Subgroups — C₃×#22582652 #41008169

Order 26973.

980 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758608	C₃	1	3	yes	maximal
2	#22582652	C₃	1	27	yes	maximal, direct product factor
3	C₃×#15688521	C₃	1	9	yes	maximal, complemented
4	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	6	no	—
6	C₃×#11758608	—	3	18	no	—
7	C₃×#11758608	C₃²	1	9	yes	—
8	C₃×#11758608	C₃²	1	27	yes	—
9	C₃²×#66524	—	3	3	no	—
10	#15688521	C₃²	1	81	yes	complemented
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×3₋¹⁺²×C₃₇	—	3	3	no	—
13	C₃×3₋¹⁺²×C₃₇	—	3	6	no	—
14	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
15	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
16	C₃³×C₃₇	—	9	3	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
18	C₃³×C₃₇	H(3,𝔽₃)	1	6	yes	—
19	C₃³×C₃₇	C₃³	1	1	yes	characteristic
20	C₃³×C₃₇	C₃³	1	3	yes	—
21	3₋¹⁺²×C₃₇	—	3	27	no	—
22	C₃×C₉×C₃₇	—	3	6	no	—
23	C₃×C₉×C₃₇	—	3	3	no	—
24	C₃×C₉×C₃₇	C₃³	1	9	yes	—
25	#11758608	—	3	54	no	—
26	#11758608	—	3	18	no	—
27	#11758608	C₃³	1	27	yes	—
28	C₃×#66524	—	3	27	no	—
29	C₃×#66524	—	3	9	no	—
30	C₃×#6912	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	3	9	no	—
32	C₃²×C₃₇	—	3	3	no	—
33	C₃²×C₃₇	—	9	6	no	—
34	C₃²×C₃₇	—	9	3	no	—
35	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
39	C₉×C₃₇	—	3	9	no	—
40	#66524	—	3	27	no	—
41	C₃²×3₋¹⁺²	—	37	3	no	—
42	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
43	#6912	—	37	27	no	—
44	C₃×#733	—	37	9	no	—
45	C₃×C₃₇	—	3	9	no	—
46	C₃×C₃₇	—	3	2	no	—
47	C₃×C₃₇	—	3	1	no	—
48	C₃×C₃₇	#6912	1	3	yes	complemented
49	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	C₃×3₋¹⁺²	—	3	6	no	—
52	C₃×3₋¹⁺²	—	3	3	no	—
53	C₃×3₋¹⁺²	—	37	27	no	—
54	C₃×3₋¹⁺²	—	37	9	no	—
55	C₃×3₋¹⁺²	—	111	18	no	—
56	C₃×3₋¹⁺²	—	111	6	no	—
57	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	9	yes	—
58	C₃²×C₉	—	111	3	no	—
59	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
60	#733	—	37	81	no	—
61	C₃₇	C₃×#6912	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	9	3	no	—
63	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
64	C₃³	C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
65	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
66	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
67	3₋¹⁺²	—	3	27	no	—
68	3₋¹⁺²	—	37	27	no	—
69	3₋¹⁺²	—	111	18	no	—
70	3₋¹⁺²	—	111	54	no	—
71	C₃×C₉	—	3	6	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	111	27	no	—
74	C₃×C₉	—	111	9	no	—
75	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
76	C₃²	—	3	9	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	9	6	no	—
79	C₃²	—	9	3	no	—
80	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
81	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
82	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
83	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
84	C₉	—	3	9	no	—
85	C₉	—	111	27	no	—
86	C₃	—	3	9	no	—
87	C₃	—	3	2	no	—
88	C₃	—	3	1	no	—
89	C₃	#22582652	1	3	yes	direct product factor
90	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6912
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #22582652	C₃	direct (N × H)
3. C₃×#15688521	C₃	semidirect
10. #15688521	C₃²	semidirect
48. C₃×C₃₇	#6912	semidirect
61. C₃₇	C₃×#6912	semidirect
89. C₃	#22582652	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#11758608	3
#22582652	3
C₃×#15688521	3
C₃²×3₋¹⁺²×C₃₇	3
C₃×#6912	37

Subgroups — C3×#22582652 #41008169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582652 #41008169