Subgroups — C₃×C₉×H(3,𝔽₃)×C₃₇ #41004648

Order 26973.

1688 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	C₃³×C₉×C₃₇	C₃	1	4	yes	maximal, complemented
3	C₉×H(3,𝔽₃)×C₃₇	C₃	1	27	yes	maximal, direct product factor
4	C₃×C₃₇×C₃²⋊C₉	C₃	1	8	yes	maximal, complemented
5	C₃⁴×C₃₇	C₃²	1	4	yes	—
6	C₃×H(3,𝔽₃)×C₃₇	C₉	1	27	yes	direct product factor
7	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
8	C₃²×C₉×C₃₇	—	3	36	no	—
9	C₃²×C₉×C₃₇	C₃²	1	8	yes	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃²×C₉×C₃₇	C₃²	1	36	yes	complemented
12	C₃₇×C₃²⋊C₉	C₃²	1	72	yes	complemented
13	C₃³×C₃₇	—	3	12	no	—
14	C₃³×C₃₇	—	3	24	no	—
15	C₃³×C₃₇	C₃×C₉	1	36	yes	complemented
16	C₃³×C₃₇	C₃³	1	12	yes	—
17	C₃³×C₃₇	C₃³	1	1	yes	characteristic
18	H(3,𝔽₃)×C₃₇	C₃×C₉	1	81	yes	direct product factor
19	C₃×C₉×C₃₇	—	3	24	no	—
20	C₃×C₉×C₃₇	—	3	108	no	—
21	C₃×C₉×C₃₇	H(3,𝔽₃)	1	9	yes	direct product factor
22	C₃×C₉×C₃₇	C₃³	1	24	yes	—
23	C₃×C₉×C₃₇	C₃³	1	3	yes	—
24	C₃×C₉×H(3,𝔽₃)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₃²×C₃₇	—	3	12	no	—
26	C₃²×C₃₇	—	3	24	no	—
27	C₃²×C₃₇	—	3	108	no	—
28	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
29	C₃²×C₃₇	C₃²×C₉	1	36	yes	complemented
30	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
31	C₃²×C₃₇	#737	1	6	yes	—
32	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
33	C₉×C₃₇	—	3	24	no	—
34	C₉×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	direct product factor
35	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
36	C₃³×C₉	C₃×C₃₇	1	4	yes	complemented
37	C₉×H(3,𝔽₃)	C₃×C₃₇	1	27	yes	direct product factor
38	C₃×C₃²⋊C₉	C₃×C₃₇	1	8	yes	complemented
39	C₃×C₃₇	—	3	36	no	—
40	C₃×C₃₇	C₉×H(3,𝔽₃)	1	9	yes	direct product factor
41	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
42	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
43	C₃×C₃₇	C₃×#737	1	2	yes	—
44	C₃⁴	C₃²×C₃₇	1	4	yes	—
45	C₃×H(3,𝔽₃)	C₉×C₃₇	1	27	yes	direct product factor
46	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	9	yes	—
47	C₃²×C₉	—	3	36	no	—
48	C₃²×C₉	C₃²×C₃₇	1	8	yes	—
49	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
50	C₃²×C₉	C₃²×C₃₇	1	36	yes	complemented
51	C₃²⋊C₉	C₃²×C₃₇	1	72	yes	complemented
52	C₃₇	C₃×C₉×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
53	C₃³	—	3	12	no	—
54	C₃³	—	3	24	no	—
55	C₃³	C₃×C₉×C₃₇	1	36	yes	complemented
56	C₃³	C₃³×C₃₇	1	12	yes	—
57	C₃³	C₃³×C₃₇	1	1	yes	characteristic
58	H(3,𝔽₃)	C₃×C₉×C₃₇	1	81	yes	direct product factor
59	C₃×C₉	—	3	24	no	—
60	C₃×C₉	—	3	108	no	—
61	C₃×C₉	H(3,𝔽₃)×C₃₇	1	9	yes	direct product factor
62	C₃×C₉	C₃³×C₃₇	1	24	yes	—
63	C₃×C₉	C₃³×C₃₇	1	3	yes	—
64	C₃²	—	3	108	no	—
65	C₃²	—	3	12	no	—
66	C₃²	—	3	24	no	—
67	C₃²	C₃²×C₉×C₃₇	1	3	yes	—
68	C₃²	C₃²×C₉×C₃₇	1	36	yes	complemented
69	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	3	yes	—
70	C₃²	C₃₇×#737	1	6	yes	—
71	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
72	C₉	—	3	24	no	—
73	C₉	C₃×H(3,𝔽₃)×C₃₇	1	9	yes	direct product factor
74	C₃	—	3	36	no	—
75	C₃	C₉×H(3,𝔽₃)×C₃₇	1	9	yes	direct product factor
76	C₃	C₃³×C₉×C₃₇	1	1	yes	characteristic
77	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
78	C₃	C₃×C₃₇×#737	1	2	yes	—

	subgroup	G/H
center	C₃²×C₉×C₃₇	C₃²
commutator	C₃	C₃³×C₉×C₃₇
Frattini, cosocle	C₃²	C₃⁴×C₃₇
socle	C₃³×C₃₇	C₃³
prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃³×C₉×C₃₇	C₃	semidirect
3. C₉×H(3,𝔽₃)×C₃₇	C₃	direct (N × H)
4. C₃×C₃₇×C₃²⋊C₉	C₃	semidirect
6. C₃×H(3,𝔽₃)×C₃₇	C₉	direct (N × H)
11. C₃²×C₉×C₃₇	C₃²	semidirect
12. C₃₇×C₃²⋊C₉	C₃²	semidirect
15. C₃³×C₃₇	C₃×C₉	semidirect
18. H(3,𝔽₃)×C₃₇	C₃×C₉	direct (N × H)
21. C₃×C₉×C₃₇	H(3,𝔽₃)	direct (N × H)
24. C₃×C₉×H(3,𝔽₃)	C₃₇	direct (N × H)
29. C₃²×C₃₇	C₃²×C₉	semidirect
34. C₉×C₃₇	C₃×H(3,𝔽₃)	direct (N × H)
36. C₃³×C₉	C₃×C₃₇	semidirect
37. C₉×H(3,𝔽₃)	C₃×C₃₇	direct (N × H)
38. C₃×C₃²⋊C₉	C₃×C₃₇	semidirect
40. C₃×C₃₇	C₉×H(3,𝔽₃)	direct (N × H)
45. C₃×H(3,𝔽₃)	C₉×C₃₇	direct (N × H)
50. C₃²×C₉	C₃²×C₃₇	semidirect
51. C₃²⋊C₉	C₃²×C₃₇	semidirect
52. C₃₇	C₃×C₉×H(3,𝔽₃)	direct (N × H)
55. C₃³	C₃×C₉×C₃₇	semidirect
58. H(3,𝔽₃)	C₃×C₉×C₃₇	direct (N × H)
61. C₃×C₉	H(3,𝔽₃)×C₃₇	direct (N × H)
68. C₃²	C₃²×C₉×C₃₇	semidirect
73. C₉	C₃×H(3,𝔽₃)×C₃₇	direct (N × H)
75. C₃	C₉×H(3,𝔽₃)×C₃₇	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
C₉×H(3,𝔽₃)×C₃₇	3
C₃×C₃₇×C₃²⋊C₉	3
C₃×C₉×H(3,𝔽₃)	37

Subgroups — C3×C9×H(3,𝔽3)×C37 #41004648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×H(3,𝔽₃)×C₃₇ #41004648