Subgroups — #41006900

Order 26973.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582568	C₃	1	3	yes	maximal
2	C₃×#15688505	C₃	1	1	yes	maximal, characteristic
3	3₋¹⁺²⋊#66524	C₃	1	6	yes	maximal
4	#22582472	C₃	1	2	yes	maximal
5	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₃²	1	2	yes	—
7	C₃²×#66524	C₃²	1	1	yes	characteristic
8	#15688505	—	3	3	no	—
9	#15688504	—	3	9	no	—
10	C₃²⋊#66524	C₃²	1	6	yes	—
11	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃₇×#733	—	3	3	no	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	3₋¹⁺²×C₃₇	—	3	3	no	—
16	3₋¹⁺²×C₃₇	—	3	6	no	—
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	3	2	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
21	C₃×#66524	—	3	6	no	—
22	C₃×#66524	—	3	2	no	—
23	C₃×#66524	—	9	1	no	—
24	C₃×#66524	—	9	2	no	—
25	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
26	#10624791	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	3	2	no	—
29	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
31	C₃²×C₃₇	#737	1	2	yes	—
32	C₉×C₃₇	—	3	3	no	—
33	C₉×C₃₇	—	9	3	no	—
34	#66524	—	3	3	no	—
35	#66524	—	9	6	no	—
36	#66524	—	9	2	no	—
37	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic
38	#6894	—	37	3	no	—
39	C₃×C₉⋊C₉	—	37	1	no	—
40	3₋¹⁺²⋊C₉	—	37	6	no	—
41	#6868	—	37	2	no	—
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	—	3	2	no	—
44	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₃₇	#6912	1	1	yes	characteristic
46	C₃×C₃₇	#6913	1	2	yes	—
47	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
48	C₃²×C₉	—	37	1	no	—
49	C₃²×C₉	—	37	2	no	—
50	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	#733	—	3	3	no	—
52	C₉⋊C₉	—	111	9	no	—
53	C₉⋊C₉	—	111	3	no	—
54	C₃²⋊C₉	—	37	6	no	—
55	C₃₇	#10624791	1	1	yes	characteristic, complemented, Sylow
56	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
57	3₋¹⁺²	—	3	3	no	—
58	3₋¹⁺²	—	3	6	no	—
59	C₃×C₉	—	3	1	no	—
60	C₃×C₉	—	3	2	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	37	3	no	—
63	C₃×C₉	—	333	2	no	—
64	C₃×C₉	—	333	1	no	—
65	C₃×C₉	—	111	6	no	—
66	C₃×C₉	—	111	2	no	—
67	C₃×C₉	#11758608	1	3	yes	—
68	C₃²	—	3	1	no	—
69	C₃²	—	3	2	no	—
70	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
71	C₃²	C₃×#11758608	1	1	yes	characteristic
72	C₃²	#15688530	1	2	yes	—
73	C₉	—	3	3	no	—
74	C₉	—	9	3	no	—
75	C₉	—	333	2	no	—
76	C₉	—	333	6	no	—
77	C₉	—	111	3	no	—
78	C₃	—	3	1	no	—
79	C₃	—	3	2	no	—
80	C₃	#22582556	1	1	yes	characteristic
81	C₃	#22582649	1	1	yes	characteristic
82	C₃	#22582662	1	2	yes	—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, prime radical, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	#10624791
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

Complements

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

N	H ≅ G/N	type
55. C₃₇	#10624791	semidirect

Maximal subgroups

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

subgroup	index
#22582568	3
C₃×#15688505	3
3₋¹⁺²⋊#66524	3
#22582472	3
C₃×C₃₇×#733	3
#10624791	37