Subgroups — C₃²×C₃₇×C₂₇⋊C₃ #41004705

Order 26973.

944 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₃²×C₉×C₃₇	C₃²
commutator	C₃	C₃³×C₉×C₃₇
Frattini, cosocle	C₉	C₃⁴×C₃₇
socle	C₃³×C₃₇	C₃×C₉
prime radical	C₃⁴×C₃₇	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₃₇×C₂₇⋊C₃	C₃	direct (N × H)
3. C₃²×C₂₇×C₃₇	C₃	semidirect
9. C₃₇×C₂₇⋊C₃	C₃²	direct (N × H)
10. C₃×C₂₇×C₃₇	C₃²	semidirect
19. C₂₇×C₃₇	C₃³	semidirect
20. C₃²×C₂₇⋊C₃	C₃₇	direct (N × H)
22. C₃²×C₃₇	C₂₇⋊C₃	direct (N × H)
29. C₃×C₂₇⋊C₃	C₃×C₃₇	direct (N × H)
30. C₃²×C₂₇	C₃×C₃₇	semidirect
32. C₃×C₃₇	C₃×C₂₇⋊C₃	direct (N × H)
39. C₂₇⋊C₃	C₃²×C₃₇	direct (N × H)
40. C₃×C₂₇	C₃²×C₃₇	semidirect
41. C₃₇	C₃²×C₂₇⋊C₃	direct (N × H)
50. C₂₇	C₃³×C₃₇	semidirect
52. C₃²	C₃₇×C₂₇⋊C₃	direct (N × H)
59. C₃	C₃×C₃₇×C₂₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₉×C₃₇	3
C₃×C₃₇×C₂₇⋊C₃	3
C₃²×C₂₇×C₃₇	3
C₃²×C₂₇⋊C₃	37

Subgroups — C32×C37×C27⋊C3 #41004705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₇×C₂₇⋊C₃ #41004705