C₃²	1	1	yes	characteristic
17	#10624833	—	19	1	no	maximal, Sylow
18	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
19	C₃²⋊(C₁₉⋊C₃)	C₂₇	1	1	yes	characteristic
20	#10587025	C₂₇	1	2	yes	—
21	#10587023	C₂₇	1	3	yes	—
22	#10587022	C₂₇	1	3	yes	—
23	C₉×C₁₉⋊C₃	C₃×C₉	1	3	yes	—
24	C₁₉×C₂₇	C₃×C₉	1	1	yes	characteristic
25	C₁₉×C₂₇	C₃×C₉	1	1	yes	characteristic
26	C₁₉×C₂₇	C₃³	1	1	yes	characteristic
27	#10587018	C₃×C₉	1	6	yes	—
28	#6903	—	19	1	no	—
29	C₈₁⋊C₃	—	19	2	no	—
30	C₈₁⋊C₃	—	19	3	no	—
31	C₈₁⋊C₃	—	19	3	no	—
32	C₃×C₈₁	—	19	3	no	—
33	C₃×C₈₁	C₁₉⋊C₃	1	1	yes	characteristic, complemented
34	C₃²×C₁₉	C₃×C₂₇	1	1	yes	characteristic
35	C₃×C₁₉⋊C₃	C₃×C₂₇	1	3	yes	—
36	C₉×C₁₉	C₃×C₂₇	1	1	yes	characteristic
37	C₉×C₁₉	C₃×C₂₇	1	1	yes	characteristic
38	C₉×C₁₉	C₃²×C₉	1	1	yes	characteristic
39	#4294	C₃×C₂₇	1	6	yes	—
40	#737	—	19	1	no	—
41	C₂₇⋊C₃	—	19	2	no	—
42	C₂₇⋊C₃	—	19	3	no	—
43	C₂₇⋊C₃	—	19	3	no	—
44	C₃×C₂₇	—	19	3	no	—
45	C₃×C₂₇	C₃×C₁₉⋊C₃	1	1	yes	characteristic
46	C₈₁	—	19	6	no	—
47	C₈₁	C₃×C₁₉⋊C₃	1	1	yes	characteristic
48	C₈₁	C₃×C₁₉⋊C₃	1	1	yes	characteristic
49	C₈₁	C₃×C₁₉⋊C₃	1	1	yes	characteristic
50	C₃×C₁₉	—	3	1	no	—
51	C₃×C₁₉	C₃²×C₂₇	1	1	yes	characteristic
52	C₁₉⋊C₃	—	3	3	no	—
53	3₋¹⁺²	—	19	2	no	—
54	3₋¹⁺²	—	19	3	no	—
55	3₋¹⁺²	—	19	3	no	—
56	H(3,𝔽₃)	—	19	1	no	—
57	C₃×C₉	—	19	3	no	—
58	C₃×C₉	C₉×C₁₉⋊C₃	1	1	yes	characteristic
59	C₂₇	—	19	6	no	—
60	C₂₇	C₉×C₁₉⋊C₃	1	1	yes	characteristic
61	C₂₇	C₉×C₁₉⋊C₃	1	1	yes	characteristic
62	C₂₇	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
63	C₁₉	#10624833	1	1	yes	characteristic, complemented, Sylow
64	C₃²	—	19	3	no	—
65	C₃²	C₂₇×C₁₉⋊C₃	1	1	yes	characteristic
66	C₉	—	19	6	no	—
67	C₉	C₂₇×C₁₉⋊C₃	1	1	yes	characteristic
68	C₉	C₂₇×C₁₉⋊C₃	1	1	yes	characteristic
69	C₉	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
70	C₃	—	3	1	no	—
71	C₃	—	57	3	no	—
72	C₃	C₃×C₂₇×C₁₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈₁	C₃×C₁₉⋊C₃
commutator, socle	C₃×C₁₉	C₃²×C₂₇
Frattini	C₂₇	C₃²×C₁₉⋊C₃
Fitting	C₃×C₁₉×C₈₁	C₃
prime radical	C₃²⋊(C₁₉⋊C₃)	C₂₇
cosocle	C₁₉×C₂₇	C₃³
hypercenter	C₃×C₈₁	C₁₉⋊C₃
hypocenter	C₁₉	#10624833
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₈₁ ◁ C₃×C₈₁

Complements

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

N	H ≅ G/N	type
2. #17987367	C₃	semidirect
3. #17987366	C₃	semidirect
4. #17987365	C₃	semidirect
5. C₁₉⋊C₃×C₈₁	C₃	semidirect
6. C₃×C₁₉×C₈₁	C₃	semidirect
33. C₃×C₈₁	C₁₉⋊C₃	semidirect
63. C₁₉	#10624833	semidirect

Maximal subgroups

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

subgroup	index
#17987479	3
#17987367	3
#17987366	3
#17987365	3
C₁₉⋊C₃×C₈₁	3
C₃×C₁₉×C₈₁	3
#10624833	19