Subgroups — #41006428

Order 26973.

250 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688521	C₃	1	1	yes	maximal, characteristic
2	C₉×#11758608	C₃	1	3	yes	maximal
3	#22582480	C₃	1	6	yes	maximal
4	#22582472	C₃	1	2	yes	maximal
5	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	C₃²	1	3	yes	—
7	#15688521	C₉	1	9	yes	complemented
8	#15688504	—	3	6	no	—
9	C₃²⋊#66524	C₃²	1	6	yes	—
10	C₉×#66524	—	3	3	no	—
11	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
13	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
14	C₃₇×C₉⋊C₉	—	3	3	no	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	2	no	—
18	C₃×C₉×C₃₇	—	3	2	no	—
19	C₃×C₉×C₃₇	—	9	1	no	—
20	C₃×C₉×C₃₇	—	9	2	no	—
21	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
22	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
23	#11758608	—	3	6	no	—
24	#11758608	C₃×C₉	1	9	yes	—
25	C₃×#66524	—	3	3	no	—
26	C₃×#66524	—	3	6	no	—
27	#10624723	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	1	no	—
29	C₃²×C₃₇	—	3	2	no	—
30	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
31	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₃²×C₃₇	#737	1	2	yes	—
33	C₉×C₃₇	—	3	3	no	—
34	C₉×C₃₇	—	3	3	no	—
35	C₉×C₃₇	—	9	2	no	—
36	#66524	—	3	9	no	—
37	#66524	—	9	6	no	—
38	C₃×#733	—	37	1	no	—
39	C₉×3₋¹⁺²	—	37	3	no	—
40	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
41	3₋¹⁺²⋊C₉	—	37	6	no	—
42	#6868	—	37	2	no	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	3	2	no	—
45	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
46	C₃×C₃₇	#6912	1	1	yes	characteristic
47	C₃×C₃₇	#6913	1	2	yes	—
48	C₃×3₋¹⁺²	—	37	3	no	—
49	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
50	C₃²×C₉	C₃×C₃₇⋊C₃	1	2	yes	—
51	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	#733	—	37	9	no	—
53	C₉⋊C₉	—	3	3	no	—
54	C₉⋊C₉	—	111	6	no	—
55	C₃²⋊C₉	—	37	6	no	—
56	C₉²	—	111	3	no	—
57	C₃₇	#10624723	1	1	yes	characteristic, complemented, Sylow
58	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
59	3₋¹⁺²	—	37	9	no	—
60	3₋¹⁺²	—	111	6	no	—
61	C₃×C₉	—	3	1	no	—
62	C₃×C₉	—	3	2	no	—
63	C₃×C₉	—	3	2	no	—
64	C₃×C₉	—	9	1	no	—
65	C₃×C₉	—	9	2	no	—
66	C₃×C₉	—	111	6	no	—
67	C₃×C₉	—	111	3	no	—
68	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
69	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	2	no	—
72	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
73	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
74	C₃²	#15688531	1	2	yes	—
75	C₉	—	3	3	no	—
76	C₉	—	3	3	no	—
77	C₉	—	9	2	no	—
78	C₉	—	333	6	no	—
79	C₉	—	111	9	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	2	no	—
82	C₃	C₉×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
83	C₃	#22582652	1	1	yes	characteristic
84	C₃	#22582661	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical, cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624723
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

Complements

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

N	H ≅ G/N	type
7. #15688521	C₉	semidirect
57. C₃₇	#10624723	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688521	3
C₉×#11758608	3
#22582480	3
#22582472	3
C₃×C₃₇×C₉⋊C₉	3
#10624723	37