Subgroups — C₃₇×#10624862 #41004651

Order 26973.

1178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₉×H(3,𝔽₃)×C₃₇	C₃	1	27	yes	maximal, complemented
3	C₃×C₃₇×C₃²⋊C₉	C₃	1	12	yes	maximal, complemented
4	C₃⁴×C₃₇	C₃²	1	4	yes	—
5	C₃×H(3,𝔽₃)×C₃₇	C₉	1	27	yes	complemented
6	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
7	C₃²×C₉×C₃₇	C₃²	1	36	yes	complemented
8	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	—
9	C₃₇×C₃²⋊C₉	C₃²	1	72	yes	complemented
10	C₃³×C₃₇	—	9	4	no	—
11	C₃³×C₃₇	—	9	8	no	—
12	C₃³×C₃₇	C₃×C₉	1	36	yes	complemented
13	C₃³×C₃₇	C₃³	1	1	yes	characteristic
14	C₃³×C₃₇	C₃³	1	12	yes	—
15	H(3,𝔽₃)×C₃₇	C₃×C₉	1	81	yes	complemented
16	C₃×C₉×C₃₇	—	9	36	no	—
17	C₃×C₉×C₃₇	C₃³	1	27	yes	—
18	#10624862	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₃²×C₃₇	—	3	12	no	—
20	C₃²×C₃₇	—	3	1	no	—
21	C₃²×C₃₇	—	3	2	no	—
22	C₃²×C₃₇	—	3	24	no	—
23	C₃²×C₃₇	—	9	36	no	—
24	C₃²×C₃₇	C₃²×C₉	1	36	yes	—
25	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
26	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
27	C₉×C₃₇	—	3	27	no	—
28	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
29	C₉×H(3,𝔽₃)	C₃×C₃₇	1	27	yes	complemented
30	C₃×C₃²⋊C₉	C₃×C₃₇	1	12	yes	complemented
31	C₃×C₃₇	—	3	3	no	—
32	C₃×C₃₇	—	3	36	no	—
33	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
34	C₃×C₃₇	H(5,𝔽₃)	1	1	yes	characteristic
35	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
36	C₃⁴	C₃²×C₃₇	1	4	yes	—
37	C₃×H(3,𝔽₃)	C₉×C₃₇	1	27	yes	complemented
38	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	9	yes	—
39	C₃²×C₉	C₃²×C₃₇	1	36	yes	complemented
40	C₃²⋊C₉	C₃²×C₃₇	1	9	yes	—
41	C₃²⋊C₉	C₃²×C₃₇	1	72	yes	complemented
42	C₃₇	#10624862	1	1	yes	characteristic, direct product factor, Sylow
43	C₃³	—	9	4	no	—
44	C₃³	—	9	8	no	—
45	C₃³	C₃×C₉×C₃₇	1	36	yes	complemented
46	C₃³	C₃³×C₃₇	1	1	yes	characteristic
47	C₃³	C₃³×C₃₇	1	12	yes	—
48	H(3,𝔽₃)	C₃×C₉×C₃₇	1	81	yes	complemented
49	C₃×C₉	—	9	36	no	—
50	C₃×C₉	C₃³×C₃₇	1	27	yes	—
51	C₃²	—	3	12	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	3	2	no	—
54	C₃²	—	3	24	no	—
55	C₃²	—	9	36	no	—
56	C₃²	C₃²×C₉×C₃₇	1	36	yes	—
57	C₃²	C₃²×C₉×C₃₇	1	3	yes	—
58	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
59	C₉	—	3	27	no	—
60	C₃	—	3	3	no	—
61	C₃	—	3	36	no	—
62	C₃	C₃³×C₉×C₃₇	1	1	yes	characteristic
63	C₃	C₃₇×H(5,𝔽₃)	1	1	yes	characteristic
64	C₃	C₃₇×3₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator	C₃	C₃³×C₉×C₃₇
Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
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×H(3,𝔽₃)×C₃₇	C₃	semidirect
3. C₃×C₃₇×C₃²⋊C₉	C₃	semidirect
5. C₃×H(3,𝔽₃)×C₃₇	C₉	semidirect
7. C₃²×C₉×C₃₇	C₃²	semidirect
9. C₃₇×C₃²⋊C₉	C₃²	semidirect
12. C₃³×C₃₇	C₃×C₉	semidirect
15. H(3,𝔽₃)×C₃₇	C₃×C₉	semidirect
18. #10624862	C₃₇	direct (N × H)
29. C₉×H(3,𝔽₃)	C₃×C₃₇	semidirect
30. C₃×C₃²⋊C₉	C₃×C₃₇	semidirect
37. C₃×H(3,𝔽₃)	C₉×C₃₇	semidirect
39. C₃²×C₉	C₃²×C₃₇	semidirect
41. C₃²⋊C₉	C₃²×C₃₇	semidirect
42. C₃₇	#10624862	direct (N × H)
45. C₃³	C₃×C₉×C₃₇	semidirect
48. H(3,𝔽₃)	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×H(3,𝔽₃)×C₃₇	3
C₉×H(3,𝔽₃)×C₃₇	3
C₃×C₃₇×C₃²⋊C₉	3
#10624862	37

Subgroups — C37×#10624862 #41004651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624862 #41004651