Subgroups — #41004790

Order 26973.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758601	C₃	1	3	yes	maximal
2	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃×#11758601	—	3	3	no	—
4	C₃×#11758601	—	3	3	no	—
5	C₃×#11758601	—	3	3	no	—
6	C₃×#11758601	C₉	1	9	yes	complemented
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	6	no	—
14	C₃×C₉×C₃₇	C₂₇	1	9	yes	complemented
15	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
16	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
18	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
19	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
20	#11758601	—	3	9	no	—
21	#10624463	—	37	1	no	maximal, Sylow
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	C₃²×C₃₇	C₃×C₂₇	1	3	yes	—
27	C₃²×C₃₇	C₂₇⋊C₃	1	6	yes	—
28	C₉×C₃₇	—	3	9	no	—
29	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
30	C₉×C₃₇	C₉⋊C₉	1	6	yes	—
31	C₃²×C₂₇	—	37	3	no	—
32	C₃×C₉²	C₃₇⋊C₃	1	1	yes	characteristic
33	C₃×C₃₇	#6855	1	1	yes	characteristic
34	C₃×C₃₇	C₉×C₂₇	1	1	yes	characteristic
35	C₃×C₃₇	#6864	1	2	yes	—
36	C₃×C₃₇	C₃²⋊C₂₇	1	3	yes	—
37	C₃×C₃₇	C₉⋊C₂₇	1	3	yes	—
38	C₃×C₃₇	C₉⋊C₂₇	1	3	yes	—
39	C₃²×C₉	#66524	1	3	yes	—
40	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
41	C₃×C₂₇	—	37	9	no	—
42	C₃×C₂₇	—	111	3	no	—
43	C₃×C₂₇	—	111	3	no	—
44	C₃×C₂₇	—	111	3	no	—
45	C₉²	—	3	2	no	—
46	C₉²	—	3	1	no	—
47	C₃₇	#10624463	1	1	yes	characteristic, complemented, Sylow
48	C₃³	C₃×#66524	1	1	yes	characteristic
49	C₃×C₉	—	3	3	no	—
50	C₃×C₉	—	3	6	no	—
51	C₃×C₉	#11758601	1	9	yes	complemented
52	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
53	C₃×C₉	#11758605	1	3	yes	—
54	C₃×C₉	#11758606	1	3	yes	—
55	C₃×C₉	#11758608	1	2	yes	—
56	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
57	C₂₇	—	111	9	no	—
58	C₃²	C₉×#66524	1	1	yes	characteristic
59	C₃²	C₃²⋊#66524	1	1	yes	characteristic
60	C₃²	#15688505	1	1	yes	characteristic
61	C₃²	#15688506	1	1	yes	characteristic
62	C₃²	C₃×#11758601	1	3	yes	—
63	C₃²	#15688511	1	6	yes	—
64	C₉	—	3	9	no	—
65	C₉	#15688502	1	3	yes	—
66	C₉	#15688504	1	6	yes	—
67	C₃	#22582441	1	1	yes	characteristic
68	C₃	C₉×#11758601	1	1	yes	characteristic
69	C₃	#22582464	1	2	yes	—
70	C₃	C₃²⋊#11758601	1	3	yes	—
71	C₃	#22582489	1	3	yes	—
72	C₃	#22582490	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
commutator	C₃×C₃₇	C₉×C₂₇
Fitting	C₃×C₉²×C₃₇	C₃
socle, prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×C₉²	C₃₇⋊C₃
hypocenter	C₃₇	#10624463
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃²×C₉ ◁ C₃×C₉²

Complements

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

N	H ≅ G/N	type
6. C₃×#11758601	C₉	semidirect
14. C₃×C₉×C₃₇	C₂₇	semidirect
47. C₃₇	#10624463	semidirect
51. C₃×C₉	#11758601	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#11758601	3
C₃×C₉²×C₃₇	3
#10624463	37