Subgroups — #28355725

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

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

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. C₃×#13095301	C₃	semidirect
6. C₁₉⋊C₃×C₈₁	C₃	semidirect
7. C₁₉×C₈₁⋊C₃	C₃	semidirect
34. C₈₁⋊C₃	C₁₉⋊C₃	semidirect
67. 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
C₃×#13095301	3
C₁₉⋊C₃×C₈₁	3
C₁₉×C₈₁⋊C₃	3
#10624833	19