Subgroups — #41005109

Order 26973.

374 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688506	C₃	1	3	yes	maximal
2	C₃×#15688505	C₃	1	3	yes	maximal
3	#22582441	C₃	1	6	yes	maximal
4	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃²×#66524	C₃²	1	9	yes	—
6	#15688506	—	3	9	no	—
7	#15688505	—	3	9	no	—
8	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₉²×C₃₇	C₉	1	9	yes	complemented
12	C₃³×C₃₇	C₃³	1	1	yes	characteristic
13	C₃×C₉×C₃₇	—	3	6	no	—
14	C₃×C₉×C₃₇	C₃×C₉	1	6	yes	—
15	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
16	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
17	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	—
18	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	—
19	C₃×#66524	—	3	9	no	—
20	C₃×#66524	—	3	9	no	—
21	C₃×#66524	—	3	18	no	—
22	#10624550	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
24	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
25	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
26	C₃²×C₃₇	#737	1	6	yes	—
27	C₉×C₃₇	—	3	6	no	—
28	C₉×C₃₇	C₉⋊C₉	1	9	yes	complemented
29	C₉×C₃₇	C₉⋊C₉	1	9	yes	complemented
30	#66524	—	9	9	no	—
31	C₃×C₉⋊C₉	—	37	3	no	—
32	C₃×C₉⋊C₉	—	37	3	no	—
33	C₃×C₉²	C₃₇⋊C₃	1	1	yes	characteristic
34	#6855	—	37	6	no	—
35	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
36	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
37	C₃×C₃₇	#6887	1	2	yes	—
38	C₃×C₃₇	#6896	1	9	yes	—
39	C₃²×C₉	—	37	9	no	—
40	C₃²×C₉	C₃×C₃₇⋊C₃	1	2	yes	—
41	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
42	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
43	C₉⋊C₉	—	111	9	no	—
44	C₉⋊C₉	—	111	9	no	—
45	C₉²	#66524	1	9	yes	complemented
46	C₃₇	#10624550	1	1	yes	characteristic, complemented, Sylow
47	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
48	C₃×C₉	—	3	6	no	—
49	C₃×C₉	—	111	9	no	—
50	C₃×C₉	—	111	9	no	—
51	C₃×C₉	—	111	18	no	—
52	C₃×C₉	#11758605	1	9	yes	—
53	C₃×C₉	#11758606	1	9	yes	—
54	C₃×C₉	C₃×#66524	1	6	yes	—
55	C₃×C₉	C₃×#66524	1	3	yes	—
56	C₃×C₉	C₃×#66524	1	3	yes	—
57	C₃²	C₃²×#66524	1	1	yes	characteristic
58	C₃²	C₃×#11758606	1	3	yes	—
59	C₃²	C₃×#11758605	1	3	yes	—
60	C₃²	#15688531	1	6	yes	—
61	C₉	—	3	6	no	—
62	C₉	—	333	9	no	—
63	C₉	#15688505	1	9	yes	complemented
64	C₉	#15688506	1	9	yes	complemented
65	C₃	C₃×#15688505	1	1	yes	characteristic
66	C₃	C₃×#15688506	1	1	yes	characteristic
67	C₃	#22582538	1	2	yes	—
68	C₃	#22582581	1	9	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃×C₉²×C₃₇	C₃
socle, prime radical, cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×C₉²	C₃₇⋊C₃
hypocenter	C₃₇	#10624550
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃×C₉²

Complements

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

N	H ≅ G/N	type
11. C₉²×C₃₇	C₉	semidirect
28. C₉×C₃₇	C₉⋊C₉	semidirect
29. C₉×C₃₇	C₉⋊C₉	semidirect
45. C₉²	#66524	semidirect
46. C₃₇	#10624550	semidirect
63. C₉	#15688505	semidirect
64. C₉	#15688506	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688506	3
C₃×#15688505	3
#22582441	3
C₃×C₉²×C₃₇	3
#10624550	37