Subgroups — C₃²×#15688509 #41008116

Order 26973.

944 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688509	C₃	1	36	yes	maximal, direct product factor
3	C₃²×#11758601	C₃	1	2	yes	maximal, complemented
4	C₃²×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	C₉	1	1	yes	characteristic
6	C₃²×#66524	C₉	1	2	yes	—
7	C₃×C₉×C₃₇⋊C₃	C₉	1	24	yes	—
8	C₃×C₉×C₃₇⋊C₃	C₃²	1	12	yes	—
9	#15688509	C₃²	1	81	yes	direct product factor
10	C₃×#11758601	C₃²	1	24	yes	complemented
11	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×C₂₇×C₃₇	C₃²	1	12	yes	complemented
13	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₃²×C₃₇⋊C₃	—	3	9	no	—
15	C₃²×C₃₇⋊C₃	C₃×C₉	1	12	yes	—
16	C₃×C₉×C₃₇	C₃×C₉	1	8	yes	—
17	C₃×C₉×C₃₇	C₃³	1	4	yes	—
18	C₃×#66524	C₃×C₉	1	24	yes	—
19	C₉×C₃₇⋊C₃	C₃×C₉	1	72	yes	—
20	C₉×C₃₇⋊C₃	C₃³	1	9	yes	—
21	C₂₇×C₃₇	C₃³	1	9	yes	complemented
22	#11758601	C₃³	1	18	yes	complemented
23	C₃²×C₂₇⋊C₃	—	37	1	no	maximal, Sylow
24	C₃²×C₃₇	C₂₇⋊C₃	1	9	yes	complemented
25	C₃²×C₃₇	C₃²×C₉	1	4	yes	—
26	C₃×C₃₇⋊C₃	—	3	36	no	—
27	C₃×C₃₇⋊C₃	C₃²×C₉	1	9	yes	—
28	C₉×C₃₇	C₃²×C₉	1	8	yes	—
29	C₉×C₃₇	C₃⁴	1	1	yes	characteristic
30	#66524	C₃²×C₉	1	18	yes	—
31	C₃³×C₉	—	37	1	no	—
32	C₃×C₂₇⋊C₃	—	37	36	no	—
33	C₃²×C₂₇	—	37	2	no	—
34	C₃²×C₂₇	C₃₇⋊C₃	1	1	yes	characteristic, complemented
35	C₃×C₃₇	C₃×C₂₇⋊C₃	1	12	yes	complemented
36	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
37	C₃₇⋊C₃	—	3	9	no	—
38	C₃⁴	—	37	1	no	—
39	C₃²×C₉	—	37	2	no	—
40	C₃²×C₉	—	37	24	no	—
41	C₃²×C₉	—	37	12	no	—
42	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
43	C₂₇⋊C₃	—	37	81	no	—
44	C₃×C₂₇	—	37	24	no	—
45	C₃×C₂₇	C₃×C₃₇⋊C₃	1	12	yes	complemented
46	C₃₇	C₃²×C₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
47	C₃³	—	37	12	no	—
48	C₃³	—	111	9	no	—
49	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
50	C₃×C₉	—	37	24	no	—
51	C₃×C₉	—	37	72	no	—
52	C₃×C₉	—	37	9	no	—
53	C₃×C₉	C₉×C₃₇⋊C₃	1	8	yes	—
54	C₃×C₉	C₃²×C₃₇⋊C₃	1	4	yes	—
55	C₂₇	—	37	18	no	—
56	C₂₇	C₃²×C₃₇⋊C₃	1	9	yes	complemented
57	C₃²	—	37	9	no	—
58	C₃²	—	111	36	no	—
59	C₃²	#15688509	1	9	yes	direct product factor
60	C₃²	C₃×C₉×C₃₇⋊C₃	1	4	yes	—
61	C₉	—	37	18	no	—
62	C₉	C₃×C₉×C₃₇⋊C₃	1	8	yes	—
63	C₉	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
64	C₃	—	111	9	no	—
65	C₃	C₃×#15688509	1	12	yes	direct product factor
66	C₃	C₃²×C₉×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₉	C₃×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
Frattini	C₉	C₃³×C₃₇⋊C₃
Fitting	C₃²×C₂₇×C₃₇	C₃
socle	C₃³×C₃₇	C₃×C₉
prime radical	C₃³×C₃₇⋊C₃	C₉
cosocle	C₉×C₃₇	C₃⁴
hypercenter	C₃²×C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	C₃²×C₂₇⋊C₃
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃²×C₉ ◁ C₃²×C₂₇

Complements

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

N	H ≅ G/N	type
2. C₃×#15688509	C₃	direct (N × H)
3. C₃²×#11758601	C₃	semidirect
4. C₃²×C₂₇×C₃₇	C₃	semidirect
9. #15688509	C₃²	direct (N × H)
10. C₃×#11758601	C₃²	semidirect
12. C₃×C₂₇×C₃₇	C₃²	semidirect
21. C₂₇×C₃₇	C₃³	semidirect
22. #11758601	C₃³	semidirect
24. C₃²×C₃₇	C₂₇⋊C₃	semidirect
34. C₃²×C₂₇	C₃₇⋊C₃	semidirect
35. C₃×C₃₇	C₃×C₂₇⋊C₃	semidirect
45. C₃×C₂₇	C₃×C₃₇⋊C₃	semidirect
46. C₃₇	C₃²×C₂₇⋊C₃	semidirect
56. C₂₇	C₃²×C₃₇⋊C₃	semidirect
59. C₃²	#15688509	direct (N × H)
65. C₃	C₃×#15688509	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×C₉×C₃₇⋊C₃	3
C₃×#15688509	3
C₃²×#11758601	3
C₃²×C₂₇×C₃₇	3
C₃²×C₂₇⋊C₃	37

Subgroups — C32×#15688509 #41008116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#15688509 #41008116