Subgroups — #41008293

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×#15688532	C₃	1	1	yes	maximal, characteristic
3	C₃×#11758603	—	3	3	no	—
4	#15688532	—	3	9	no	—
5	C₃×#11758608	C₉	1	1	yes	characteristic
6	C₃×#11758605	—	3	1	no	—
7	C₃×#11758606	—	3	1	no	—
8	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
9	C₃²×#66524	C₃²	1	1	yes	characteristic
10	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
11	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	—	3	1	no	—
14	3₋¹⁺²×C₃₇	—	3	3	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
17	#11758608	—	3	3	no	—
18	C₃×#66524	—	3	1	no	—
19	C₃×#66524	—	3	3	no	—
20	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
22	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
23	#11758606	—	3	9	no	—
24	#11758605	—	3	9	no	—
25	C₉×C₃₇⋊C₃	—	3	9	no	—
26	#11758603	—	9	3	no	—
27	#10624456	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	1	no	—
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	—	3	3	no	—
32	C₃×C₃₇⋊C₃	—	9	3	no	—
33	C₉×C₃₇	—	3	3	no	—
34	#66524	—	3	3	no	—
35	#66524	—	3	1	no	—
36	#66524	—	3	1	no	—
37	#66524	AΣL(1,𝔽₂₇)	1	3	yes	—
38	C₃×#737	—	37	1	no	—
39	C₃²⋊C₂₇	—	37	3	no	—
40	C₃×C₃₇	—	9	1	no	—
41	C₃×C₃₇	#6866	1	1	yes	characteristic
42	C₃×C₃₇	#6872	1	3	yes	—
43	C₃₇⋊C₃	—	9	3	no	—
44	#737	—	111	9	no	—
45	C₃×3₋¹⁺²	—	37	1	no	—
46	C₃×3₋¹⁺²	—	111	1	no	—
47	C₃×3₋¹⁺²	—	111	1	no	—
48	C₃×3₋¹⁺²	#66526	1	1	yes	characteristic
49	C₃×H(3,𝔽₃)	—	37	1	no	—
50	C₃²×C₉	—	37	1	no	—
51	C₃²×C₉	—	111	1	no	—
52	C₃×C₂₇	—	111	3	no	—
53	C₃₇	#10624456	1	1	yes	characteristic, complemented, Sylow
54	C₃³	—	111	1	no	—
55	C₃³	C₃×#66526	1	1	yes	characteristic
56	3₋¹⁺²	—	3	3	no	—
57	3₋¹⁺²	—	111	9	no	—
58	3₋¹⁺²	—	111	9	no	—
59	3₋¹⁺²	—	111	3	no	—
60	H(3,𝔽₃)	—	111	3	no	—
61	C₃×C₉	—	3	1	no	—
62	C₃×C₉	—	37	1	no	—
63	C₃×C₉	—	37	1	no	—
64	C₃×C₉	—	37	1	no	—
65	C₃×C₉	—	111	9	no	—
66	C₃×C₉	—	111	3	no	—
67	C₃×C₉	—	111	1	no	—
68	C₂₇	—	333	3	no	—
69	C₃²	—	3	1	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	333	3	no	—
72	C₃²	—	111	3	no	—
73	C₃²	C₃²⋊#66526	1	1	yes	characteristic
74	C₉	—	3	3	no	—
75	C₉	—	37	3	no	—
76	C₉	—	111	1	no	—
77	C₉	—	111	1	no	—
78	C₉	—	111	3	no	—
79	C₃	—	9	1	no	—
80	C₃	—	333	3	no	—
81	C₃	#22582691	1	1	yes	characteristic
82	C₃	#22582700	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃³	C₃×#66526
Fitting	C₃×3₋¹⁺²×C₃₇	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃²×#66524	C₃²
hypercenter	C₃×3₋¹⁺²	#66526
hypocenter	C₃₇	#10624456
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃²⋊#11758603	C₃	semidirect
53. C₃₇	#10624456	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×#15688532	3
#10624456	37