Subgroups — #28357011

Order 13851.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#10587020	C₃	1	1	yes	maximal, characteristic
2	#17987573	C₃	1	3	yes	maximal, complemented
3	#17987572	C₃	1	3	yes	maximal, complemented
4	C₃²⋊#10587020	C₃	1	2	yes	maximal
5	C₉×#10587020	C₃	1	3	yes	maximal, complemented
6	#17987415	C₃	1	1	yes	maximal, characteristic
7	C₃×#10587020	—	3	3	no	—
8	C₃×#10587020	C₃²	1	1	yes	characteristic
9	C₃×#10587020	C₃²	1	1	yes	characteristic
10	C₃×#10587020	C₃²	1	6	yes	—
11	C₃×#10587020	C₃²	1	1	yes	characteristic
12	C₃×#10587025	C₉	1	1	yes	characteristic
13	C₃²×#4294	C₃²	1	1	yes	characteristic
14	#13095305	C₉	1	6	yes	—
15	C₃²⋊#4294	C₉	1	1	yes	characteristic
16	C₉×#4294	C₃²	1	3	yes	—
17	C₁₉×C₃²⋊C₉	C₉	1	1	yes	characteristic
18	C₂₇×3₋¹⁺²	—	19	1	no	maximal, Sylow
19	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
21	#10587025	C₂₇	1	9	yes	complemented
22	C₃×#4294	—	3	1	no	—
23	C₃×#4294	—	3	2	no	—
24	C₃×#4294	C₃×C₉	1	3	yes	—
25	C₃×#4294	C₃×C₉	1	2	yes	—
26	C₃×#4294	C₃×C₉	1	3	yes	—
27	C₃×#4294	C₃³	1	1	yes	characteristic
28	#10587020	—	3	1	no	—
29	#10587020	—	3	1	no	—
30	#10587020	—	3	6	no	—
31	#10587020	3₋¹⁺²	1	3	yes	complemented
32	C₃²×C₂₇	—	19	1	no	—
33	C₉×3₋¹⁺²	—	19	1	no	—
34	C₉⋊C₂₇	—	19	3	no	—
35	C₉⋊C₂₇	—	19	3	no	—
36	C₃²⋊C₂₇	—	19	2	no	—
37	C₉×C₂₇	—	19	3	no	—
38	C₃²×C₁₉	—	3	1	no	—
39	C₃²×C₁₉	—	3	1	no	—
40	C₃²×C₁₉	—	3	1	no	—
41	C₃²×C₁₉	C₃×C₂₇	1	3	yes	—
42	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
43	C₉×C₁₉	—	3	3	no	—
44	#4294	—	3	2	no	—
45	#4294	—	3	3	no	—
46	#4294	C₃×C₂₇	1	9	yes	complemented
47	#4294	C₃×3₋¹⁺²	1	1	yes	characteristic
48	#4294	#737	1	2	yes	—
49	C₃×3₋¹⁺²	—	19	1	no	—
50	C₃²×C₉	—	19	1	no	—
51	C₃×C₂₇	—	19	1	no	—
52	C₃×C₂₇	—	19	6	no	—
53	C₃×C₂₇	—	19	1	no	—
54	C₃×C₂₇	—	19	1	no	—
55	C₃×C₂₇	—	57	3	no	—
56	C₉⋊C₉	—	19	6	no	—
57	C₃²⋊C₉	—	19	1	no	—
58	C₃²⋊C₉	#4296	1	1	yes	characteristic
59	C₉²	—	19	3	no	—
60	C₃×C₁₉	—	3	3	no	—
61	C₃×C₁₉	C₉×3₋¹⁺²	1	1	yes	characteristic
62	C₃×C₁₉	C₃²×C₂₇	1	1	yes	characteristic
63	C₃×C₁₉	#6903	1	1	yes	characteristic
64	C₃×C₁₉	#6903	1	1	yes	characteristic
65	C₃³	C₃×#4296	1	1	yes	characteristic
66	3₋¹⁺²	—	19	9	no	—
67	C₃×C₉	—	19	3	no	—
68	C₃×C₉	—	19	1	no	—
69	C₃×C₉	—	19	3	no	—
70	C₃×C₉	—	19	2	no	—
71	C₃×C₉	—	57	1	no	—
72	C₃×C₉	—	57	2	no	—
73	C₃×C₉	C₃×#4296	1	3	yes	—
74	C₂₇	—	19	3	no	—
75	C₂₇	—	57	6	no	—
76	C₂₇	—	57	1	no	—
77	C₂₇	—	57	1	no	—
78	C₁₉	C₂₇×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
79	C₃²	—	3	1	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	1	no	—
82	C₃²	(C₁₉⋊C₃)⋊C₂₇	1	3	yes	—
83	C₃²	C₃²×#4296	1	1	yes	characteristic
84	C₉	—	3	3	no	—
85	C₉	—	19	1	no	—
86	C₉	—	19	2	no	—
87	C₉	—	19	9	no	—
88	C₉	—	57	2	no	—
89	C₉	—	57	3	no	—
90	C₃	—	3	3	no	—
91	C₃	3₋¹⁺²×#4296	1	1	yes	characteristic
92	C₃	C₃×(C₁₉⋊C₃)⋊C₂₇	1	1	yes	characteristic
93	C₃	#17987598	1	1	yes	characteristic
94	C₃	#17987599	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×#4296
commutator	C₃×C₁₉	C₃²×C₂₇
Fitting	C₁₉×C₃²⋊C₉	C₉
socle	C₃²×C₁₉	C₃²×C₉
prime radical	C₃³×C₁₉	C₃×C₉
cosocle	C₃×#4294	C₃³
hypercenter	C₃²⋊C₉	#4296
hypocenter	C₁₉	C₂₇×3₋¹⁺²
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. #17987573	C₃	semidirect
3. #17987572	C₃	semidirect
5. C₉×#10587020	C₃	semidirect
21. #10587025	C₂₇	semidirect
31. #10587020	3₋¹⁺²	semidirect
46. #4294	C₃×C₂₇	semidirect
78. 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×#10587020	3
#17987573	3
#17987572	3
C₃²⋊#10587020	3
C₉×#10587020	3
#17987415	3
C₂₇×3₋¹⁺²	19