Subgroups — #28354861

Order 13851.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#10587018	C₃	1	1	yes	maximal, characteristic
2	C₉×#10587025	C₃	1	1	yes	maximal, characteristic
3	#17987355	C₃	1	6	yes	maximal, complemented
4	C₃²⋊#10587018	C₃	1	1	yes	maximal, characteristic
5	C₂₇×#4294	C₃	1	3	yes	maximal, complemented
6	C₁₉×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
7	C₃×#10587025	C₉	1	1	yes	characteristic
8	C₃×#10587018	—	3	3	no	—
9	C₃×#10587018	C₃²	1	3	yes	—
10	C₃×#10587018	C₃²	1	2	yes	—
11	C₃×#10587018	C₃²	1	1	yes	characteristic
12	#13095305	C₉	1	6	yes	—
13	C₃²⋊#4294	C₉	1	2	yes	—
14	C₉×#4294	C₃²	1	3	yes	—
15	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
16	C₃×C₁₉×C₂₇	C₃²	1	3	yes	—
17	C₂₇×3₋¹⁺²	—	19	1	no	maximal, Sylow
18	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₁₉	—	3	1	no	—
20	C₃×C₉×C₁₉	—	3	1	no	—
21	C₃×C₉×C₁₉	—	3	1	no	—
22	C₃×C₉×C₁₉	C₃×C₉	1	2	yes	—
23	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
24	#10587025	C₂₇	1	9	yes	complemented
25	C₃×#4294	C₃×C₉	1	6	yes	—
26	C₃×#4294	C₃×C₉	1	3	yes	—
27	C₁₉×C₂₇	—	3	3	no	—
28	#10587018	—	3	2	no	—
29	#10587018	—	3	3	no	—
30	#10587018	3₋¹⁺²	1	3	yes	complemented
31	C₃²×C₂₇	—	19	1	no	—
32	C₉×3₋¹⁺²	—	19	1	no	—
33	C₉⋊C₂₇	—	19	6	no	—
34	C₃²⋊C₂₇	—	19	1	no	—
35	C₃²⋊C₂₇	C₁₉⋊C₃	1	1	yes	characteristic
36	C₉×C₂₇	—	19	3	no	—
37	C₃²×C₁₉	—	3	1	no	—
38	C₃²×C₁₉	—	3	2	no	—
39	C₃²×C₁₉	C₃×C₂₇	1	3	yes	—
40	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
41	C₉×C₁₉	—	3	2	no	—
42	C₉×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
43	C₉×C₁₉	#737	1	1	yes	characteristic
44	C₉×C₁₉	#737	1	1	yes	characteristic
45	#4294	—	3	6	no	—
46	#4294	C₃×C₂₇	1	9	yes	complemented
47	C₃×3₋¹⁺²	—	19	1	no	—
48	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
49	C₃×C₂₇	—	19	1	no	—
50	C₃×C₂₇	—	19	3	no	—
51	C₃×C₂₇	—	19	2	no	—
52	C₃×C₂₇	—	57	3	no	—
53	C₃×C₂₇	C₃×C₁₉⋊C₃	1	3	yes	—
54	C₉⋊C₉	—	19	6	no	—
55	C₃²⋊C₉	—	19	2	no	—
56	C₉²	—	19	3	no	—
57	C₃×C₁₉	—	3	3	no	—
58	C₃×C₁₉	C₉×3₋¹⁺²	1	1	yes	characteristic
59	C₃×C₁₉	C₃²×C₂₇	1	1	yes	characteristic
60	C₃×C₁₉	#6903	1	2	yes	—
61	C₃³	C₉×C₁₉⋊C₃	1	1	yes	characteristic
62	3₋¹⁺²	—	19	9	no	—
63	C₃×C₉	—	3	1	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	19	3	no	—
67	C₃×C₉	—	19	6	no	—
68	C₃×C₉	C₉×C₁₉⋊C₃	1	2	yes	—
69	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
70	C₂₇	—	3	3	no	—
71	C₂₇	—	19	3	no	—
72	C₂₇	—	57	2	no	—
73	C₂₇	—	57	3	no	—
74	C₁₉	C₂₇×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	C₂₇×C₁₉⋊C₃	1	3	yes	—
78	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
79	C₉	—	3	2	no	—
80	C₉	—	19	9	no	—
81	C₉	—	57	6	no	—
82	C₉	3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
83	C₉	#13095331	1	1	yes	characteristic
84	C₉	#13095333	1	1	yes	characteristic
85	C₃	—	3	3	no	—
86	C₃	#17987415	1	1	yes	characteristic
87	C₃	C₃×C₂₇×C₁₉⋊C₃	1	1	yes	characteristic
88	C₃	#17987481	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
commutator	C₃×C₁₉	C₃²×C₂₇
Fitting	C₁₉×C₃²⋊C₂₇	C₃
socle	C₃²×C₁₉	C₃²×C₉
prime radical	C₃³×C₁₉	C₃×C₉
cosocle	C₃×C₉×C₁₉	C₃³
hypercenter	C₃²⋊C₂₇	C₁₉⋊C₃
hypocenter	C₁₉	C₂₇×3₋¹⁺²
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃×C₉ ◁ C₃²⋊C₂₇

Complements

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

N	H ≅ G/N	type
3. #17987355	C₃	semidirect
5. C₂₇×#4294	C₃	semidirect
24. #10587025	C₂₇	semidirect
30. #10587018	3₋¹⁺²	semidirect
46. #4294	C₃×C₂₇	semidirect
74. C₁₉	C₂₇×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#10587018	3
C₉×#10587025	3
#17987355	3
C₃²⋊#10587018	3
C₂₇×#4294	3
C₁₉×C₃²⋊C₂₇	3
C₂₇×3₋¹⁺²	19