Subgroups — #41004951

Order 26973.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582612	C₃	1	1	yes	maximal, characteristic
2	#22582500	C₃	1	2	yes	maximal, complemented
3	C₃×C₃₇×C₈₁	C₃	1	1	yes	maximal, characteristic, complemented
4	#15688531	C₉	1	1	yes	characteristic
5	#15688511	C₉	1	2	yes	—
6	#15688510	—	3	1	no	—
7	#15688509	—	3	1	no	—
8	C₂₇×C₃₇⋊C₃	—	3	1	no	—
9	#15688500	—	3	2	no	—
10	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
11	C₃₇×C₈₁	—	3	1	no	—
12	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃²⋊(C₃₇⋊C₃)	C₂₇	1	1	yes	characteristic
14	#11758608	C₂₇	1	2	yes	—
15	#11758606	—	3	1	no	—
16	#11758605	—	3	1	no	—
17	C₉×C₃₇⋊C₃	—	3	1	no	—
18	C₂₇×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
19	C₂₇×C₃₇	3₋¹⁺²	1	1	yes	characteristic
20	C₂₇×C₃₇	3₋¹⁺²	1	1	yes	characteristic
21	#11758601	—	3	2	no	—
22	#10624504	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
24	C₃×C₃₇⋊C₃	—	3	1	no	—
25	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
26	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
27	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
28	#66524	—	3	2	no	—
29	#6903	—	37	1	no	—
30	C₈₁⋊C₃	—	37	2	no	—
31	C₃×C₈₁	C₃₇⋊C₃	1	1	yes	characteristic, complemented
32	C₃×C₃₇	—	3	1	no	—
33	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
34	C₃₇⋊C₃	—	9	1	no	—
35	#737	—	37	1	no	—
36	C₂₇⋊C₃	—	37	2	no	—
37	C₂₇⋊C₃	—	111	1	no	—
38	C₂₇⋊C₃	—	111	1	no	—
39	C₃×C₂₇	—	111	1	no	—
40	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
41	C₈₁	—	3	1	no	—
42	C₈₁	—	111	2	no	—
43	C₃₇	#10624504	1	1	yes	characteristic, complemented, Sylow
44	3₋¹⁺²	—	37	2	no	—
45	3₋¹⁺²	—	111	1	no	—
46	3₋¹⁺²	—	111	1	no	—
47	H(3,𝔽₃)	—	37	1	no	—
48	C₃×C₉	—	111	1	no	—
49	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
50	C₂₇	—	111	2	no	—
51	C₂₇	#11758605	1	1	yes	characteristic
52	C₂₇	#11758606	1	1	yes	characteristic
53	C₂₇	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
54	C₃²	—	111	1	no	—
55	C₃²	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
56	C₉	—	111	2	no	—
57	C₉	#15688502	1	1	yes	characteristic
58	C₉	#15688509	1	1	yes	characteristic
59	C₉	#15688510	1	1	yes	characteristic
60	C₃	—	3	1	no	—
61	C₃	—	333	1	no	—
62	C₃	#22582465	1	1	yes	characteristic

	subgroup	G/H
center	C₂₇	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃×C₂₇
Frattini	C₃×C₂₇	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₈₁	C₃
socle	C₃×C₃₇	C₃²⋊C₂₇
prime radical	C₃²⋊(C₃₇⋊C₃)	C₂₇
cosocle	C₃×C₂₇×C₃₇	C₃²
hypercenter	C₃×C₈₁	C₃₇⋊C₃
hypocenter	C₃₇	#10624504
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂₇ ◁ C₃×C₂₇ ◁ C₃×C₈₁

Complements

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

N	H ≅ G/N	type
2. #22582500	C₃	semidirect
3. C₃×C₃₇×C₈₁	C₃	semidirect
31. C₃×C₈₁	C₃₇⋊C₃	semidirect
43. C₃₇	#10624504	semidirect

Maximal subgroups

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

subgroup	index
#22582612	3
#22582500	3
C₃×C₃₇×C₈₁	3
#10624504	37