Subgroups — C₃₇×#10624886 #41004675

Order 26973.

728 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	C₃₇×#6895	C₃	1	9	yes	maximal, complemented
3	C₃₇×#6894	C₃	1	27	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	—	3	9	no	—
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
8	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	9	yes	—
10	C₃₇×C₉⋊C₉	C₃²	1	81	yes	complemented
11	C₃₇×C₃²⋊C₉	C₃²	1	18	yes	—
12	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	—
13	C₃³×C₃₇	—	9	3	no	—
14	C₃³×C₃₇	3₋¹⁺²	1	9	yes	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃³×C₃₇	C₃³	1	3	yes	—
17	3₋¹⁺²×C₃₇	—	3	27	no	—
18	C₃×C₉×C₃₇	—	3	9	no	—
19	C₃×C₉×C₃₇	—	9	9	no	—
20	C₃×C₉×C₃₇	C₃³	1	27	yes	—
21	C₃×C₉×C₃₇	C₃³	1	9	yes	—
22	#10624886	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	3	no	—
25	C₃²×C₃₇	—	3	6	no	—
26	C₃²×C₃₇	—	9	9	no	—
27	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
28	C₃²×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
29	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
30	C₉×C₃₇	—	3	9	no	—
31	C₉×C₃₇	—	3	27	no	—
32	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
33	#6895	C₃×C₃₇	1	9	yes	complemented
34	#6894	C₃×C₃₇	1	27	yes	complemented
35	C₃×C₃²⋊C₉	C₃×C₃₇	1	3	yes	—
36	C₃×C₃₇	—	3	9	no	—
37	C₃×C₃₇	—	3	3	no	—
38	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
39	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
40	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
41	C₃×3₋¹⁺²	—	3	9	no	—
42	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
43	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
44	C₃²×C₉	C₃²×C₃₇	1	9	yes	—
45	C₉⋊C₉	C₃²×C₃₇	1	81	yes	complemented
46	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	—
47	C₃²⋊C₉	C₃²×C₃₇	1	9	yes	—
48	C₃₇	#10624886	1	1	yes	characteristic, direct product factor, Sylow
49	C₃³	—	9	3	no	—
50	C₃³	3₋¹⁺²×C₃₇	1	9	yes	—
51	C₃³	C₃³×C₃₇	1	1	yes	characteristic
52	C₃³	C₃³×C₃₇	1	3	yes	—
53	3₋¹⁺²	—	3	27	no	—
54	C₃×C₉	—	3	9	no	—
55	C₃×C₉	—	9	9	no	—
56	C₃×C₉	C₃³×C₃₇	1	27	yes	—
57	C₃×C₉	C₃³×C₃₇	1	9	yes	—
58	C₃²	—	3	3	no	—
59	C₃²	—	3	3	no	—
60	C₃²	—	3	6	no	—
61	C₃²	—	9	9	no	—
62	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
63	C₃²	C₃×3₋¹⁺²×C₃₇	1	9	yes	—
64	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
65	C₉	—	3	9	no	—
66	C₉	—	3	27	no	—
67	C₃	—	3	9	no	—
68	C₃	—	3	3	no	—
69	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
70	C₃	C₃₇×3₋¹⁺⁴	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
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
2. C₃₇×#6895	C₃	semidirect
3. C₃₇×#6894	C₃	semidirect
10. C₃₇×C₉⋊C₉	C₃²	semidirect
22. #10624886	C₃₇	direct (N × H)
33. #6895	C₃×C₃₇	semidirect
34. #6894	C₃×C₃₇	semidirect
45. C₉⋊C₉	C₃²×C₃₇	semidirect
48. C₃₇	#10624886	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
C₃₇×#6895	3
C₃₇×#6894	3
C₃×C₃₇×C₃²⋊C₉	3
#10624886	37

Subgroups — C37×#10624886 #41004675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624886 #41004675