Subgroups — #41004798

Order 26973.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688511	C₃	1	3	yes	maximal
2	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	#15688511	—	3	3	no	—
4	#15688511	—	3	3	no	—
5	#15688511	C₉	1	9	yes	complemented
6	C₃×#11758601	—	3	3	no	—
7	C₃²×C₉×C₃₇	C₉	1	3	yes	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₉²×C₃₇	—	3	1	no	—
10	C₉²×C₃₇	—	3	2	no	—
11	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
12	C₃×C₉×C₃₇	—	3	3	no	—
13	C₃×C₉×C₃₇	—	3	9	no	—
14	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
15	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
17	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
18	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
19	#11758601	—	3	9	no	—
20	#10624467	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	—	3	3	no	—
22	C₃²×C₃₇	C₉²	1	1	yes	characteristic
23	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
24	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
25	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
26	C₉×C₃₇	—	3	9	no	—
27	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
28	C₉×C₃₇	C₉⋊C₉	1	6	yes	—
29	C₃×C₂₇⋊C₃	—	37	3	no	—
30	C₃×C₉²	C₃₇⋊C₃	1	1	yes	characteristic
31	C₃×C₃₇	—	3	1	no	—
32	C₃×C₃₇	—	3	1	no	—
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	#6855	1	1	yes	characteristic
35	C₃×C₃₇	#6869	1	3	yes	—
36	C₃²×C₉	#66524	1	3	yes	—
37	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
38	C₂₇⋊C₃	—	37	9	no	—
39	C₂₇⋊C₃	—	111	3	no	—
40	C₂₇⋊C₃	—	111	3	no	—
41	C₃×C₂₇	—	111	3	no	—
42	C₉²	—	3	2	no	—
43	C₉²	—	3	1	no	—
44	C₃₇	#10624467	1	1	yes	characteristic, complemented, Sylow
45	C₃³	C₃×#66524	1	1	yes	characteristic
46	C₃×C₉	—	3	3	no	—
47	C₃×C₉	—	3	9	no	—
48	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
49	C₃×C₉	#11758605	1	3	yes	—
50	C₃×C₉	#11758606	1	3	yes	—
51	C₃×C₉	#11758608	1	2	yes	—
52	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
53	C₂₇	—	111	9	no	—
54	C₃²	—	3	3	no	—
55	C₃²	C₉×#66524	1	1	yes	characteristic
56	C₃²	C₃²⋊#66524	1	1	yes	characteristic
57	C₃²	#15688505	1	1	yes	characteristic
58	C₃²	#15688506	1	1	yes	characteristic
59	C₉	—	3	9	no	—
60	C₉	#15688502	1	3	yes	—
61	C₉	#15688504	1	6	yes	—
62	C₃	—	3	1	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	1	no	—
65	C₃	#22582441	1	1	yes	characteristic
66	C₃	#22582474	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₉²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₉²×C₃₇	C₃
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×C₉²	C₃₇⋊C₃
hypocenter	C₃₇	#10624467
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×C₉²

Complements

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

N	H ≅ G/N	type
5. #15688511	C₉	semidirect
44. C₃₇	#10624467	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688511	3
C₃×C₉²×C₃₇	3
#10624467	37