Subgroups — C₃²×C₃₇×#732 #41004710

Order 26973.

1376 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	3	yes	maximal, complemented
2	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃₇×#732	C₃	1	36	yes	maximal, direct product factor
4	C₃⁴×C₃₇	—	3	3	no	—
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×H(3,𝔽₃)×C₃₇	—	3	24	no	—
7	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	36	yes	complemented
8	C₃²×C₉×C₃₇	—	3	8	no	—
9	C₃²×C₉×C₃₇	—	3	1	no	—
10	C₃²×C₉×C₃₇	C₃²	1	12	yes	complemented
11	C₃₇×#732	C₃²	1	81	yes	direct product factor
12	C₃³×C₃₇	—	3	9	no	—
13	C₃³×C₃₇	—	3	36	no	—
14	C₃³×C₃₇	—	9	27	no	—
15	C₃³×C₃₇	H(3,𝔽₃)	1	8	yes	—
16	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃³×C₃₇	C₃³	1	4	yes	—
18	H(3,𝔽₃)×C₃₇	—	3	72	no	—
19	H(3,𝔽₃)×C₃₇	C₃³	1	27	yes	complemented
20	C₃×C₉×C₃₇	—	3	24	no	—
21	C₃×C₉×C₃₇	—	3	12	no	—
22	C₃×C₉×C₃₇	C₃³	1	9	yes	complemented
23	C₃²×#732	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₃²×C₃₇	—	3	24	no	—
25	C₃²×C₃₇	—	3	12	no	—
26	C₃²×C₃₇	—	3	27	no	—
27	C₃²×C₃₇	—	9	108	no	—
28	C₃²×C₃₇	#732	1	9	yes	direct product factor
29	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	8	yes	—
30	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
31	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
32	C₉×C₃₇	—	3	9	no	—
33	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	3	yes	complemented
34	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic, complemented
35	C₃×#732	C₃×C₃₇	1	36	yes	direct product factor
36	C₃×C₃₇	—	3	8	no	—
37	C₃×C₃₇	—	3	1	no	—
38	C₃×C₃₇	—	9	27	no	—
39	C₃×C₃₇	C₃×#732	1	12	yes	direct product factor
40	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃⁴	—	3	3	no	—
42	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
43	C₃×H(3,𝔽₃)	—	3	24	no	—
44	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	36	yes	complemented
45	C₃²×C₉	—	3	8	no	—
46	C₃²×C₉	—	3	1	no	—
47	C₃²×C₉	C₃²×C₃₇	1	12	yes	complemented
48	#732	C₃²×C₃₇	1	81	yes	direct product factor
49	C₃₇	C₃²×#732	1	1	yes	characteristic, direct product factor, Sylow
50	C₃³	—	3	9	no	—
51	C₃³	—	3	36	no	—
52	C₃³	—	9	27	no	—
53	C₃³	H(3,𝔽₃)×C₃₇	1	8	yes	—
54	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
55	C₃³	C₃³×C₃₇	1	4	yes	—
56	H(3,𝔽₃)	—	3	72	no	—
57	H(3,𝔽₃)	C₃³×C₃₇	1	27	yes	complemented
58	C₃×C₉	—	3	24	no	—
59	C₃×C₉	—	3	12	no	—
60	C₃×C₉	C₃³×C₃₇	1	9	yes	complemented
61	C₃²	—	3	24	no	—
62	C₃²	—	3	12	no	—
63	C₃²	—	3	27	no	—
64	C₃²	—	9	108	no	—
65	C₃²	C₃₇×#732	1	9	yes	direct product factor
66	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	8	yes	—
67	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	4	yes	—
68	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
69	C₉	—	3	9	no	—
70	C₃	—	3	8	no	—
71	C₃	—	3	1	no	—
72	C₃	—	9	27	no	—
73	C₃	C₃×C₃₇×#732	1	12	yes	direct product factor
74	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃³×C₃₇	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
2. C₃³×C₉×C₃₇	C₃	semidirect
3. C₃×C₃₇×#732	C₃	direct (N × H)
7. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
10. C₃²×C₉×C₃₇	C₃²	semidirect
11. C₃₇×#732	C₃²	direct (N × H)
19. H(3,𝔽₃)×C₃₇	C₃³	semidirect
22. C₃×C₉×C₃₇	C₃³	semidirect
23. C₃²×#732	C₃₇	direct (N × H)
28. C₃²×C₃₇	#732	direct (N × H)
33. C₃²×H(3,𝔽₃)	C₃×C₃₇	semidirect
34. C₃³×C₉	C₃×C₃₇	semidirect
35. C₃×#732	C₃×C₃₇	direct (N × H)
39. C₃×C₃₇	C₃×#732	direct (N × H)
44. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
47. C₃²×C₉	C₃²×C₃₇	semidirect
48. #732	C₃²×C₃₇	direct (N × H)
49. C₃₇	C₃²×#732	direct (N × H)
57. H(3,𝔽₃)	C₃³×C₃₇	semidirect
60. C₃×C₉	C₃³×C₃₇	semidirect
65. C₃²	C₃₇×#732	direct (N × H)
73. C₃	C₃×C₃₇×#732	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×H(3,𝔽₃)×C₃₇	3
C₃³×C₉×C₃₇	3
C₃×C₃₇×#732	3
C₃²×#732	37

Subgroups — C32×C37×#732 #41004710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₇×#732 #41004710