Subgroups — #41005019

Order 26973.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
2	#22582472	C₃	1	2	yes	maximal
3	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	—	3	1	no	—
5	C₃²⋊#66524	—	3	2	no	—
6	C₉×#66524	—	3	3	no	—
7	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₃₇×#733	—	3	2	no	—
10	C₃₇×#733	C₉	1	3	yes	complemented
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	3₋¹⁺²×C₃₇	—	3	3	no	—
13	3₋¹⁺²×C₃₇	—	9	1	no	—
14	3₋¹⁺²×C₃₇	—	9	1	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	9	1	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
21	C₃×#66524	—	3	6	no	—
22	C₃×#66524	—	3	3	no	—
23	C₃×#66524	—	3	3	no	—
24	C₃×#66524	—	9	2	no	—
25	#10624525	—	37	1	no	maximal, Sylow
26	C₃²×C₃₇	—	3	2	no	—
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
29	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
30	C₃²×C₃₇	#731	1	1	yes	characteristic
31	C₃²×C₃₇	#732	1	1	yes	characteristic
32	C₉×C₃₇	—	3	1	no	—
33	C₉×C₃₇	—	3	2	no	—
34	C₉×C₃₇	—	9	3	no	—
35	#66524	—	3	9	no	—
36	#66524	—	27	2	no	—
37	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic
38	C₃×C₉²	—	37	1	no	—
39	#6868	—	37	2	no	—
40	C₃×C₃₇	—	3	1	no	—
41	C₃×C₃₇	—	3	1	no	—
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
44	C₃×C₃₇	#6878	1	2	yes	—
45	C₃×C₃₇	#6880	1	1	yes	characteristic
46	C₃×3₋¹⁺²	—	3	1	no	—
47	C₃²×C₉	—	111	1	no	—
48	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
49	#733	—	3	2	no	—
50	#733	#66524	1	3	yes	complemented
51	C₃²⋊C₉	—	111	2	no	—
52	C₉²	—	111	3	no	—
53	C₃₇	#10624525	1	1	yes	characteristic, complemented, Sylow
54	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
55	3₋¹⁺²	—	3	3	no	—
56	3₋¹⁺²	—	9	1	no	—
57	3₋¹⁺²	—	9	1	no	—
58	C₃×C₉	—	3	1	no	—
59	C₃×C₉	—	3	1	no	—
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	9	1	no	—
62	C₃×C₉	—	333	2	no	—
63	C₃×C₉	—	111	6	no	—
64	C₃×C₉	—	111	3	no	—
65	C₃×C₉	—	111	3	no	—
66	C₃×C₉	C₃×#66524	1	1	yes	characteristic
67	C₃×C₉	#11758608	1	2	yes	—
68	C₃²	—	3	2	no	—
69	C₃²	—	3	1	no	—
70	C₃²	C₃²⋊#66524	1	1	yes	characteristic
71	C₃²	#15688512	1	1	yes	characteristic
72	C₃²	#15688515	1	1	yes	characteristic
73	C₃²	#15688518	1	1	yes	characteristic
74	C₉	—	3	1	no	—
75	C₉	—	3	2	no	—
76	C₉	—	9	3	no	—
77	C₉	—	111	9	no	—
78	C₉	—	999	2	no	—
79	C₃	—	3	1	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	1	no	—
82	C₃	H(3,𝔽₃)⋊#66524	1	1	yes	characteristic
83	C₃	#22582505	1	2	yes	—
84	C₃	#22582508	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688518
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	#10624525
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#733

Complements

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

N	H ≅ G/N	type
10. C₃₇×#733	C₉	semidirect
50. #733	#66524	semidirect
53. C₃₇	#10624525	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#66524	3
#22582472	3
C₃×C₃₇×#733	3
#10624525	37