Subgroups — #41008201

Order 26973.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582680	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃₇⋊H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#15688530	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582644	C₃	1	24	yes	maximal, complemented
5	C₃×#15688517	C₃	1	12	yes	maximal, complemented
6	C₃₇×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, complemented
7	#15688530	—	3	9	no	—
8	#15688530	—	3	9	no	—
9	C₃×#11758608	C₃²	1	12	yes	complemented
10	C₃×#11758608	C₃²	1	8	yes	complemented
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	12	yes	—
12	H(3,𝔽₃)×C₃₇⋊C₃	—	3	9	no	—
13	C₃²×#66524	C₃²	1	4	yes	complemented
14	#15688517	C₃²	1	9	yes	complemented
15	#15688517	C₃²	1	72	yes	complemented
16	C₃₇×#737	—	3	9	no	—
17	C₃×3₋¹⁺²×C₃₇	C₃²	1	12	yes	complemented
18	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
19	C₃³×C₃₇	C₃³	1	4	yes	—
20	C₃²×C₃₇⋊C₃	—	3	12	no	—
21	3₋¹⁺²×C₃₇	—	3	24	no	—
22	3₋¹⁺²×C₃₇	C₃³	1	9	yes	—
23	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
24	C₃×C₉×C₃₇	—	3	12	no	—
25	C₃²⋊(C₃₇⋊C₃)	—	3	24	no	—
26	C₃²⋊(C₃₇⋊C₃)	C₃³	1	9	yes	—
27	#11758608	—	3	24	no	—
28	#11758608	—	3	24	no	—
29	#11758608	C₃³	1	9	yes	—
30	C₃×#66524	—	3	12	no	—
31	C₃×#66524	—	3	12	no	—
32	C₃×#66524	C₃³	1	1	yes	characteristic
33	C₃×#66524	C₃³	1	8	yes	—
34	#10624931	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	—	9	4	no	—
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
37	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
38	C₃×C₃₇⋊C₃	—	3	9	no	—
39	C₃×C₃₇⋊C₃	—	27	12	no	—
40	C₉×C₃₇	—	3	9	no	—
41	#66524	—	3	1	no	—
42	#66524	—	3	9	no	—
43	#66524	—	3	8	no	—
44	3₋¹⁺⁴	—	37	1	no	—
45	3₋¹⁺⁴	C₃₇⋊C₃	1	1	yes	characteristic, complemented
46	H(5,𝔽₃)	—	37	1	no	—
47	C₃×#737	—	37	1	no	—
48	#6910	—	37	24	no	—
49	C₃×#731	—	37	12	no	—
50	C₃×C₃₇	—	3	12	no	—
51	C₃×C₃₇	—	3	1	no	—
52	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃₇⋊C₃	—	9	9	no	—
54	#737	—	3	9	no	—
55	#737	—	111	9	no	—
56	#737	—	111	9	no	—
57	C₃×3₋¹⁺²	—	37	8	no	—
58	C₃×3₋¹⁺²	—	37	12	no	—
59	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	12	yes	complemented
60	C₃×H(3,𝔽₃)	—	37	12	no	—
61	C₃×H(3,𝔽₃)	—	111	9	no	—
62	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
63	C₃²×C₉	—	37	4	no	—
64	#731	—	37	9	no	—
65	#731	—	37	72	no	—
66	C₃₇	#10624931	1	1	yes	characteristic, complemented, Sylow
67	C₃³	—	111	12	no	—
68	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
69	3₋¹⁺²	—	3	24	no	—
70	3₋¹⁺²	—	37	9	no	—
71	3₋¹⁺²	—	111	24	no	—
72	3₋¹⁺²	—	111	24	no	—
73	3₋¹⁺²	C₃²×C₃₇⋊C₃	1	9	yes	—
74	H(3,𝔽₃)	—	37	9	no	—
75	H(3,𝔽₃)	—	111	24	no	—
76	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
77	C₃×C₉	—	3	12	no	—
78	C₃×C₉	—	37	1	no	—
79	C₃×C₉	—	37	8	no	—
80	C₃×C₉	—	111	12	no	—
81	C₃×C₉	—	111	12	no	—
82	C₃²	—	9	4	no	—
83	C₃²	—	111	9	no	—
84	C₃²	—	999	12	no	—
85	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
86	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
87	C₉	—	3	9	no	—
88	C₉	—	111	1	no	—
89	C₉	—	111	8	no	—
90	C₉	—	111	9	no	—
91	C₃	—	3	1	no	—
92	C₃	—	3	12	no	—
93	C₃	—	333	9	no	—
94	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	C₃₇×3₋¹⁺⁴	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
prime radical	C₃₇⋊H(5,𝔽₃)	C₃
hypercenter	3₋¹⁺⁴	C₃₇⋊C₃
hypocenter	C₃₇	#10624931
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ 3₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #22582680	C₃	semidirect
3. C₃×#15688530	C₃	semidirect
4. #22582644	C₃	semidirect
5. C₃×#15688517	C₃	semidirect
6. C₃₇×3₋¹⁺⁴	C₃	semidirect
9. C₃×#11758608	C₃²	semidirect
10. C₃×#11758608	C₃²	semidirect
13. C₃²×#66524	C₃²	semidirect
14. #15688517	C₃²	semidirect
15. #15688517	C₃²	semidirect
17. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
45. 3₋¹⁺⁴	C₃₇⋊C₃	semidirect
59. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
66. 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
C₃₇⋊H(5,𝔽₃)	3
C₃×#15688530	3
#22582644	3
C₃×#15688517	3
C₃₇×3₋¹⁺⁴	3
#10624931	37