Subgroups — #41005319

Order 26973.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#11758608	C₃	1	3	yes	maximal
2	#22582458	C₃	1	3	yes	maximal
3	#22582455	C₃	1	3	yes	maximal
4	#22582449	C₃	1	3	yes	maximal
5	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	C₃²	1	3	yes	—
7	#15688504	—	3	6	no	—
8	C₃²⋊#66524	C₃²	1	3	yes	—
9	C₃²⋊#66524	C₃²	1	3	yes	—
10	C₉×#66524	—	3	3	no	—
11	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
12	C₃₇×C₉⋊C₉	—	3	2	no	—
13	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
14	C₉²×C₃₇	—	3	1	no	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃×C₉×C₃₇	—	3	3	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
19	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
20	#11758608	—	3	9	no	—
21	C₃×#66524	—	3	3	no	—
22	C₃×#66524	—	3	3	no	—
23	C₃×#66524	—	3	3	no	—
24	#10624588	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	1	no	—
26	C₃²×C₃₇	—	3	3	no	—
27	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₉×C₃₇	—	3	2	no	—
29	C₉×C₃₇	—	9	3	no	—
30	C₉×C₃₇	#733	1	3	yes	—
31	#66524	—	3	9	no	—
32	#66524	—	9	3	no	—
33	#66524	—	9	3	no	—
34	C₉×3₋¹⁺²	—	37	3	no	—
35	#6887	C₃₇⋊C₃	1	1	yes	characteristic
36	#6861	—	37	3	no	—
37	#6860	—	37	3	no	—
38	#6858	—	37	3	no	—
39	C₃×C₃₇	—	9	1	no	—
40	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
41	C₃×C₃₇	#6908	1	3	yes	—
42	C₃×3₋¹⁺²	—	37	3	no	—
43	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
44	C₉⋊C₉	—	3	2	no	—
45	C₉⋊C₉	—	111	6	no	—
46	C₃²⋊C₉	—	37	3	no	—
47	C₃²⋊C₉	—	37	3	no	—
48	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
49	C₉²	—	3	1	no	—
50	C₉²	—	111	3	no	—
51	C₃₇	#10624588	1	1	yes	characteristic, complemented, Sylow
52	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
53	3₋¹⁺²	—	111	9	no	—
54	C₃×C₉	—	3	3	no	—
55	C₃×C₉	—	3	3	no	—
56	C₃×C₉	—	111	3	no	—
57	C₃×C₉	—	111	3	no	—
58	C₃×C₉	—	111	3	no	—
59	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
60	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
61	C₃²	—	3	1	no	—
62	C₃²	—	3	3	no	—
63	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
64	C₉	—	3	2	no	—
65	C₉	—	9	3	no	—
66	C₉	—	333	3	no	—
67	C₉	—	333	3	no	—
68	C₉	—	111	9	no	—
69	C₉	#15688521	1	3	yes	—
70	C₃	—	9	1	no	—
71	C₃	C₃×#15688521	1	1	yes	characteristic
72	C₃	#22582630	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator, prime radical, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624588
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
51. C₃₇	#10624588	semidirect

Maximal subgroups

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

subgroup	index
C₉×#11758608	3
#22582458	3
#22582455	3
#22582449	3
C₃₇×#6887	3
#10624588	37