Subgroups — #41004950

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	1	yes	maximal, characteristic, complemented
3	C₃×#15688500	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×C₈₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
5	#15688531	C₉	1	1	yes	characteristic
6	#15688511	C₉	1	2	yes	—
7	#15688510	—	3	1	no	—
8	#15688509	—	3	1	no	—
9	C₂₇×C₃₇⋊C₃	—	3	1	no	—
10	#15688500	—	3	1	no	—
11	#15688500	—	3	1	no	—
12	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
13	C₃₇×C₈₁	—	3	1	no	—
14	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃²⋊(C₃₇⋊C₃)	C₂₇	1	1	yes	characteristic
16	#11758608	C₂₇	1	2	yes	—
17	#11758606	—	3	1	no	—
18	#11758605	—	3	1	no	—
19	C₉×C₃₇⋊C₃	—	3	1	no	—
20	C₂₇×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₂₇×C₃₇	3₋¹⁺²	1	1	yes	characteristic
22	C₂₇×C₃₇	3₋¹⁺²	1	1	yes	characteristic
23	#11758601	—	3	2	no	—
24	#10624503	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
26	C₃×C₃₇⋊C₃	—	3	1	no	—
27	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
28	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
29	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
30	#66524	—	3	2	no	—
31	#6903	—	37	1	no	—
32	C₈₁⋊C₃	—	37	1	no	—
33	C₈₁⋊C₃	C₃₇⋊C₃	1	1	yes	characteristic, complemented
34	C₃×C₈₁	—	37	1	no	—
35	C₃×C₃₇	—	3	1	no	—
36	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
37	C₃₇⋊C₃	—	9	1	no	—
38	#737	—	37	1	no	—
39	C₂₇⋊C₃	—	37	2	no	—
40	C₂₇⋊C₃	—	111	1	no	—
41	C₂₇⋊C₃	—	111	1	no	—
42	C₃×C₂₇	—	111	1	no	—
43	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
44	C₈₁	—	3	1	no	—
45	C₈₁	—	111	1	no	—
46	C₈₁	—	111	1	no	—
47	C₃₇	#10624503	1	1	yes	characteristic, complemented, Sylow
48	3₋¹⁺²	—	37	2	no	—
49	3₋¹⁺²	—	111	1	no	—
50	3₋¹⁺²	—	111	1	no	—
51	H(3,𝔽₃)	—	37	1	no	—
52	C₃×C₉	—	111	1	no	—
53	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
54	C₂₇	—	111	2	no	—
55	C₂₇	#11758605	1	1	yes	characteristic
56	C₂₇	#11758606	1	1	yes	characteristic
57	C₂₇	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
58	C₃²	—	111	1	no	—
59	C₃²	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
60	C₉	—	111	2	no	—
61	C₉	#15688502	1	1	yes	characteristic
62	C₉	#15688509	1	1	yes	characteristic
63	C₉	#15688510	1	1	yes	characteristic
64	C₃	—	3	1	no	—
65	C₃	—	333	1	no	—
66	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	C₃₇×C₈₁⋊C₃	C₃
socle	C₃×C₃₇	C₃²⋊C₂₇
prime radical	C₃²⋊(C₃₇⋊C₃)	C₂₇
cosocle	C₃×C₂₇×C₃₇	C₃²
hypercenter	C₈₁⋊C₃	C₃₇⋊C₃
hypocenter	C₃₇	#10624503
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂₇ ◁ C₃×C₂₇ ◁ C₈₁⋊C₃

Complements

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

N	H ≅ G/N	type
2. #22582500	C₃	semidirect
3. C₃×#15688500	C₃	semidirect
4. C₃₇×C₈₁⋊C₃	C₃	semidirect
33. C₈₁⋊C₃	C₃₇⋊C₃	semidirect
47. C₃₇	#10624503	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×#15688500	3
C₃₇×C₈₁⋊C₃	3
#10624503	37