Subgroups — #41006845

Order 26973.

238 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×C₉×#66524	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	#15688504	—	3	6	no	—
9	#15688504	—	3	3	no	—
10	C₃²⋊#66524	C₃²	1	6	yes	—
11	C₉×#66524	—	3	2	no	—
12	C₉×#66524	—	3	1	no	—
13	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
14	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
15	C₃₇×#733	—	3	3	no	—
16	C₃³×C₃₇	C₃³	1	1	yes	characteristic
17	3₋¹⁺²×C₃₇	—	3	3	no	—
18	3₋¹⁺²×C₃₇	—	3	6	no	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	—	3	2	no	—
21	C₃×C₉×C₃₇	—	3	3	no	—
22	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
23	C₃×#66524	—	3	2	no	—
24	C₃×#66524	—	3	3	no	—
25	C₃×#66524	—	3	6	no	—
26	C₃×#66524	—	3	6	no	—
27	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
28	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
29	#10624787	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
34	C₃²×C₃₇	#737	1	2	yes	—
35	C₉×C₃₇	—	3	3	no	—
36	C₉×C₃₇	—	9	3	no	—
37	#66524	—	3	2	no	—
38	#66524	—	3	6	no	—
39	#66524	—	9	6	no	—
40	#66524	#733	1	3	yes	complemented
41	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic
42	#6894	—	37	3	no	—
43	C₃×C₉²	—	37	1	no	—
44	3₋¹⁺²⋊C₉	—	37	6	no	—
45	#6868	—	37	2	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	—	3	2	no	—
48	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
50	C₃×C₃₇	#6908	1	2	yes	—
51	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
52	C₃²×C₉	—	37	1	no	—
53	C₃²×C₉	—	37	2	no	—
54	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
55	#733	—	3	3	no	—
56	C₉⋊C₉	—	111	3	no	—
57	C₉⋊C₉	—	111	6	no	—
58	C₃²⋊C₉	—	37	6	no	—
59	C₉²	—	111	1	no	—
60	C₉²	—	111	2	no	—
61	C₃₇	#10624787	1	1	yes	characteristic, complemented, Sylow
62	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
63	3₋¹⁺²	—	3	3	no	—
64	3₋¹⁺²	—	3	6	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	2	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	37	1	no	—
69	C₃×C₉	—	37	2	no	—
70	C₃×C₉	—	111	3	no	—
71	C₃×C₉	—	111	6	no	—
72	C₃×C₉	—	111	2	no	—
73	C₃×C₉	—	111	6	no	—
74	C₃×C₉	#11758608	1	3	yes	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
78	C₃²	C₃×#11758608	1	1	yes	characteristic
79	C₃²	#15688530	1	2	yes	—
80	C₉	—	3	3	no	—
81	C₉	—	9	3	no	—
82	C₉	—	37	3	no	—
83	C₉	—	333	6	no	—
84	C₉	—	111	2	no	—
85	C₉	—	111	6	no	—
86	C₃	—	3	1	no	—
87	C₃	—	3	2	no	—
88	C₃	#22582556	1	1	yes	characteristic
89	C₃	#733×C₃₇⋊C₃	1	1	yes	characteristic
90	C₃	#22582634	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	#10624787
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
40. #66524	#733	semidirect
61. C₃₇	#10624787	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×C₉×#66524	3
3₋¹⁺²⋊#66524	3
#22582472	3
C₃×C₃₇×#733	3
#10624787	37