Subgroups — #41008296

Order 26973.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²⋊#11758603	C₃	1	3	yes	maximal, complemented
2	C₃×#15688530	C₃	1	1	yes	maximal, characteristic
3	C₃×#11758603	—	3	3	no	—
4	#15688530	—	3	9	no	—
5	C₃×#11758608	—	3	2	no	—
6	C₃×#11758608	C₉	1	2	yes	—
7	C₃²×#66524	—	3	1	no	—
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
10	C₃³×C₃₇	—	3	1	no	—
11	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
12	H(3,𝔽₃)×C₃₇	—	3	3	no	—
13	#11758608	—	3	18	no	—
14	#11758608	—	3	6	no	—
15	C₃×#66524	—	3	2	no	—
16	C₃×#66524	—	3	9	no	—
17	C₃×#66524	—	3	3	no	—
18	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×#66524	3₋¹⁺²	1	2	yes	—
20	#11758603	—	9	3	no	—
21	#10624459	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	3	1	no	—
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	3	no	—
25	C₃²×C₃₇	—	9	3	no	—
26	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
27	#66524	—	3	6	no	—
28	#66524	—	3	2	no	—
29	#66524	AΣL(1,𝔽₂₇)	1	3	yes	—
30	C₃×#737	—	37	1	no	—
31	C₃²⋊C₂₇	—	37	3	no	—
32	C₃×C₃₇	—	9	1	no	—
33	C₃×C₃₇	—	9	3	no	—
34	C₃×C₃₇	#6866	1	1	yes	characteristic
35	C₃×C₃₇	#6873	1	3	yes	—
36	#737	—	111	9	no	—
37	C₃×3₋¹⁺²	—	37	2	no	—
38	C₃×3₋¹⁺²	—	111	2	no	—
39	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic
40	C₃²×C₉	—	37	1	no	—
41	C₃²×C₉	—	111	1	no	—
42	C₃×C₂₇	—	111	3	no	—
43	C₃₇	#10624459	1	1	yes	characteristic, complemented, Sylow
44	C₃³	—	3	1	no	—
45	C₃³	C₃×#66526	1	1	yes	characteristic
46	3₋¹⁺²	—	111	18	no	—
47	3₋¹⁺²	—	111	6	no	—
48	H(3,𝔽₃)	—	3	3	no	—
49	C₃×C₉	—	37	1	no	—
50	C₃×C₉	—	37	2	no	—
51	C₃×C₉	—	111	3	no	—
52	C₃×C₉	—	111	9	no	—
53	C₃×C₉	—	111	2	no	—
54	C₂₇	—	333	3	no	—
55	C₃²	—	3	1	no	—
56	C₃²	—	3	3	no	—
57	C₃²	—	3	3	no	—
58	C₃²	—	9	3	no	—
59	C₃²	C₃²⋊#66526	1	1	yes	characteristic
60	C₉	—	37	3	no	—
61	C₉	—	111	2	no	—
62	C₉	—	111	6	no	—
63	C₃	—	9	1	no	—
64	C₃	—	9	3	no	—
65	C₃	#22582691	1	1	yes	characteristic
66	C₃	#22582701	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃³	C₃×#66526
Fitting, prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
cosocle	C₃²×#66524	C₃²
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	#10624459
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²⋊#11758603	C₃	semidirect
43. C₃₇	#10624459	semidirect

Maximal subgroups

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

subgroup	index
C₃²⋊#11758603	3
C₃×#15688530	3
#10624459	37