Subgroups — #41004841

Order 26973.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688521	C₃	1	1	yes	maximal, characteristic
2	#22582470	C₃	1	2	yes	maximal
3	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	—	3	1	no	—
5	#15688521	—	3	3	no	—
6	C₃²⋊#66524	—	3	2	no	—
7	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₃₇×#733	—	3	3	no	—
10	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
11	3₋¹⁺²×C₃₇	—	3	3	no	—
12	3₋¹⁺²×C₃₇	—	9	2	no	—
13	C₃×C₉×C₃₇	—	3	1	no	—
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	2	no	—
16	C₃×C₉×C₃₇	—	9	1	no	—
17	#11758608	—	3	3	no	—
18	#11758608	—	9	2	no	—
19	C₃×#66524	—	9	1	no	—
20	C₃×#66524	—	9	2	no	—
21	#10624479	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	9	1	no	—
23	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
24	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
25	C₃²×C₃₇	#732	1	2	yes	—
26	C₉×C₃₇	—	9	1	no	—
27	C₉×C₃₇	—	9	3	no	—
28	#66524	—	9	3	no	—
29	#66524	—	27	2	no	—
30	C₃×#733	—	37	1	no	—
31	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic
32	#6867	—	37	2	no	—
33	C₃×C₃₇	—	3	2	no	—
34	C₃×C₃₇	—	3	1	no	—
35	C₃×C₃₇	—	3	1	no	—
36	C₃×C₃₇	#6856	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	3	1	no	—
38	C₃×3₋¹⁺²	—	111	1	no	—
39	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
40	#733	—	3	3	no	—
41	#733	—	111	3	no	—
42	C₃²⋊C₉	—	111	2	no	—
43	C₃₇	#10624479	1	1	yes	characteristic, complemented, Sylow
44	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
45	3₋¹⁺²	—	3	3	no	—
46	3₋¹⁺²	—	9	2	no	—
47	3₋¹⁺²	—	333	2	no	—
48	3₋¹⁺²	—	111	3	no	—
49	C₃×C₉	—	3	1	no	—
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	3	2	no	—
52	C₃×C₉	—	9	1	no	—
53	C₃×C₉	—	333	1	no	—
54	C₃×C₉	—	333	2	no	—
55	C₃²	—	9	1	no	—
56	C₃²	#15688512	1	1	yes	characteristic
57	C₃²	#15688513	1	1	yes	characteristic
58	C₃²	#15688518	1	2	yes	—
59	C₉	—	9	1	no	—
60	C₉	—	9	3	no	—
61	C₉	—	333	3	no	—
62	C₉	—	999	2	no	—
63	C₃	—	3	2	no	—
64	C₃	—	3	1	no	—
65	C₃	—	3	1	no	—
66	C₃	#22582442	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582442
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃×C₃₇	#6856
prime radical	C₃³×C₃₇	H(3,𝔽₃)
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	#10624479
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#733

Complements

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

N	H ≅ G/N	type
43. C₃₇	#10624479	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688521	3
#22582470	3
C₃×C₃₇×#733	3
#10624479	37