Subgroups — #41008375

Order 26973.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582691	C₃	1	3	yes	maximal
2	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66526	—	3	3	no	—
4	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
5	C₃×#11758608	C₉	1	2	yes	complemented
6	C₃×#11758605	—	3	1	no	—
7	C₃×#11758606	—	3	1	no	—
8	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
9	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
10	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic, complemented
11	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
12	C₃²×C₃₇⋊C₃	—	3	3	no	—
13	C₃²×C₃₇⋊C₃	—	9	3	no	—
14	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
16	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
17	C₃×#66526	—	9	3	no	—
18	3₋¹⁺²×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	#11758608	—	3	6	no	—
21	C₃×#66524	—	3	2	no	—
22	#11758606	—	3	9	no	—
23	#11758605	—	3	9	no	—
24	C₉×C₃₇⋊C₃	—	3	9	no	—
25	#10624509	—	37	1	no	maximal, Sylow
26	C₃²×C₃₇	—	3	1	no	—
27	C₃²×C₃₇	—	9	1	no	—
28	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
29	C₃×C₃₇⋊C₃	—	3	1	no	—
30	C₃×C₃₇⋊C₃	—	3	3	no	—
31	C₃×C₃₇⋊C₃	—	3	1	no	—
32	C₃×C₃₇⋊C₃	—	9	1	no	—
33	C₃×C₃₇⋊C₃	—	9	9	no	—
34	C₃×C₃₇⋊C₃	—	9	1	no	—
35	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
36	#66526	—	27	3	no	—
37	C₉×C₃₇	—	3	3	no	—
38	#66524	—	3	6	no	—
39	C₃²×3₋¹⁺²	—	37	1	no	—
40	#6866	—	37	3	no	—
41	C₃×C₃₇	—	3	1	no	—
42	C₃×C₃₇	—	9	1	no	—
43	C₃×C₃₇	#6866	1	1	yes	characteristic
44	C₃₇⋊C₃	—	3	3	no	—
45	C₃₇⋊C₃	—	3	3	no	—
46	C₃₇⋊C₃	—	9	1	no	—
47	C₃⁴	—	37	1	no	—
48	C₃×3₋¹⁺²	—	37	2	no	—
49	C₃×3₋¹⁺²	—	111	1	no	—
50	C₃×3₋¹⁺²	—	111	1	no	—
51	C₃×3₋¹⁺²	—	111	9	no	—
52	C₃×3₋¹⁺²	#66526	1	1	yes	characteristic, complemented
53	C₃²×C₉	—	111	1	no	—
54	C₃²⋊C₉	—	111	3	no	—
55	C₃₇	#10624509	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	37	1	no	—
57	C₃³	—	37	1	no	—
58	C₃³	—	37	1	no	—
59	C₃³	—	333	3	no	—
60	C₃³	—	111	3	no	—
61	C₃³	C₃×#66526	1	1	yes	characteristic
62	3₋¹⁺²	—	3	3	no	—
63	3₋¹⁺²	—	111	9	no	—
64	3₋¹⁺²	—	111	9	no	—
65	3₋¹⁺²	—	111	6	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	333	3	no	—
68	C₃×C₉	—	111	9	no	—
69	C₃×C₉	—	111	2	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	9	1	no	—
72	C₃²	—	37	3	no	—
73	C₃²	—	333	1	no	—
74	C₃²	—	333	1	no	—
75	C₃²	—	333	9	no	—
76	C₃²	—	111	3	no	—
77	C₃²	—	111	1	no	—
78	C₃²	—	111	1	no	—
79	C₃²	C₃²⋊#66526	1	1	yes	characteristic
80	C₉	—	3	3	no	—
81	C₉	—	111	6	no	—
82	C₉	—	999	3	no	—
83	C₃	—	3	1	no	—
84	C₃	—	9	1	no	—
85	C₃	—	333	1	no	—
86	C₃	—	111	3	no	—
87	C₃	—	111	3	no	—
88	C₃	#22582691	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582691
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃³	C₃×#66526
Fitting	C₃×3₋¹⁺²×C₃₇	C₉
socle	C₃×C₃₇	#6866
prime radical, cosocle	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₃×3₋¹⁺²	#66526
hypocenter	C₃₇	#10624509
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
5. C₃×#11758608	C₉	semidirect
10. C₃×3₋¹⁺²×C₃₇	C₉	semidirect
52. C₃×3₋¹⁺²	#66526	semidirect
55. C₃₇	#10624509	semidirect

Maximal subgroups

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

subgroup	index
#22582691	3
C₃×3₋¹⁺²×C₃₇⋊C₃	3
#10624509	37