Subgroups — #41008205

Order 26973.

1062 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇⋊H(5,𝔽₃)	C₃	1	3	yes	maximal, complemented
2	#22582648	C₃	1	24	yes	maximal, complemented
3	C₃×#15688519	C₃	1	12	yes	maximal, complemented
4	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	36	yes	complemented
6	H(3,𝔽₃)×C₃₇⋊C₃	—	3	27	no	—
7	#15688519	C₃²	1	9	yes	complemented
8	#15688519	C₃²	1	72	yes	complemented
9	C₃₇×#737	—	3	9	no	—
10	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	complemented
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₃₇	C₃²	1	4	yes	complemented
13	C₃³×C₃₇	C₃³	1	4	yes	—
14	C₃²×C₃₇⋊C₃	—	3	36	no	—
15	3₋¹⁺²×C₃₇	—	3	24	no	—
16	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
17	C₃×C₉×C₃₇	—	3	12	no	—
18	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
19	C₃×C₉×C₃₇	C₃³	1	8	yes	—
20	C₃²⋊(C₃₇⋊C₃)	—	3	72	no	—
21	C₃²⋊(C₃₇⋊C₃)	C₃³	1	27	yes	—
22	#10624932	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	9	4	no	—
24	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
25	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
26	C₃×C₃₇⋊C₃	—	3	27	no	—
27	C₃×C₃₇⋊C₃	—	27	36	no	—
28	C₉×C₃₇	—	3	1	no	—
29	C₉×C₃₇	—	3	8	no	—
30	H(5,𝔽₃)	—	37	3	no	—
31	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic, complemented
32	#6911	—	37	24	no	—
33	C₃×#732	—	37	12	no	—
34	C₃×C₃₇	—	3	12	no	—
35	C₃×C₃₇	—	3	1	no	—
36	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃₇⋊C₃	—	9	27	no	—
38	#737	—	3	9	no	—
39	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	8	yes	complemented
40	C₃×H(3,𝔽₃)	—	37	36	no	—
41	C₃×H(3,𝔽₃)	—	111	27	no	—
42	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
43	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	complemented
44	#732	—	37	9	no	—
45	#732	—	37	72	no	—
46	C₃₇	#10624932	1	1	yes	characteristic, complemented, Sylow
47	C₃³	—	111	36	no	—
48	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
49	3₋¹⁺²	—	3	24	no	—
50	H(3,𝔽₃)	—	37	27	no	—
51	H(3,𝔽₃)	—	111	72	no	—
52	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
53	C₃×C₉	—	3	12	no	—
54	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
55	C₃×C₉	C₃²×C₃₇⋊C₃	1	8	yes	—
56	C₃²	—	9	4	no	—
57	C₃²	—	111	27	no	—
58	C₃²	—	999	36	no	—
59	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
60	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
61	C₉	—	3	1	no	—
62	C₉	—	3	8	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	12	no	—
65	C₃	—	333	27	no	—
66	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	C₃×C₃₇×#737	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624932
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. C₃₇⋊H(5,𝔽₃)	C₃	semidirect
2. #22582648	C₃	semidirect
3. C₃×#15688519	C₃	semidirect
4. C₃×C₃₇×#737	C₃	semidirect
5. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
7. #15688519	C₃²	semidirect
8. #15688519	C₃²	semidirect
10. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
12. C₃²×C₉×C₃₇	C₃²	semidirect
31. C₃×#737	C₃₇⋊C₃	semidirect
39. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
43. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
46. C₃₇	#10624932	semidirect

Maximal subgroups

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

subgroup	index
C₃₇⋊H(5,𝔽₃)	3
#22582648	3
C₃×#15688519	3
C₃×C₃₇×#737	3
#10624932	37