Subgroups — #41005009

Order 26973.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582474	C₃	1	3	yes	maximal
2	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic
3	#15688511	—	3	3	no	—
4	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
5	C₃₇×C₉⋊C₉	—	3	1	no	—
6	C₃₇×C₉⋊C₉	C₉	1	3	yes	—
7	C₃₇×C₃²⋊C₉	—	3	1	no	—
8	C₉²×C₃₇	—	3	1	no	—
9	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
10	C₃×C₉×C₃₇	—	3	1	no	—
11	C₃×C₉×C₃₇	—	3	2	no	—
12	C₃×C₉×C₃₇	—	3	3	no	—
13	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
16	#11758601	—	9	3	no	—
17	#10624522	—	37	1	no	maximal, Sylow
18	C₃²×C₃₇	—	3	1	no	—
19	C₃²×C₃₇	—	9	1	no	—
20	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
21	C₉×C₃₇	—	3	1	no	—
22	C₉×C₃₇	—	3	1	no	—
23	C₉×C₃₇	—	9	3	no	—
24	C₉×C₃₇	#732	1	1	yes	characteristic
25	C₉×C₃₇	#733	1	2	yes	—
26	#6887	C₃₇⋊C₃	1	1	yes	characteristic
27	#6869	—	37	3	no	—
28	C₃×C₃₇	—	3	1	no	—
29	C₃×C₃₇	—	9	1	no	—
30	C₃×C₃₇	#6868	1	1	yes	characteristic
31	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
32	C₂₇⋊C₃	—	111	3	no	—
33	C₉⋊C₉	—	3	1	no	—
34	C₉⋊C₉	#66524	1	3	yes	—
35	C₃²⋊C₉	—	3	1	no	—
36	C₉²	—	3	1	no	—
37	C₃₇	#10624522	1	1	yes	characteristic, complemented, Sylow
38	C₃³	#11758608	1	1	yes	characteristic
39	C₃×C₉	—	3	1	no	—
40	C₃×C₉	—	3	2	no	—
41	C₃×C₉	—	3	3	no	—
42	C₃×C₉	C₃×#66524	1	1	yes	characteristic
43	C₃×C₉	#11758608	1	1	yes	characteristic
44	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
45	C₂₇	—	333	3	no	—
46	C₃²	—	3	1	no	—
47	C₃²	—	9	1	no	—
48	C₃²	C₃²⋊#66524	1	1	yes	characteristic
49	C₉	—	3	1	no	—
50	C₉	—	3	1	no	—
51	C₉	—	9	3	no	—
52	C₉	#15688519	1	1	yes	characteristic
53	C₉	#15688521	1	2	yes	—
54	C₃	—	3	1	no	—
55	C₃	—	9	1	no	—
56	C₃	#22582472	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#15688519
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃×C₃₇	#6868
prime radical	C₃³×C₃₇	3₋¹⁺²
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624522
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
37. C₃₇	#10624522	semidirect

Maximal subgroups

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

subgroup	index
#22582474	3
C₃₇×#6887	3
#10624522	37