Subgroups — C₃×C₂₇×#66526 #41008462

Order 26973.

404 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×(C₃₇⋊C₃)⋊C₂₇	C₃	1	2	yes	maximal
2	C₃×C₉×#66526	C₃	1	1	yes	maximal, characteristic
3	C₂₇×#66526	C₃	1	9	yes	maximal, direct product factor
4	C₃×C₂₇×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
5	C₃²×#66526	C₉	1	3	yes	—
6	(C₃₇⋊C₃)⋊C₂₇	C₉	1	12	yes	complemented
7	(C₃₇⋊C₃)⋊C₂₇	C₃²	1	6	yes	—
8	C₃×#11758603	C₉	1	6	yes	complemented
9	C₉×#66526	C₉	1	6	yes	—
10	C₉×#66526	C₃²	1	3	yes	—
11	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
12	C₃×#11758601	C₉	1	2	yes	complemented
13	C₂₇×C₃₇⋊C₃	C₉	1	6	yes	complemented
14	C₂₇×C₃₇⋊C₃	C₃²	1	3	yes	—
15	C₃×C₂₇×C₃₇	C₉	1	1	yes	characteristic, complemented
16	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
17	C₃×#66526	C₂₇	1	27	yes	direct product factor
18	C₃×#66526	C₃×C₉	1	9	yes	—
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×#66524	C₃×C₉	1	2	yes	—
21	C₉×C₃₇⋊C₃	C₃×C₉	1	2	yes	—
22	C₉×C₃₇⋊C₃	C₃×C₉	1	2	yes	—
23	C₉×C₃₇⋊C₃	C₃×C₉	1	4	yes	—
24	C₉×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
25	#11758603	C₃×C₉	1	18	yes	complemented
26	C₂₇×C₃₇	C₃×C₉	1	3	yes	complemented
27	#11758601	C₃×C₉	1	6	yes	complemented
28	C₃×C₉×C₂₇	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	C₉²	1	1	yes	characteristic
30	C₃×C₃₇⋊C₃	C₉²	1	2	yes	—
31	C₃×C₃₇⋊C₃	C₃×C₂₇	1	9	yes	—
32	C₃×C₃₇⋊C₃	C₃²×C₉	1	1	yes	characteristic
33	#66526	C₃×C₂₇	1	27	yes	direct product factor
34	C₉×C₃₇	C₉²	1	2	yes	—
35	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
36	#66524	C₉²	1	4	yes	—
37	#66524	C₃²×C₉	1	2	yes	—
38	C₃²×C₂₇	—	37	2	no	—
39	C₃²×C₂₇	—	37	1	no	—
40	C₃×C₉²	—	37	1	no	—
41	C₉×C₂₇	—	37	9	no	—
42	C₃×C₃₇	C₉×C₂₇	1	3	yes	complemented
43	C₃×C₃₇	C₃×C₉²	1	1	yes	characteristic
44	C₃₇⋊C₃	C₉×C₂₇	1	6	yes	complemented
45	C₃₇⋊C₃	C₃²×C₂₇	1	3	yes	—
46	C₃²×C₉	—	37	3	no	—
47	C₃²×C₉	—	37	1	no	—
48	C₃×C₂₇	—	37	6	no	—
49	C₃×C₂₇	—	37	12	no	—
50	C₃×C₂₇	—	37	6	no	—
51	C₃×C₂₇	—	37	2	no	—
52	C₃×C₂₇	—	37	6	no	—
53	C₃×C₂₇	—	37	3	no	—
54	C₃×C₂₇	#66526	1	1	yes	characteristic, direct product factor
55	C₉²	—	37	6	no	—
56	C₉²	—	37	3	no	—
57	C₃₇	C₃×C₉×C₂₇	1	1	yes	characteristic, complemented, Sylow
58	C₃³	—	37	1	no	—
59	C₃×C₉	—	37	27	no	—
60	C₃×C₉	—	37	9	no	—
61	C₃×C₉	—	37	2	no	—
62	C₃×C₉	—	37	2	no	—
63	C₃×C₉	—	37	4	no	—
64	C₃×C₉	—	37	2	no	—
65	C₃×C₉	—	37	1	no	—
66	C₃×C₉	C₃×#66526	1	1	yes	characteristic
67	C₂₇	—	37	18	no	—
68	C₂₇	—	37	6	no	—
69	C₂₇	C₃×#66526	1	3	yes	direct product factor
70	C₃²	—	37	9	no	—
71	C₃²	—	37	1	no	—
72	C₃²	—	37	2	no	—
73	C₃²	C₉×#66526	1	1	yes	characteristic
74	C₉	—	37	27	no	—
75	C₉	—	37	2	no	—
76	C₉	—	37	4	no	—
77	C₉	C₉×#66526	1	2	yes	—
78	C₉	C₃²×#66526	1	1	yes	characteristic
79	C₃	—	37	3	no	—
80	C₃	—	37	6	no	—
81	C₃	C₂₇×#66526	1	3	yes	direct product factor
82	C₃	C₃×C₉×#66526	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃×C₂₇	#66526
commutator, hypocenter	C₃₇	C₃×C₉×C₂₇
Frattini	C₉	C₃²×#66526
Fitting	C₃×C₂₇×C₃₇	C₉
socle	C₃²×C₃₇	C₉²
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₉×C₃₇⋊C₃	C₃³
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₃×C₂₇

Complements

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

N	H ≅ G/N	type
3. C₂₇×#66526	C₃	direct (N × H)
6. (C₃₇⋊C₃)⋊C₂₇	C₉	semidirect
8. C₃×#11758603	C₉	semidirect
12. C₃×#11758601	C₉	semidirect
13. C₂₇×C₃₇⋊C₃	C₉	semidirect
15. C₃×C₂₇×C₃₇	C₉	semidirect
17. C₃×#66526	C₂₇	direct (N × H)
25. #11758603	C₃×C₉	semidirect
26. C₂₇×C₃₇	C₃×C₉	semidirect
27. #11758601	C₃×C₉	semidirect
33. #66526	C₃×C₂₇	direct (N × H)
42. C₃×C₃₇	C₉×C₂₇	semidirect
44. C₃₇⋊C₃	C₉×C₂₇	semidirect
54. C₃×C₂₇	#66526	direct (N × H)
57. C₃₇	C₃×C₉×C₂₇	semidirect
69. C₂₇	C₃×#66526	direct (N × H)
81. C₃	C₂₇×#66526	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×(C₃₇⋊C₃)⋊C₂₇	3
C₃×C₉×#66526	3
C₂₇×#66526	3
C₃×C₂₇×C₃₇⋊C₃	3
C₃×C₉×C₂₇	37

Subgroups — C3×C27×#66526 #41008462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂₇×#66526 #41008462