Subgroups — #41007318

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

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

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. C₃×#15688500	C₃	semidirect
6. C₈₁×C₃₇⋊C₃	C₃	semidirect
7. C₃₇×C₈₁⋊C₃	C₃	semidirect
40. C₈₁⋊C₃	C₃₇⋊C₃	semidirect
57. 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
C₃×#15688500	3
C₈₁×C₃₇⋊C₃	3
C₃₇×C₈₁⋊C₃	3
#10624833	37