Subgroups — C₃×C₃₇×#6912 #41004716

Order 26973.

980 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	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃₇×#6912	C₃	1	27	yes	maximal, direct product factor
4	C₃×C₃₇×#733	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₃₇	—	3	18	no	—
7	C₃×3₋¹⁺²×C₃₇	—	3	6	no	—
8	C₃×3₋¹⁺²×C₃₇	—	3	6	no	—
9	C₃×3₋¹⁺²×C₃₇	—	3	3	no	—
10	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
11	C₃×3₋¹⁺²×C₃₇	C₃²	1	27	yes	—
12	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
13	C₃²×C₉×C₃₇	—	3	3	no	—
14	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
15	C₃₇×#733	C₃²	1	81	yes	complemented
16	C₃³×C₃₇	—	9	3	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
18	C₃³×C₃₇	H(3,𝔽₃)	1	6	yes	—
19	C₃³×C₃₇	C₃³	1	1	yes	characteristic
20	C₃³×C₃₇	C₃³	1	3	yes	—
21	3₋¹⁺²×C₃₇	—	3	18	no	—
22	3₋¹⁺²×C₃₇	—	3	54	no	—
23	3₋¹⁺²×C₃₇	—	3	27	no	—
24	3₋¹⁺²×C₃₇	C₃³	1	27	yes	—
25	C₃×C₉×C₃₇	—	3	27	no	—
26	C₃×C₉×C₃₇	—	3	6	no	—
27	C₃×C₉×C₃₇	—	3	9	no	—
28	C₃×C₉×C₃₇	—	3	3	no	—
29	C₃×C₉×C₃₇	C₃³	1	9	yes	—
30	C₃×#6912	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₃²×C₃₇	—	3	9	no	—
32	C₃²×C₃₇	—	3	3	no	—
33	C₃²×C₃₇	—	9	6	no	—
34	C₃²×C₃₇	—	9	3	no	—
35	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
39	C₉×C₃₇	—	3	27	no	—
40	C₉×C₃₇	—	3	9	no	—
41	C₃²×3₋¹⁺²	C₃×C₃₇	1	3	yes	—
42	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
43	#6912	C₃×C₃₇	1	27	yes	direct product factor
44	C₃×#733	C₃×C₃₇	1	9	yes	complemented
45	C₃×C₃₇	—	3	9	no	—
46	C₃×C₃₇	—	3	2	no	—
47	C₃×C₃₇	—	3	1	no	—
48	C₃×C₃₇	#6912	1	3	yes	direct product factor
49	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
51	C₃×3₋¹⁺²	—	3	6	no	—
52	C₃×3₋¹⁺²	—	3	18	no	—
53	C₃×3₋¹⁺²	—	3	3	no	—
54	C₃×3₋¹⁺²	—	3	6	no	—
55	C₃×3₋¹⁺²	C₃²×C₃₇	1	27	yes	—
56	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
57	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
58	C₃²×C₉	—	3	3	no	—
59	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
60	#733	C₃²×C₃₇	1	81	yes	complemented
61	C₃₇	C₃×#6912	1	1	yes	characteristic, direct product factor, Sylow
62	C₃³	—	9	3	no	—
63	C₃³	H(3,𝔽₃)×C₃₇	1	3	yes	—
64	C₃³	H(3,𝔽₃)×C₃₇	1	6	yes	—
65	C₃³	C₃³×C₃₇	1	1	yes	characteristic
66	C₃³	C₃³×C₃₇	1	3	yes	—
67	3₋¹⁺²	—	3	54	no	—
68	3₋¹⁺²	—	3	18	no	—
69	3₋¹⁺²	—	3	27	no	—
70	3₋¹⁺²	C₃³×C₃₇	1	27	yes	—
71	C₃×C₉	—	3	27	no	—
72	C₃×C₉	—	3	9	no	—
73	C₃×C₉	—	3	6	no	—
74	C₃×C₉	—	3	3	no	—
75	C₃×C₉	C₃³×C₃₇	1	9	yes	—
76	C₃²	—	3	9	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	9	6	no	—
79	C₃²	—	9	3	no	—
80	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	9	yes	—
81	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	2	yes	—
82	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
83	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
84	C₉	—	3	27	no	—
85	C₉	—	3	9	no	—
86	C₃	—	3	9	no	—
87	C₃	—	3	2	no	—
88	C₃	—	3	1	no	—
89	C₃	C₃₇×#6912	1	3	yes	direct product factor
90	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃₇×#6912	C₃	direct (N × H)
4. C₃×C₃₇×#733	C₃	semidirect
15. C₃₇×#733	C₃²	semidirect
30. C₃×#6912	C₃₇	direct (N × H)
43. #6912	C₃×C₃₇	direct (N × H)
44. C₃×#733	C₃×C₃₇	semidirect
48. C₃×C₃₇	#6912	direct (N × H)
60. #733	C₃²×C₃₇	semidirect
61. C₃₇	C₃×#6912	direct (N × H)
89. C₃	C₃₇×#6912	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×#6912	3
C₃×C₃₇×#733	3
C₃×#6912	37

Subgroups — C3×C37×#6912 #41004716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6912 #41004716