Subgroups — #28354607

Order 13851.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#13095311	C₃	1	3	yes	maximal
2	C₉²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
3	#17987331	C₃	1	2	yes	maximal, complemented
4	#17987329	C₃	1	3	yes	maximal, complemented
5	C₂₇×#4294	C₃	1	3	yes	maximal, complemented
6	C₁₉×#6864	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₉×C₁₉⋊C₃	C₉	1	3	yes	—
8	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	#13095311	C₉	1	27	yes	complemented
10	C₃×#10587018	C₃²	1	6	yes	—
11	C₉×#4294	C₉	1	6	yes	—
12	C₉×#4294	C₃²	1	1	yes	characteristic
13	C₉×#4294	C₃²	1	1	yes	characteristic
14	C₃×C₁₉×C₂₇	C₃²	1	3	yes	—
15	C₉²×C₁₉	C₃²	1	1	yes	characteristic
16	C₉×C₂₇⋊C₃	—	19	1	no	maximal, Sylow
17	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
19	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
20	C₃×#4294	C₃×C₉	1	3	yes	—
21	C₃×#4294	C₃×C₉	1	2	yes	—
22	C₃×#4294	C₃×C₉	1	3	yes	—
23	C₉×C₁₉⋊C₃	—	3	9	no	—
24	C₉×C₁₉⋊C₃	C₃×C₉	1	9	yes	—
25	C₁₉×C₂₇	C₃×C₉	1	9	yes	complemented
26	#10587018	C₃×C₉	1	18	yes	complemented
27	C₃×C₂₇⋊C₃	—	19	3	no	—
28	C₃×C₉²	—	19	1	no	—
29	#6864	—	19	3	no	—
30	#6864	—	19	2	no	—
31	#6864	C₁₉⋊C₃	1	1	yes	characteristic, complemented
32	C₉×C₂₇	—	19	3	no	—
33	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
34	C₃×C₁₉⋊C₃	—	3	3	no	—
35	C₃×C₁₉⋊C₃	C₉²	1	3	yes	—
36	C₉×C₁₉	—	3	3	no	—
37	C₉×C₁₉	C₃²×C₉	1	3	yes	—
38	#4294	—	3	3	no	—
39	#4294	C₉²	1	6	yes	—
40	#4294	C₂₇⋊C₃	1	9	yes	complemented
41	C₃²×C₉	—	19	1	no	—
42	C₃²×C₉	—	19	3	no	—
43	C₂₇⋊C₃	—	19	27	no	—
44	C₃×C₂₇	—	19	6	no	—
45	C₃×C₂₇	C₃×C₁₉⋊C₃	1	3	yes	—
46	C₉²	—	19	6	no	—
47	C₉²	—	19	1	no	—
48	C₉²	—	19	1	no	—
49	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
50	C₃×C₁₉	C₃×C₉²	1	1	yes	characteristic
51	C₃×C₁₉	C₃×C₂₇⋊C₃	1	3	yes	—
52	C₁₉⋊C₃	—	3	3	no	—
53	C₃³	—	19	1	no	—
54	C₃×C₉	—	19	2	no	—
55	C₃×C₉	—	19	3	no	—
56	C₃×C₉	—	19	9	no	—
57	C₃×C₉	—	19	3	no	—
58	C₃×C₉	—	57	9	no	—
59	C₃×C₉	C₉×C₁₉⋊C₃	1	3	yes	—
60	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
61	C₂₇	—	19	18	no	—
62	C₂₇	C₉×C₁₉⋊C₃	1	9	yes	complemented
63	C₁₉	C₉×C₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
64	C₃²	—	19	3	no	—
65	C₃²	—	57	3	no	—
66	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
67	C₉	—	3	3	no	—
68	C₉	—	19	6	no	—
69	C₉	—	19	9	no	—
70	C₉	—	57	3	no	—
71	C₉	C₃×C₉×C₁₉⋊C₃	1	3	yes	—
72	C₃	—	57	3	no	—
73	C₃	C₉²×C₁₉⋊C₃	1	1	yes	characteristic
74	C₃	C₁₉⋊C₃×C₂₇⋊C₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
commutator	C₃×C₁₉	C₃×C₉²
Fitting	C₁₉×#6864	C₃
socle	C₃²×C₁₉	C₃²×C₉
prime radical	C₃²×C₁₉⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₁₉	C₃³
hypercenter	#6864	C₁₉⋊C₃
hypocenter	C₁₉	C₉×C₂₇⋊C₃
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃×C₉ ◁ #6864

Complements

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

N	H ≅ G/N	type
3. #17987331	C₃	semidirect
4. #17987329	C₃	semidirect
5. C₂₇×#4294	C₃	semidirect
6. C₁₉×#6864	C₃	semidirect
9. #13095311	C₉	semidirect
25. C₁₉×C₂₇	C₃×C₉	semidirect
26. #10587018	C₃×C₉	semidirect
31. #6864	C₁₉⋊C₃	semidirect
40. #4294	C₂₇⋊C₃	semidirect
62. C₂₇	C₉×C₁₉⋊C₃	semidirect
63. C₁₉	C₉×C₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095311	3
C₉²×C₁₉⋊C₃	3
#17987331	3
#17987329	3
C₂₇×#4294	3
C₁₉×#6864	3
C₉×C₂₇⋊C₃	19