Subgroups — C₃²×C₃₇×#733 #41004711

Order 26973.

1052 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
2	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₇×#733	C₃	1	36	yes	maximal, direct product factor
4	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²×C₃₇	—	3	24	no	—
6	C₃×3₋¹⁺²×C₃₇	C₃²	1	36	yes	—
7	C₃²×C₉×C₃₇	—	3	8	no	—
8	C₃²×C₉×C₃₇	—	3	1	no	—
9	C₃²×C₉×C₃₇	—	3	3	no	—
10	C₃²×C₉×C₃₇	C₃²	1	12	yes	—
11	C₃₇×#733	C₃²	1	81	yes	direct product factor
12	C₃³×C₃₇	—	3	9	no	—
13	C₃³×C₃₇	H(3,𝔽₃)	1	8	yes	—
14	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃³×C₃₇	C₃³	1	4	yes	—
16	3₋¹⁺²×C₃₇	—	3	72	no	—
17	3₋¹⁺²×C₃₇	C₃³	1	27	yes	—
18	C₃×C₉×C₃₇	—	3	24	no	—
19	C₃×C₉×C₃₇	—	3	12	no	—
20	C₃×C₉×C₃₇	—	3	36	no	—
21	C₃×C₉×C₃₇	C₃³	1	9	yes	—
22	C₃²×#733	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃²×C₃₇	—	3	24	no	—
24	C₃²×C₃₇	—	3	12	no	—
25	C₃²×C₃₇	#733	1	9	yes	direct product factor
26	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	8	yes	—
27	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
28	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
29	C₉×C₃₇	—	3	9	no	—
30	C₉×C₃₇	—	3	27	no	—
31	C₃²×3₋¹⁺²	C₃×C₃₇	1	3	yes	—
32	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic
33	C₃×#733	C₃×C₃₇	1	36	yes	direct product factor
34	C₃×C₃₇	—	3	8	no	—
35	C₃×C₃₇	—	3	1	no	—
36	C₃×C₃₇	C₃×#733	1	12	yes	direct product factor
37	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	3	24	no	—
40	C₃×3₋¹⁺²	C₃²×C₃₇	1	36	yes	—
41	C₃²×C₉	—	3	8	no	—
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	—	3	3	no	—
44	C₃²×C₉	C₃²×C₃₇	1	12	yes	—
45	#733	C₃²×C₃₇	1	81	yes	direct product factor
46	C₃₇	C₃²×#733	1	1	yes	characteristic, direct product factor, Sylow
47	C₃³	—	3	9	no	—
48	C₃³	H(3,𝔽₃)×C₃₇	1	8	yes	—
49	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
50	C₃³	C₃³×C₃₇	1	4	yes	—
51	3₋¹⁺²	—	3	72	no	—
52	3₋¹⁺²	C₃³×C₃₇	1	27	yes	—
53	C₃×C₉	—	3	24	no	—
54	C₃×C₉	—	3	12	no	—
55	C₃×C₉	—	3	36	no	—
56	C₃×C₉	C₃³×C₃₇	1	9	yes	—
57	C₃²	—	3	24	no	—
58	C₃²	—	3	12	no	—
59	C₃²	C₃₇×#733	1	9	yes	direct product factor
60	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	8	yes	—
61	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	4	yes	—
62	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
63	C₉	—	3	9	no	—
64	C₉	—	3	27	no	—
65	C₃	—	3	8	no	—
66	C₃	—	3	1	no	—
67	C₃	C₃×C₃₇×#733	1	12	yes	direct product factor
68	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
prime radical	C₃⁴×C₃₇	C₃²
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
3. C₃×C₃₇×#733	C₃	direct (N × H)
11. C₃₇×#733	C₃²	direct (N × H)
22. C₃²×#733	C₃₇	direct (N × H)
25. C₃²×C₃₇	#733	direct (N × H)
33. C₃×#733	C₃×C₃₇	direct (N × H)
36. C₃×C₃₇	C₃×#733	direct (N × H)
45. #733	C₃²×C₃₇	direct (N × H)
46. C₃₇	C₃²×#733	direct (N × H)
59. C₃²	C₃₇×#733	direct (N × H)
67. C₃	C₃×C₃₇×#733	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
C₃³×C₉×C₃₇	3
C₃×C₃₇×#733	3
C₃²×#733	37

Subgroups — C32×C37×#733 #41004711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₇×#733 #41004711