Subgroups — C₃₇×#10624881 #41004670

Order 26973.

1124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	3	yes	maximal, complemented
2	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃₇×#6893	C₃	1	27	yes	maximal, complemented
4	C₃₇×#6891	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₃₇	C₃²	1	3	yes	—
6	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
7	C₃×3₋¹⁺²×C₃₇	—	3	27	no	—
8	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
9	C₃×3₋¹⁺²×C₃₇	C₃²	1	18	yes	complemented
10	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	9	yes	complemented
13	C₃₇×C₃²⋊C₉	C₃²	1	81	yes	complemented
14	C₃³×C₃₇	—	3	9	no	—
15	C₃³×C₃₇	—	9	3	no	—
16	C₃³×C₃₇	—	9	9	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	9	yes	—
18	C₃³×C₃₇	C₃³	1	9	yes	—
19	C₃³×C₃₇	C₃³	1	3	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	3₋¹⁺²×C₃₇	—	3	27	no	—
22	3₋¹⁺²×C₃₇	—	3	54	no	—
23	H(3,𝔽₃)×C₃₇	—	3	27	no	—
24	C₃×C₉×C₃₇	—	3	27	no	—
25	C₃×C₉×C₃₇	C₃³	1	27	yes	—
26	#10624881	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	—	3	9	no	—
29	C₃²×C₃₇	—	3	9	no	—
30	C₃²×C₃₇	—	3	27	no	—
31	C₃²×C₃₇	—	9	9	no	—
32	C₃²×C₃₇	—	9	27	no	—
33	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
34	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
35	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
36	C₉×C₃₇	—	3	27	no	—
37	C₃²×3₋¹⁺²	C₃×C₃₇	1	3	yes	complemented
38	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
39	#6893	C₃×C₃₇	1	27	yes	complemented
40	#6891	C₃×C₃₇	1	9	yes	complemented
41	C₃×C₃₇	—	3	3	no	—
42	C₃×C₃₇	—	3	9	no	—
43	C₃×C₃₇	—	9	9	no	—
44	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
46	C₃⁴	C₃²×C₃₇	1	3	yes	—
47	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
48	C₃×3₋¹⁺²	—	3	27	no	—
49	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
50	C₃×3₋¹⁺²	C₃²×C₃₇	1	18	yes	complemented
51	C₃×H(3,𝔽₃)	—	3	9	no	—
52	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	9	yes	—
53	C₃²×C₉	C₃²×C₃₇	1	9	yes	complemented
54	C₃²⋊C₉	C₃²×C₃₇	1	81	yes	complemented
55	C₃₇	#10624881	1	1	yes	characteristic, direct product factor, Sylow
56	C₃³	—	3	9	no	—
57	C₃³	—	9	3	no	—
58	C₃³	—	9	9	no	—
59	C₃³	H(3,𝔽₃)×C₃₇	1	9	yes	—
60	C₃³	C₃³×C₃₇	1	9	yes	—
61	C₃³	C₃³×C₃₇	1	3	yes	—
62	C₃³	C₃³×C₃₇	1	1	yes	characteristic
63	3₋¹⁺²	—	3	27	no	—
64	3₋¹⁺²	—	3	54	no	—
65	H(3,𝔽₃)	—	3	27	no	—
66	C₃×C₉	—	3	27	no	—
67	C₃×C₉	C₃³×C₃₇	1	27	yes	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	9	no	—
70	C₃²	—	3	9	no	—
71	C₃²	—	3	27	no	—
72	C₃²	—	9	9	no	—
73	C₃²	—	9	27	no	—
74	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	9	yes	—
75	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	3	yes	—
76	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
77	C₉	—	3	27	no	—
78	C₃	—	3	3	no	—
79	C₃	—	3	9	no	—
80	C₃	—	9	9	no	—
81	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
82	C₃	C₃₇×3₋¹⁺⁴	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
3. C₃₇×#6893	C₃	semidirect
4. C₃₇×#6891	C₃	semidirect
9. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
12. C₃²×C₉×C₃₇	C₃²	semidirect
13. C₃₇×C₃²⋊C₉	C₃²	semidirect
26. #10624881	C₃₇	direct (N × H)
37. C₃²×3₋¹⁺²	C₃×C₃₇	semidirect
39. #6893	C₃×C₃₇	semidirect
40. #6891	C₃×C₃₇	semidirect
50. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
53. C₃²×C₉	C₃²×C₃₇	semidirect
54. C₃²⋊C₉	C₃²×C₃₇	semidirect
55. C₃₇	#10624881	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×3₋¹⁺²×C₃₇	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃₇×#6893	3
C₃₇×#6891	3
#10624881	37

Subgroups — C37×#10624881 #41004670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624881 #41004670