Subgroups — #41008326

Order 26973.

174 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582722	C₃	1	3	yes	maximal
2	#22582536	C₃	1	1	yes	maximal, characteristic
3	#15688538	—	3	3	no	—
4	#15688537	C₉	1	9	yes	complemented
5	C₃²⋊#66526	—	3	3	no	—
6	C₉×#66526	—	3	3	no	—
7	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	#15688504	—	3	2	no	—
9	#15688502	—	3	1	no	—
10	C₉×#66524	C₉	1	2	yes	complemented
11	C₉²×C₃₇	C₉	1	1	yes	characteristic, complemented
12	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
13	C₃×#66526	—	3	9	no	—
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×#66524	—	3	2	no	—
17	C₃×#66524	3₋¹⁺²	1	2	yes	—
18	C₉×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
19	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
20	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
21	#10624470	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
23	C₃×C₃₇⋊C₃	—	9	1	no	—
24	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	3	yes	—
25	#66526	—	9	9	no	—
26	C₉×C₃₇	—	9	1	no	—
27	C₉×C₃₇	C₉²	1	1	yes	characteristic
28	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
29	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
30	#66524	—	9	2	no	—
31	#66524	C₉⋊C₉	1	6	yes	—
32	#6887	—	37	1	no	—
33	#6887	—	37	3	no	—
34	C₃×C₃₇	—	3	1	no	—
35	C₃×C₃₇	#6855	1	1	yes	characteristic
36	C₃₇⋊C₃	—	3	3	no	—
37	C₃²×C₉	—	37	1	no	—
38	C₉⋊C₉	—	37	9	no	—
39	C₉⋊C₉	—	111	2	no	—
40	C₉⋊C₉	—	111	3	no	—
41	C₃²⋊C₉	—	111	1	no	—
42	C₃²⋊C₉	—	111	3	no	—
43	C₉²	—	37	2	no	—
44	C₉²	—	111	3	no	—
45	C₉²	#66526	1	1	yes	characteristic, complemented
46	C₃₇	#10624470	1	1	yes	characteristic, complemented, Sylow
47	C₃³	—	37	1	no	—
48	C₃×C₉	—	3	1	no	—
49	C₃×C₉	—	37	2	no	—
50	C₃×C₉	—	37	3	no	—
51	C₃×C₉	—	37	3	no	—
52	C₃×C₉	—	37	3	no	—
53	C₃×C₉	—	111	9	no	—
54	C₃×C₉	—	111	2	no	—
55	C₃×C₉	C₃×#66526	1	1	yes	characteristic
56	C₃²	—	37	3	no	—
57	C₃²	—	333	1	no	—
58	C₃²	C₃²⋊#66526	1	1	yes	characteristic
59	C₉	—	9	1	no	—
60	C₉	—	37	6	no	—
61	C₉	—	333	2	no	—
62	C₉	—	333	9	no	—
63	C₉	C₉×#66526	1	1	yes	characteristic
64	C₉	#15688537	1	1	yes	characteristic
65	C₉	#15688538	1	1	yes	characteristic
66	C₃	—	3	1	no	—
67	C₃	—	111	3	no	—
68	C₃	(C₃×#66526)⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#15688537
commutator	C₉×C₃₇	C₉²
Frattini	C₃×C₉	C₃×#66526
Fitting	C₉²×C₃₇	C₉
socle	C₃×C₃₇	#6855
prime radical	C₃²×C₃₇⋊C₃	H(3,𝔽₃)
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	C₉²	#66526
hypocenter	C₃₇	#10624470
LCS	G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
4. #15688537	C₉	semidirect
10. C₉×#66524	C₉	semidirect
11. C₉²×C₃₇	C₉	semidirect
45. C₉²	#66526	semidirect
46. C₃₇	#10624470	semidirect

Maximal subgroups

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

subgroup	index
#22582722	3
#22582536	3
#10624470	37