Subgroups — #41008198

Order 26973.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582680	C₃	1	2	yes	maximal, complemented
2	C₃×#15688530	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582642	C₃	1	24	yes	maximal, complemented
4	C₃×#15688515	C₃	1	12	yes	maximal, complemented
5	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
6	#15688530	—	3	9	no	—
7	#15688530	—	3	18	no	—
8	C₃×#11758608	C₃²	1	24	yes	complemented
9	C₃×#11758608	C₃²	1	8	yes	complemented
10	C₃²×#66524	C₃²	1	4	yes	complemented
11	#15688515	C₃²	1	9	yes	complemented
12	#15688515	C₃²	1	72	yes	complemented
13	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
14	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	12	yes	—
15	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
16	C₃³×C₃₇	—	3	12	no	—
17	C₃³×C₃₇	C₃³	1	4	yes	—
18	H(3,𝔽₃)×C₃₇	—	3	24	no	—
19	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
20	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
21	#11758608	—	3	48	no	—
22	#11758608	—	3	24	no	—
23	#11758608	C₃³	1	18	yes	—
24	C₃×#66524	—	3	12	no	—
25	C₃×#66524	—	3	24	no	—
26	C₃×#66524	C₃³	1	1	yes	characteristic
27	C₃×#66524	C₃³	1	8	yes	—
28	#10624931	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	9	no	—
30	C₃²×C₃₇	—	9	4	no	—
31	C₃²×C₃₇	—	27	12	no	—
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
33	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
34	#66524	—	3	1	no	—
35	#66524	—	3	18	no	—
36	#66524	—	3	8	no	—
37	3₋¹⁺⁴	—	37	2	no	—
38	H(5,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
39	C₃×#737	—	37	1	no	—
40	#6910	—	37	24	no	—
41	C₃×#731	—	37	12	no	—
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	—	3	12	no	—
44	C₃×C₃₇	—	9	9	no	—
45	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
46	#737	—	111	9	no	—
47	#737	—	111	18	no	—
48	C₃×3₋¹⁺²	—	37	8	no	—
49	C₃×3₋¹⁺²	—	37	24	no	—
50	C₃×H(3,𝔽₃)	—	3	9	no	—
51	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	12	yes	—
52	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃²×C₉	—	37	4	no	—
54	#731	—	37	9	no	—
55	#731	—	37	72	no	—
56	C₃₇	#10624931	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	3	12	no	—
58	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
59	3₋¹⁺²	—	37	18	no	—
60	3₋¹⁺²	—	111	24	no	—
61	3₋¹⁺²	—	111	48	no	—
62	H(3,𝔽₃)	—	3	24	no	—
63	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
64	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	9	yes	—
65	C₃×C₉	—	37	1	no	—
66	C₃×C₉	—	37	8	no	—
67	C₃×C₉	—	111	12	no	—
68	C₃×C₉	—	111	24	no	—
69	C₃²	—	3	9	no	—
70	C₃²	—	9	4	no	—
71	C₃²	—	27	12	no	—
72	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
73	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
74	C₉	—	111	1	no	—
75	C₉	—	111	8	no	—
76	C₉	—	111	18	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	12	no	—
79	C₃	—	9	9	no	—
80	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting, prime radical	C₃₇×H(5,𝔽₃)	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624931
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ H(5,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #22582680	C₃	semidirect
2. C₃×#15688530	C₃	semidirect
3. #22582642	C₃	semidirect
4. C₃×#15688515	C₃	semidirect
8. C₃×#11758608	C₃²	semidirect
9. C₃×#11758608	C₃²	semidirect
10. C₃²×#66524	C₃²	semidirect
11. #15688515	C₃²	semidirect
12. #15688515	C₃²	semidirect
56. C₃₇	#10624931	semidirect

Maximal subgroups

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

subgroup	index
#22582680	3
C₃×#15688530	3
#22582642	3
C₃×#15688515	3
C₃₇×H(5,𝔽₃)	3
#10624931	37