Subgroups — C₃₇×#10624931 #41004720

Order 26973.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×3₋¹⁺⁴	C₃	1	2	yes	maximal, complemented
2	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#6910	C₃	1	24	yes	maximal, complemented
5	C₃×C₃₇×#731	C₃	1	12	yes	maximal, complemented
6	C₃₇×#737	—	3	9	no	—
7	C₃₇×#737	—	3	18	no	—
8	C₃×3₋¹⁺²×C₃₇	C₃²	1	24	yes	complemented
9	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	complemented
10	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	12	yes	—
12	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
13	C₃²×C₉×C₃₇	C₃²	1	4	yes	complemented
14	C₃₇×#731	C₃²	1	9	yes	complemented
15	C₃₇×#731	C₃²	1	72	yes	complemented
16	C₃³×C₃₇	—	3	12	no	—
17	C₃³×C₃₇	C₃³	1	4	yes	—
18	3₋¹⁺²×C₃₇	—	3	48	no	—
19	3₋¹⁺²×C₃₇	—	3	24	no	—
20	3₋¹⁺²×C₃₇	C₃³	1	18	yes	—
21	H(3,𝔽₃)×C₃₇	—	3	24	no	—
22	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
23	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
24	C₃×C₉×C₃₇	—	3	12	no	—
25	C₃×C₉×C₃₇	—	3	24	no	—
26	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
27	C₃×C₉×C₃₇	C₃³	1	8	yes	—
28	#10624931	C₃₇	1	1	yes	maximal, characteristic, direct product factor, 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	C₉×C₃₇	—	3	1	no	—
35	C₉×C₃₇	—	3	18	no	—
36	C₉×C₃₇	—	3	8	no	—
37	3₋¹⁺⁴	C₃×C₃₇	1	2	yes	complemented
38	H(5,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
39	C₃×#737	C₃×C₃₇	1	1	yes	characteristic, complemented
40	#6910	C₃×C₃₇	1	24	yes	complemented
41	C₃×#731	C₃×C₃₇	1	12	yes	complemented
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	—	3	9	no	—
47	#737	—	3	18	no	—
48	C₃×3₋¹⁺²	C₃²×C₃₇	1	24	yes	complemented
49	C₃×3₋¹⁺²	C₃²×C₃₇	1	8	yes	complemented
50	C₃×H(3,𝔽₃)	—	3	9	no	—
51	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	12	yes	—
52	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
53	C₃²×C₉	C₃²×C₃₇	1	4	yes	complemented
54	#731	C₃²×C₃₇	1	9	yes	complemented
55	#731	C₃²×C₃₇	1	72	yes	complemented
56	C₃₇	#10624931	1	1	yes	characteristic, direct product factor, Sylow
57	C₃³	—	3	12	no	—
58	C₃³	C₃³×C₃₇	1	4	yes	—
59	3₋¹⁺²	—	3	48	no	—
60	3₋¹⁺²	—	3	24	no	—
61	3₋¹⁺²	C₃³×C₃₇	1	18	yes	—
62	H(3,𝔽₃)	—	3	24	no	—
63	H(3,𝔽₃)	H(3,𝔽₃)×C₃₇	1	9	yes	—
64	H(3,𝔽₃)	C₃³×C₃₇	1	9	yes	—
65	C₃×C₉	—	3	12	no	—
66	C₃×C₉	—	3	24	no	—
67	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
68	C₃×C₉	C₃³×C₃₇	1	8	yes	—
69	C₃²	—	3	9	no	—
70	C₃²	—	9	4	no	—
71	C₃²	—	27	12	no	—
72	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	12	yes	—
73	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
74	C₉	—	3	1	no	—
75	C₉	—	3	18	no	—
76	C₉	—	3	8	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	12	no	—
79	C₃	—	9	9	no	—
80	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
prime radical	C₃₇×H(5,𝔽₃)	C₃
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. C₃₇×3₋¹⁺⁴	C₃	semidirect
3. C₃×C₃₇×#737	C₃	semidirect
4. C₃₇×#6910	C₃	semidirect
5. C₃×C₃₇×#731	C₃	semidirect
8. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
9. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
13. C₃²×C₉×C₃₇	C₃²	semidirect
14. C₃₇×#731	C₃²	semidirect
15. C₃₇×#731	C₃²	semidirect
28. #10624931	C₃₇	direct (N × H)
37. 3₋¹⁺⁴	C₃×C₃₇	semidirect
39. C₃×#737	C₃×C₃₇	semidirect
40. #6910	C₃×C₃₇	semidirect
41. C₃×#731	C₃×C₃₇	semidirect
48. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
49. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
53. C₃²×C₉	C₃²×C₃₇	semidirect
54. #731	C₃²×C₃₇	semidirect
55. #731	C₃²×C₃₇	semidirect
56. C₃₇	#10624931	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×3₋¹⁺⁴	3
C₃₇×H(5,𝔽₃)	3
C₃×C₃₇×#737	3
C₃₇×#6910	3
C₃×C₃₇×#731	3
#10624931	37

Subgroups — C37×#10624931 #41004720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624931 #41004720