Subgroups — C₃₇⋊C₃×C₉⋊C₂₇ #41006312

Order 26973.

284 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₇×C₃₇⋊C₃	C₃	1	3	yes	maximal
2	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	#22582490	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582489	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582488	C₃	1	6	yes	maximal, complemented
6	C₃₇×C₉⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₉×C₃₇⋊C₃	C₉	1	2	yes	—
8	C₃×C₉×C₃₇⋊C₃	C₉	1	1	yes	characteristic
9	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
10	C₃×#11758601	C₃²	1	3	yes	—
11	C₃×#11758601	C₃²	1	3	yes	—
12	C₂₇×C₃₇⋊C₃	—	3	9	no	—
13	C₉×#66524	C₉	1	2	yes	—
14	C₉×#66524	C₉	1	4	yes	—
15	C₉×#66524	C₃²	1	2	yes	—
16	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
17	C₉²×C₃₇	C₃²	1	1	yes	characteristic
18	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
20	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
21	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
22	C₃×#66524	C₃×C₉	1	4	yes	—
23	C₃×#66524	C₃×C₉	1	2	yes	—
24	C₃×#66524	C₃×C₉	1	2	yes	—
25	C₉×C₃₇⋊C₃	—	3	6	no	—
26	C₉×C₃₇⋊C₃	C₂₇	1	9	yes	complemented
27	C₉×C₃₇⋊C₃	3₋¹⁺²	1	9	yes	—
28	C₂₇×C₃₇	—	3	3	no	—
29	#11758601	—	3	3	no	—
30	#11758601	—	3	3	no	—
31	C₃×C₉⋊C₂₇	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
33	C₃×C₃₇⋊C₃	C₉⋊C₉	1	3	yes	—
34	C₃×C₃₇⋊C₃	C₃×C₂₇	1	3	yes	—
35	C₃×C₃₇⋊C₃	C₂₇⋊C₃	1	6	yes	—
36	C₉×C₃₇	—	3	2	no	—
37	C₉×C₃₇	C₃×C₂₇	1	3	yes	complemented
38	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
39	#66524	—	3	4	no	—
40	#66524	C₉⋊C₉	1	6	yes	—
41	#66524	C₃×C₂₇	1	6	yes	complemented
42	C₃²×C₂₇	—	37	3	no	—
43	C₃×C₉²	—	37	1	no	—
44	C₉⋊C₂₇	—	37	6	no	—
45	C₉⋊C₂₇	—	37	1	no	—
46	C₉⋊C₂₇	—	37	1	no	—
47	C₉⋊C₂₇	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
48	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
49	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
50	C₃×C₃₇	C₃×C₂₇⋊C₃	1	2	yes	—
51	C₃₇⋊C₃	C₉⋊C₂₇	1	9	yes	direct product factor
52	C₃²×C₉	—	37	1	no	—
53	C₃²×C₉	—	37	1	no	—
54	C₃²×C₉	—	37	2	no	—
55	C₃×C₂₇	—	37	3	no	—
56	C₃×C₂₇	—	37	3	no	—
57	C₃×C₂₇	—	111	9	no	—
58	C₃×C₂₇	C₃×C₃₇⋊C₃	1	3	yes	—
59	C₉²	—	37	4	no	—
60	C₉²	—	37	2	no	—
61	C₉²	—	37	2	no	—
62	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
63	C₃₇	C₃×C₉⋊C₂₇	1	1	yes	characteristic, complemented, Sylow
64	C₃³	—	37	1	no	—
65	C₃×C₉	—	37	2	no	—
66	C₃×C₉	—	37	2	no	—
67	C₃×C₉	—	37	9	no	—
68	C₃×C₉	—	37	9	no	—
69	C₃×C₉	—	37	4	no	—
70	C₃×C₉	—	111	6	no	—
71	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
72	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
73	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
74	C₂₇	—	3	3	no	—
75	C₂₇	—	111	3	no	—
76	C₂₇	—	111	3	no	—
77	C₃²	—	37	3	no	—
78	C₃²	—	37	3	no	—
79	C₃²	—	37	6	no	—
80	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
81	C₉	—	3	2	no	—
82	C₉	—	37	6	no	—
83	C₉	—	37	6	no	—
84	C₉	—	111	4	no	—
85	C₉	C₂₇×C₃₇⋊C₃	1	3	yes	complemented
86	C₉	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
87	C₃	—	37	9	no	—
88	C₃	C₉⋊C₉×C₃₇⋊C₃	1	1	yes	characteristic
89	C₃	C₃×C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
90	C₃	C₂₇⋊C₃×C₃₇⋊C₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #22582490	C₃	semidirect
4. #22582489	C₃	semidirect
5. #22582488	C₃	semidirect
6. C₃₇×C₉⋊C₂₇	C₃	semidirect
26. C₉×C₃₇⋊C₃	C₂₇	semidirect
37. C₉×C₃₇	C₃×C₂₇	semidirect
41. #66524	C₃×C₂₇	semidirect
47. C₉⋊C₂₇	C₃₇⋊C₃	direct (N × H)
51. C₃₇⋊C₃	C₉⋊C₂₇	direct (N × H)
63. C₃₇	C₃×C₉⋊C₂₇	semidirect
85. C₉	C₂₇×C₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₂₇×C₃₇⋊C₃	3
C₉²×C₃₇⋊C₃	3
#22582490	3
#22582489	3
#22582488	3
C₃₇×C₉⋊C₂₇	3
C₃×C₉⋊C₂₇	37

Subgroups — C37⋊C3×C9⋊C27 #41006312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇⋊C₃×C₉⋊C₂₇ #41006312