Subgroups — #41004981

Order 26973.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582538	C₃	1	1	yes	maximal, characteristic
2	#22582474	C₃	1	2	yes	maximal
3	C₃₇×#6869	C₃	1	1	yes	maximal, characteristic
4	#15688511	—	3	2	no	—
5	#15688506	—	3	1	no	—
6	#15688505	—	3	1	no	—
7	C₃²⋊#66524	—	3	1	no	—
8	C₉×#66524	C₉	1	3	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
10	C₃₇×C₂₇⋊C₃	—	3	1	no	—
11	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
12	C₃×C₉×C₃₇	—	3	1	no	—
13	C₃×C₉×C₃₇	—	3	2	no	—
14	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
17	C₃×#66524	—	3	3	no	—
18	C₂₇×C₃₇	—	9	1	no	—
19	#11758601	—	9	2	no	—
20	#10624514	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	—	3	1	no	—
22	C₃²×C₃₇	—	9	1	no	—
23	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
24	C₉×C₃₇	—	3	1	no	—
25	C₉×C₃₇	—	3	1	no	—
26	C₉×C₃₇	#731	1	1	yes	characteristic
27	C₉×C₃₇	#731	1	2	yes	—
28	#66524	—	9	3	no	—
29	#6887	—	37	1	no	—
30	#6869	—	37	2	no	—
31	#6869	C₃₇⋊C₃	1	1	yes	characteristic
32	C₃×C₃₇	—	3	1	no	—
33	C₃×C₃₇	—	9	1	no	—
34	C₃×C₃₇	#6867	1	1	yes	characteristic
35	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
36	C₂₇⋊C₃	—	3	1	no	—
37	C₂₇⋊C₃	—	111	2	no	—
38	C₉⋊C₉	—	111	1	no	—
39	C₉⋊C₉	—	111	1	no	—
40	C₃²⋊C₉	—	111	1	no	—
41	C₉²	—	37	3	no	—
42	C₃₇	#10624514	1	1	yes	characteristic, complemented, Sylow
43	C₃³	#11758605	1	1	yes	characteristic
44	C₃×C₉	—	3	1	no	—
45	C₃×C₉	—	3	2	no	—
46	C₃×C₉	—	111	3	no	—
47	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
48	C₃×C₉	#11758606	1	1	yes	characteristic
49	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
50	C₂₇	—	9	1	no	—
51	C₂₇	—	333	2	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	9	1	no	—
54	C₃²	#15688502	1	1	yes	characteristic
55	C₉	—	3	1	no	—
56	C₉	—	3	1	no	—
57	C₉	—	333	3	no	—
58	C₉	#15688515	1	1	yes	characteristic
59	C₉	#15688517	1	2	yes	—
60	C₃	—	3	1	no	—
61	C₃	—	9	1	no	—
62	C₃	#22582469	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582469
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6869	C₃
socle	C₃×C₃₇	#6867
prime radical	C₃³×C₃₇	3₋¹⁺²
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6869	C₃₇⋊C₃
hypocenter	C₃₇	#10624514
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6869

Complements

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

N	H ≅ G/N	type
42. C₃₇	#10624514	semidirect

Maximal subgroups

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

subgroup	index
#22582538	3
#22582474	3
C₃₇×#6869	3
#10624514	37