Subgroups — C₃×#22582731 #41008650

Order 26973.

848 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582731	C₃	1	27	yes	maximal, direct product factor
2	C₃×#15688541	C₃	1	8	yes	maximal, complemented
3	C₃²×#11758603	C₃	1	4	yes	maximal, complemented
4	C₃×#15688530	C₃	1	1	yes	maximal, characteristic
5	#15688541	C₃²	1	72	yes	complemented
6	C₃×#11758603	C₃²	1	1	yes	characteristic
7	C₃×#11758603	C₃²	1	8	yes	—
8	C₃×#11758603	C₃²	1	36	yes	complemented
9	#15688530	C₉	1	18	yes	—
10	#15688530	C₃²	1	9	yes	—
11	C₃×#11758608	C₉	1	8	yes	—
12	C₃²×#66524	C₃²	1	4	yes	—
13	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
14	C₃³×C₃₇	C₃×C₉	1	4	yes	—
15	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	—
16	#11758608	C₃×C₉	1	72	yes	—
17	C₃×#66524	C₃×C₉	1	8	yes	—
18	C₃×#66524	C₃×C₉	1	24	yes	—
19	C₃×#66524	C₃³	1	1	yes	characteristic
20	C₃×#66524	C₃³	1	12	yes	—
21	#11758603	C₃³	1	3	yes	—
22	#11758603	C₃³	1	24	yes	—
23	C₃×#6903	—	37	1	no	maximal, Sylow
24	C₃²×C₃₇	—	3	12	no	—
25	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
26	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
27	#66524	C₃²×C₉	1	2	yes	—
28	#66524	C₃²×C₉	1	24	yes	—
29	#66524	C₃⁴	1	1	yes	characteristic
30	C₃×#737	—	37	1	no	—
31	#6903	—	37	27	no	—
32	C₃×C₂₇⋊C₃	—	37	8	no	—
33	C₃²×C₂₇	—	37	4	no	—
34	C₃×C₃₇	—	3	12	no	—
35	C₃×C₃₇	#6903	1	3	yes	complemented
36	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
37	#737	—	37	18	no	—
38	#737	—	37	9	no	—
39	C₃×3₋¹⁺²	—	37	8	no	—
40	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic
41	C₃²×C₉	—	37	4	no	—
42	C₂₇⋊C₃	—	37	72	no	—
43	C₃×C₂₇	—	37	1	no	—
44	C₃×C₂₇	—	37	8	no	—
45	C₃×C₂₇	—	37	36	no	—
46	C₃₇	C₃×#6903	1	1	yes	characteristic, complemented, Sylow
47	C₃³	C₃×#66526	1	4	yes	—
48	3₋¹⁺²	—	37	72	no	—
49	H(3,𝔽₃)	C₃×#66526	1	9	yes	—
50	C₃×C₉	—	37	24	no	—
51	C₃×C₉	—	37	12	no	—
52	C₃×C₉	—	37	8	no	—
53	C₃×C₉	—	37	1	no	—
54	C₂₇	—	37	3	no	—
55	C₂₇	—	37	24	no	—
56	C₃²	—	3	12	no	—
57	C₃²	C₃²×#66526	1	12	yes	—
58	C₃²	C₃²×#66526	1	1	yes	characteristic
59	C₉	—	37	1	no	—
60	C₉	—	37	2	no	—
61	C₉	—	37	24	no	—
62	C₃	—	3	12	no	—
63	C₃	#22582731	1	3	yes	direct product factor
64	C₃	C₃³×#66526	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²×#66526
commutator	C₃×C₃₇	C₃³×C₉
Frattini	C₃	C₃³×#66526
Fitting, prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
socle	C₃²×C₃₇	C₃²×C₉
cosocle	#66524	C₃⁴
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	C₃×#6903
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #22582731	C₃	direct (N × H)
2. C₃×#15688541	C₃	semidirect
3. C₃²×#11758603	C₃	semidirect
5. #15688541	C₃²	semidirect
8. C₃×#11758603	C₃²	semidirect
35. C₃×C₃₇	#6903	semidirect
46. C₃₇	C₃×#6903	semidirect
63. C₃	#22582731	direct (N × H)

Maximal subgroups

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

subgroup	index
#22582731	3
C₃×#15688541	3
C₃²×#11758603	3
C₃×#15688530	3
C₃×#6903	37

Subgroups — C3×#22582731 #41008650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582731 #41008650