Subgroups — C₃₇×#10624863 #41004652

Order 26973.

944 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₉×3₋¹⁺²×C₃₇	C₃	1	27	yes	maximal, complemented
3	C₃₇×#6887	C₃	1	9	yes	maximal, complemented
4	C₃×C₃₇×C₃²⋊C₉	C₃	1	3	yes	maximal
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₃₇	C₉	1	27	yes	complemented
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
8	C₃²×C₉×C₃₇	C₃²	1	9	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
10	C₃₇×C₉⋊C₉	C₃²	1	54	yes	complemented
11	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	—
12	C₃₇×C₃²⋊C₉	C₃²	1	18	yes	—
13	C₉²×C₃₇	C₃²	1	27	yes	complemented
14	C₃³×C₃₇	—	9	3	no	—
15	C₃³×C₃₇	C₃×C₉	1	9	yes	—
16	C₃³×C₃₇	C₃³	1	1	yes	characteristic
17	C₃³×C₃₇	C₃³	1	3	yes	—
18	3₋¹⁺²×C₃₇	C₃×C₉	1	81	yes	complemented
19	C₃×C₉×C₃₇	—	9	9	no	—
20	C₃×C₉×C₃₇	C₃×C₉	1	27	yes	complemented
21	C₃×C₉×C₃₇	C₃³	1	27	yes	—
22	C₃×C₉×C₃₇	C₃³	1	9	yes	—
23	#10624863	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₃²×C₃₇	—	3	3	no	—
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	3	6	no	—
27	C₃²×C₃₇	—	9	9	no	—
28	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
29	C₃²×C₃₇	C₃²×C₉	1	9	yes	—
30	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
31	C₉×C₃₇	—	3	27	no	—
32	C₉×C₃₇	C₃²×C₉	1	27	yes	—
33	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
34	C₉×3₋¹⁺²	C₃×C₃₇	1	27	yes	complemented
35	#6887	C₃×C₃₇	1	9	yes	complemented
36	C₃×C₃²⋊C₉	C₃×C₃₇	1	3	yes	—
37	C₃×C₃₇	—	3	9	no	—
38	C₃×C₃₇	—	3	3	no	—
39	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
40	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
41	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
42	C₃×3₋¹⁺²	C₉×C₃₇	1	27	yes	complemented
43	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
44	C₃²×C₉	C₃²×C₃₇	1	9	yes	—
45	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
46	C₉⋊C₉	C₃²×C₃₇	1	54	yes	complemented
47	C₃²⋊C₉	C₃²×C₃₇	1	9	yes	—
48	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	—
49	C₉²	C₃²×C₃₇	1	27	yes	complemented
50	C₃₇	#10624863	1	1	yes	characteristic, direct product factor, Sylow
51	C₃³	—	9	3	no	—
52	C₃³	C₃×C₉×C₃₇	1	9	yes	—
53	C₃³	C₃³×C₃₇	1	1	yes	characteristic
54	C₃³	C₃³×C₃₇	1	3	yes	—
55	3₋¹⁺²	C₃×C₉×C₃₇	1	81	yes	complemented
56	C₃×C₉	—	9	9	no	—
57	C₃×C₉	C₃×C₉×C₃₇	1	27	yes	complemented
58	C₃×C₉	C₃³×C₃₇	1	27	yes	—
59	C₃×C₉	C₃³×C₃₇	1	9	yes	—
60	C₃²	—	3	3	no	—
61	C₃²	—	3	3	no	—
62	C₃²	—	3	6	no	—
63	C₃²	—	9	9	no	—
64	C₃²	C₃²×C₉×C₃₇	1	3	yes	—
65	C₃²	C₃²×C₉×C₃₇	1	9	yes	—
66	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
67	C₉	—	3	27	no	—
68	C₉	C₃²×C₉×C₃₇	1	27	yes	—
69	C₃	—	3	9	no	—
70	C₃	—	3	3	no	—
71	C₃	C₃³×C₉×C₃₇	1	1	yes	characteristic
72	C₃	C₃₇×3₋¹⁺⁴	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator	C₃	C₃³×C₉×C₃₇
Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
3. C₃₇×#6887	C₃	semidirect
6. C₃×3₋¹⁺²×C₃₇	C₉	semidirect
10. C₃₇×C₉⋊C₉	C₃²	semidirect
13. C₉²×C₃₇	C₃²	semidirect
18. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
20. C₃×C₉×C₃₇	C₃×C₉	semidirect
23. #10624863	C₃₇	direct (N × H)
34. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
35. #6887	C₃×C₃₇	semidirect
42. C₃×3₋¹⁺²	C₉×C₃₇	semidirect
46. C₉⋊C₉	C₃²×C₃₇	semidirect
49. C₉²	C₃²×C₃₇	semidirect
50. C₃₇	#10624863	direct (N × H)
55. 3₋¹⁺²	C₃×C₉×C₃₇	semidirect
57. C₃×C₉	C₃×C₉×C₃₇	semidirect

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
C₉×3₋¹⁺²×C₃₇	3
C₃₇×#6887	3
C₃×C₃₇×C₃²⋊C₉	3
#10624863	37

Subgroups — C37×#10624863 #41004652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624863 #41004652