Subgroups — #41004995

Order 26973.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582472	C₃	1	3	yes	maximal
2	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66524	—	3	3	no	—
4	C₃²×C₉×C₃₇	—	3	1	no	—
5	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
6	C₉²×C₃₇	—	3	2	no	—
7	C₉²×C₃₇	C₉	1	3	yes	complemented
8	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
9	C₃×C₉×C₃₇	—	3	9	no	—
10	C₃×C₉×C₃₇	—	3	3	no	—
11	C₃×C₉×C₃₇	—	3	1	no	—
12	C₃×C₉×C₃₇	—	3	2	no	—
13	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
15	C₃×#66524	—	9	3	no	—
16	#10624518	—	37	1	no	maximal, Sylow
17	C₃²×C₃₇	—	3	3	no	—
18	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
19	C₃²×C₃₇	#732	1	1	yes	characteristic
20	C₃²×C₃₇	#733	1	2	yes	—
21	C₉×C₃₇	—	3	9	no	—
22	C₉×C₃₇	—	3	1	no	—
23	C₉×C₃₇	—	3	2	no	—
24	#66524	—	27	3	no	—
25	C₃×C₉²	C₃₇⋊C₃	1	1	yes	characteristic
26	#6868	—	37	3	no	—
27	C₃×C₃₇	—	3	1	no	—
28	C₃×C₃₇	—	3	2	no	—
29	C₃×C₃₇	#6868	1	1	yes	characteristic
30	C₃×C₃₇	#6878	1	3	yes	—
31	C₃²×C₉	—	3	1	no	—
32	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
33	C₃²⋊C₉	—	111	3	no	—
34	C₉²	—	3	2	no	—
35	C₉²	#66524	1	3	yes	complemented
36	C₃₇	#10624518	1	1	yes	characteristic, complemented, Sylow
37	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
38	C₃×C₉	—	3	9	no	—
39	C₃×C₉	—	3	3	no	—
40	C₃×C₉	—	3	1	no	—
41	C₃×C₉	—	3	2	no	—
42	C₃×C₉	—	333	3	no	—
43	C₃×C₉	C₃×#66524	1	1	yes	characteristic
44	C₃×C₉	#11758608	1	2	yes	—
45	C₃²	—	3	3	no	—
46	C₃²	C₃²⋊#66524	1	1	yes	characteristic
47	C₃²	#15688519	1	1	yes	characteristic
48	C₃²	#15688521	1	2	yes	—
49	C₉	—	3	9	no	—
50	C₉	—	3	1	no	—
51	C₉	—	3	2	no	—
52	C₉	—	999	3	no	—
53	C₃	—	3	1	no	—
54	C₃	—	3	2	no	—
55	C₃	#22582472	1	1	yes	characteristic
56	C₃	#22582502	1	3	yes	—

	subgroup	G/H
center	C₃²	#15688519
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₉²×C₃₇	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×C₉²	C₃₇⋊C₃
hypocenter	C₃₇	#10624518
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×C₉²

Complements

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

N	H ≅ G/N	type
7. C₉²×C₃₇	C₉	semidirect
35. C₉²	#66524	semidirect
36. C₃₇	#10624518	semidirect

Maximal subgroups

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

subgroup	index
#22582472	3
C₃×C₉²×C₃₇	3
#10624518	37