Subgroups — #41008527

Order 26973.

284 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×(C₃₇⋊C₃)⋊C₂₇	C₃	1	3	yes	maximal
2	#22582705	C₃	1	6	yes	maximal, complemented
3	#22582703	C₃	1	3	yes	maximal, complemented
4	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
5	(C₃₇⋊C₃)⋊C₂₇	—	3	9	no	—
6	C₃×#11758603	C₃²	1	9	yes	—
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	C₉×#66524	C₉	1	1	yes	characteristic
11	C₉×#66524	C₉	1	2	yes	—
12	C₉×#66524	C₉	1	2	yes	—
13	C₉×#66524	C₃²	1	1	yes	characteristic
14	C₉×#66524	C₃²	1	2	yes	—
15	C₉²×C₃₇	C₉	1	1	yes	characteristic
16	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
17	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
18	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×#66524	C₃×C₉	1	2	yes	—
21	C₃×#66524	C₃×C₉	1	2	yes	—
22	C₃×#66524	C₃×C₉	1	1	yes	characteristic
23	C₃×#66524	C₃×C₉	1	2	yes	—
24	C₃×#66524	C₃³	1	1	yes	characteristic
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	#11758603	—	3	9	no	—
29	C₃×C₉⋊C₂₇	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	C₉⋊C₉	1	3	yes	—
32	C₃×C₃₇⋊C₃	C₃×C₂₇	1	3	yes	—
33	C₃×C₃₇⋊C₃	C₂₇⋊C₃	1	6	yes	—
34	C₉×C₃₇	—	3	2	no	—
35	C₉×C₃₇	C₉⋊C₉	1	3	yes	—
36	C₉×C₃₇	C₃×C₂₇	1	3	yes	complemented
37	#66524	—	3	2	no	—
38	#66524	—	3	2	no	—
39	#66524	C₉⋊C₉	1	3	yes	—
40	#66524	C₃×C₂₇	1	6	yes	complemented
41	#66524	C₃×3₋¹⁺²	1	3	yes	—
42	C₃²×C₂₇	—	37	3	no	—
43	C₃×C₉²	—	37	1	no	—
44	C₉⋊C₂₇	—	37	3	no	—
45	C₉⋊C₂₇	—	37	6	no	—
46	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
47	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
48	C₃×C₃₇	C₃×C₂₇⋊C₃	1	2	yes	—
49	C₃₇⋊C₃	C₉⋊C₂₇	1	9	yes	complemented
50	C₃²×C₉	—	37	1	no	—
51	C₃²×C₉	—	37	1	no	—
52	C₃²×C₉	—	37	2	no	—
53	C₃×C₂₇	—	37	9	no	—
54	C₃×C₂₇	—	111	9	no	—
55	C₉²	—	37	2	no	—
56	C₉²	—	37	1	no	—
57	C₉²	—	37	2	no	—
58	C₉²	—	37	1	no	—
59	C₉²	—	37	2	no	—
60	C₉²	#66526	1	1	yes	characteristic
61	C₃₇	C₃×C₉⋊C₂₇	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	37	1	no	—
63	C₃×C₉	—	37	1	no	—
64	C₃×C₉	—	37	2	no	—
65	C₃×C₉	—	37	1	no	—
66	C₃×C₉	—	37	9	no	—
67	C₃×C₉	—	37	9	no	—
68	C₃×C₉	—	37	2	no	—
69	C₃×C₉	—	37	2	no	—
70	C₃×C₉	—	111	6	no	—
71	C₃×C₉	C₃×#66526	1	2	yes	—
72	C₃×C₉	C₃×#66526	1	1	yes	characteristic
73	C₃×C₉	C₃×#66526	1	1	yes	characteristic
74	C₂₇	—	111	9	no	—
75	C₃²	—	37	3	no	—
76	C₃²	—	37	3	no	—
77	C₃²	—	37	6	no	—
78	C₃²	C₃²×#66526	1	1	yes	characteristic
79	C₉	—	3	2	no	—
80	C₉	—	37	3	no	—
81	C₉	—	37	3	no	—
82	C₉	—	37	6	no	—
83	C₉	—	111	2	no	—
84	C₉	—	111	2	no	—
85	C₉	#15688537	1	3	yes	—
86	C₉	(C₃₇⋊C₃)⋊C₂₇	1	3	yes	complemented
87	C₃	—	37	9	no	—
88	C₃	C₃×#15688537	1	1	yes	characteristic
89	C₃	C₃×(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
90	C₃	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	1	2	yes	—

	subgroup	G/H
center	C₃×C₉	C₃×#66526
commutator	C₃×C₃₇	C₃²×C₂₇
Frattini	C₃²	C₃²×#66526
Fitting	C₉²×C₃₇	C₉
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₉²	#66526
hypocenter	C₃₇	C₃×C₉⋊C₂₇
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
2. #22582705	C₃	semidirect
3. #22582703	C₃	semidirect
26. C₉×C₃₇⋊C₃	C₂₇	semidirect
36. C₉×C₃₇	C₃×C₂₇	semidirect
40. #66524	C₃×C₂₇	semidirect
49. C₃₇⋊C₃	C₉⋊C₂₇	semidirect
61. C₃₇	C₃×C₉⋊C₂₇	semidirect
86. C₉	(C₃₇⋊C₃)⋊C₂₇	semidirect

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
#22582705	3
#22582703	3
C₉²×C₃₇⋊C₃	3
C₃×C₉⋊C₂₇	37