Subgroups — #41004947

Order 26973.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582500	C₃	1	3	yes	maximal, complemented
2	C₃²×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic
3	#15688500	—	3	3	no	—
4	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic
5	C₃×C₂₇×C₃₇	—	3	1	no	—
6	C₃×C₂₇×C₃₇	—	3	2	no	—
7	C₃×C₂₇×C₃₇	C₉	1	2	yes	—
8	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
9	C₃³×C₃₇	C₂₇	1	1	yes	characteristic
10	C₃×C₉×C₃₇	—	3	1	no	—
11	C₃×C₉×C₃₇	—	3	2	no	—
12	C₃×C₉×C₃₇	C₂₇	1	2	yes	—
13	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₂₇×C₃₇	—	3	2	no	—
15	C₂₇×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₂₇×C₃₇	3₋¹⁺²	1	2	yes	—
17	#10624502	—	37	1	no	maximal, Sylow
18	C₃²×C₃₇	—	3	3	no	—
19	C₃²×C₃₇	—	3	1	no	—
20	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
21	C₉×C₃₇	—	3	2	no	—
22	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
23	C₉×C₃₇	C₂₇⋊C₃	1	2	yes	—
24	C₃²×C₂₇	C₃₇⋊C₃	1	1	yes	characteristic
25	C₈₁⋊C₃	—	37	3	no	—
26	C₃×C₃₇	—	3	3	no	—
27	C₃×C₃₇	—	3	1	no	—
28	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
29	C₃²×C₉	#66524	1	1	yes	characteristic
30	C₃×C₂₇	—	3	2	no	—
31	C₃×C₂₇	—	3	1	no	—
32	C₃×C₂₇	#66524	1	2	yes	—
33	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
34	C₈₁	—	111	3	no	—
35	C₃₇	#10624502	1	1	yes	characteristic, complemented, Sylow
36	C₃³	#11758601	1	1	yes	characteristic
37	C₃×C₉	—	3	1	no	—
38	C₃×C₉	—	3	2	no	—
39	C₃×C₉	#11758601	1	2	yes	—
40	C₃×C₉	C₃×#66524	1	1	yes	characteristic
41	C₂₇	—	3	2	no	—
42	C₂₇	#11758608	1	2	yes	—
43	C₂₇	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
44	C₃²	—	3	3	no	—
45	C₃²	—	3	1	no	—
46	C₃²	C₃×#11758601	1	1	yes	characteristic
47	C₉	—	3	2	no	—
48	C₉	C₃²⋊#66524	1	1	yes	characteristic
49	C₉	#15688511	1	2	yes	—
50	C₃	—	3	1	no	—
51	C₃	—	3	3	no	—
52	C₃	C₃²⋊#11758601	1	1	yes	characteristic

	subgroup	G/H
center	C₂₇	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃×C₂₇
Frattini	C₃×C₂₇	C₃×C₃₇⋊C₃
Fitting	C₃²×C₂₇×C₃₇	C₃
socle	C₃×C₃₇	C₃²⋊C₂₇
prime radical	C₃³×C₃₇	C₂₇
cosocle	C₃×C₂₇×C₃₇	C₃²
hypercenter	C₃²×C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624502
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂₇ ◁ C₃×C₂₇ ◁ C₃²×C₂₇

Complements

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

N	H ≅ G/N	type
1. #22582500	C₃	semidirect
35. C₃₇	#10624502	semidirect

Maximal subgroups

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

subgroup	index
#22582500	3
C₃²×C₂₇×C₃₇	3
#10624502	37