Subgroups — #41007316

Order 26973.

154 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	#22582499	C₃	1	3	yes	maximal, complemented
4	#22582498	C₃	1	3	yes	maximal, complemented
5	C₈₁×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
6	C₃×C₃₇×C₈₁	C₃	1	1	yes	maximal, characteristic, complemented
7	#15688531	C₉	1	1	yes	characteristic
8	#15688511	C₉	1	2	yes	—
9	#15688510	C₉	1	3	yes	—
10	#15688509	C₉	1	3	yes	—
11	C₂₇×C₃₇⋊C₃	C₃²	1	3	yes	—
12	#15688500	C₃²	1	6	yes	—
13	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
14	C₃₇×C₈₁	C₃²	1	1	yes	characteristic
15	C₃₇×C₈₁	C₃²	1	1	yes	characteristic
16	C₃₇×C₈₁	C₃²	1	1	yes	characteristic
17	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
18	C₃²⋊(C₃₇⋊C₃)	C₂₇	1	1	yes	characteristic
19	#11758608	C₂₇	1	2	yes	—
20	#11758606	C₂₇	1	3	yes	—
21	#11758605	C₂₇	1	3	yes	—
22	C₉×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
23	C₂₇×C₃₇	C₃×C₉	1	1	yes	characteristic
24	C₂₇×C₃₇	C₃×C₉	1	1	yes	characteristic
25	C₂₇×C₃₇	C₃³	1	1	yes	characteristic
26	#11758601	C₃×C₉	1	6	yes	—
27	#10624833	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
29	C₃×C₃₇⋊C₃	C₃×C₂₇	1	3	yes	—
30	C₉×C₃₇	C₃×C₂₇	1	1	yes	characteristic
31	C₉×C₃₇	C₃×C₂₇	1	1	yes	characteristic
32	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
33	#66524	C₃×C₂₇	1	6	yes	—
34	#6903	—	37	1	no	—
35	C₈₁⋊C₃	—	37	2	no	—
36	C₈₁⋊C₃	—	37	3	no	—
37	C₈₁⋊C₃	—	37	3	no	—
38	C₃×C₈₁	—	37	3	no	—
39	C₃×C₈₁	C₃₇⋊C₃	1	1	yes	characteristic, complemented
40	C₃×C₃₇	—	3	1	no	—
41	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
42	C₃₇⋊C₃	—	3	3	no	—
43	#737	—	37	1	no	—
44	C₂₇⋊C₃	—	37	2	no	—
45	C₂₇⋊C₃	—	37	3	no	—
46	C₂₇⋊C₃	—	37	3	no	—
47	C₃×C₂₇	—	37	3	no	—
48	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
49	C₈₁	—	37	6	no	—
50	C₈₁	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	C₈₁	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	C₈₁	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃₇	#10624833	1	1	yes	characteristic, complemented, Sylow
54	3₋¹⁺²	—	37	2	no	—
55	3₋¹⁺²	—	37	3	no	—
56	3₋¹⁺²	—	37	3	no	—
57	H(3,𝔽₃)	—	37	1	no	—
58	C₃×C₉	—	37	3	no	—
59	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
60	C₂₇	—	37	6	no	—
61	C₂₇	C₉×C₃₇⋊C₃	1	1	yes	characteristic
62	C₂₇	C₉×C₃₇⋊C₃	1	1	yes	characteristic
63	C₂₇	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
64	C₃²	—	37	3	no	—
65	C₃²	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
66	C₉	—	37	6	no	—
67	C₉	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
68	C₉	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
69	C₉	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
70	C₃	—	3	1	no	—
71	C₃	—	111	3	no	—
72	C₃	C₃×C₂₇×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈₁	C₃×C₃₇⋊C₃
commutator, socle	C₃×C₃₇	C₃²×C₂₇
Frattini	C₂₇	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₈₁	C₃
prime radical	C₃²⋊(C₃₇⋊C₃)	C₂₇
cosocle	C₂₇×C₃₇	C₃³
hypercenter	C₃×C₈₁	C₃₇⋊C₃
hypocenter	C₃₇	#10624833
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₈₁ ◁ C₃×C₈₁

Complements

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

N	H ≅ G/N	type
2. #22582500	C₃	semidirect
3. #22582499	C₃	semidirect
4. #22582498	C₃	semidirect
5. C₈₁×C₃₇⋊C₃	C₃	semidirect
6. C₃×C₃₇×C₈₁	C₃	semidirect
39. C₃×C₈₁	C₃₇⋊C₃	semidirect
53. C₃₇	#10624833	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
#22582499	3
#22582498	3
C₈₁×C₃₇⋊C₃	3
C₃×C₃₇×C₈₁	3
#10624833	37