Subgroups — C₃₇×#10624932 #41004721

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	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#6911	C₃	1	24	yes	maximal, complemented
4	C₃×C₃₇×#732	C₃	1	12	yes	maximal, complemented
5	C₃₇×#737	—	3	9	no	—
6	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	complemented
7	C₃×H(3,𝔽₃)×C₃₇	—	3	27	no	—
8	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
9	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	36	yes	complemented
10	C₃²×C₉×C₃₇	C₃²	1	4	yes	complemented
11	C₃₇×#732	C₃²	1	9	yes	complemented
12	C₃₇×#732	C₃²	1	72	yes	complemented
13	C₃³×C₃₇	—	3	36	no	—
14	C₃³×C₃₇	C₃³	1	4	yes	—
15	3₋¹⁺²×C₃₇	—	3	24	no	—
16	H(3,𝔽₃)×C₃₇	—	3	72	no	—
17	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
18	H(3,𝔽₃)×C₃₇	C₃³	1	27	yes	—
19	C₃×C₉×C₃₇	—	3	12	no	—
20	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
21	C₃×C₉×C₃₇	C₃³	1	8	yes	—
22	#10624932	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃²×C₃₇	—	3	27	no	—
24	C₃²×C₃₇	—	9	4	no	—
25	C₃²×C₃₇	—	27	36	no	—
26	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
27	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
28	C₉×C₃₇	—	3	1	no	—
29	C₉×C₃₇	—	3	8	no	—
30	H(5,𝔽₃)	C₃×C₃₇	1	3	yes	complemented
31	C₃×#737	C₃×C₃₇	1	1	yes	characteristic, complemented
32	#6911	C₃×C₃₇	1	24	yes	complemented
33	C₃×#732	C₃×C₃₇	1	12	yes	complemented
34	C₃×C₃₇	—	3	1	no	—
35	C₃×C₃₇	—	3	12	no	—
36	C₃×C₃₇	—	9	27	no	—
37	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
38	#737	—	3	9	no	—
39	C₃×3₋¹⁺²	C₃²×C₃₇	1	8	yes	complemented
40	C₃×H(3,𝔽₃)	—	3	27	no	—
41	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
42	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	36	yes	complemented
43	C₃²×C₉	C₃²×C₃₇	1	4	yes	complemented
44	#732	C₃²×C₃₇	1	9	yes	complemented
45	#732	C₃²×C₃₇	1	72	yes	complemented
46	C₃₇	#10624932	1	1	yes	characteristic, direct product factor, Sylow
47	C₃³	—	3	36	no	—
48	C₃³	C₃³×C₃₇	1	4	yes	—
49	3₋¹⁺²	—	3	24	no	—
50	H(3,𝔽₃)	—	3	72	no	—
51	H(3,𝔽₃)	H(3,𝔽₃)×C₃₇	1	9	yes	—
52	H(3,𝔽₃)	C₃³×C₃₇	1	27	yes	—
53	C₃×C₉	—	3	12	no	—
54	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
55	C₃×C₉	C₃³×C₃₇	1	8	yes	—
56	C₃²	—	3	27	no	—
57	C₃²	—	9	4	no	—
58	C₃²	—	27	36	no	—
59	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	12	yes	—
60	C₃²	C₃⁴×C₃₇	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	—	9	27	no	—
66	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₃₇ ◁ C₃×H(3,𝔽₃)×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×H(5,𝔽₃)	C₃	semidirect
2. C₃×C₃₇×#737	C₃	semidirect
3. C₃₇×#6911	C₃	semidirect
4. C₃×C₃₇×#732	C₃	semidirect
6. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
9. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
10. C₃²×C₉×C₃₇	C₃²	semidirect
11. C₃₇×#732	C₃²	semidirect
12. C₃₇×#732	C₃²	semidirect
22. #10624932	C₃₇	direct (N × H)
30. H(5,𝔽₃)	C₃×C₃₇	semidirect
31. C₃×#737	C₃×C₃₇	semidirect
32. #6911	C₃×C₃₇	semidirect
33. C₃×#732	C₃×C₃₇	semidirect
39. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
42. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
43. C₃²×C₉	C₃²×C₃₇	semidirect
44. #732	C₃²×C₃₇	semidirect
45. #732	C₃²×C₃₇	semidirect
46. C₃₇	#10624932	direct (N × H)

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
C₃×C₃₇×#737	3
C₃₇×#6911	3
C₃×C₃₇×#732	3
#10624932	37

Subgroups — C37×#10624932 #41004721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624932 #41004721