Subgroups — C₃²×C₃₇×#731 #41004709

Order 26973.

1160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	2	yes	maximal, complemented
2	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃₇×#731	C₃	1	36	yes	maximal, direct product factor
5	C₃⁴×C₃₇	—	3	1	no	—
6	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
7	C₃×3₋¹⁺²×C₃₇	—	3	16	no	—
8	C₃×3₋¹⁺²×C₃₇	C₃²	1	24	yes	complemented
9	C₃×H(3,𝔽₃)×C₃₇	—	3	8	no	—
10	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	12	yes	—
11	C₃²×C₉×C₃₇	—	3	8	no	—
12	C₃²×C₉×C₃₇	—	3	1	no	—
13	C₃²×C₉×C₃₇	—	3	2	no	—
14	C₃²×C₉×C₃₇	C₃²	1	12	yes	complemented
15	C₃₇×#731	C₃²	1	81	yes	direct product factor
16	C₃³×C₃₇	—	3	9	no	—
17	C₃³×C₃₇	—	3	12	no	—
18	C₃³×C₃₇	—	9	9	no	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	8	yes	—
20	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃³×C₃₇	C₃³	1	4	yes	—
22	3₋¹⁺²×C₃₇	—	3	48	no	—
23	3₋¹⁺²×C₃₇	C₃³	1	18	yes	complemented
24	H(3,𝔽₃)×C₃₇	—	3	24	no	—
25	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
26	C₃×C₉×C₃₇	—	3	24	no	—
27	C₃×C₉×C₃₇	—	3	12	no	—
28	C₃×C₉×C₃₇	—	3	24	no	—
29	C₃×C₉×C₃₇	C₃³	1	9	yes	complemented
30	C₃²×#731	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₃²×C₃₇	—	3	24	no	—
32	C₃²×C₃₇	—	3	12	no	—
33	C₃²×C₃₇	—	3	9	no	—
34	C₃²×C₃₇	—	9	36	no	—
35	C₃²×C₃₇	#731	1	9	yes	direct product factor
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	8	yes	—
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
38	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
39	C₉×C₃₇	—	3	9	no	—
40	C₉×C₃₇	—	3	18	no	—
41	C₃²×3₋¹⁺²	C₃×C₃₇	1	2	yes	complemented
42	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
43	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic, complemented
44	C₃×#731	C₃×C₃₇	1	36	yes	direct product factor
45	C₃×C₃₇	—	3	8	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	—	9	9	no	—
48	C₃×C₃₇	C₃×#731	1	12	yes	direct product factor
49	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃⁴	—	3	1	no	—
51	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
52	C₃×3₋¹⁺²	—	3	16	no	—
53	C₃×3₋¹⁺²	C₃²×C₃₇	1	24	yes	complemented
54	C₃×H(3,𝔽₃)	—	3	8	no	—
55	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	12	yes	—
56	C₃²×C₉	—	3	8	no	—
57	C₃²×C₉	—	3	1	no	—
58	C₃²×C₉	—	3	2	no	—
59	C₃²×C₉	C₃²×C₃₇	1	12	yes	complemented
60	#731	C₃²×C₃₇	1	81	yes	direct product factor
61	C₃₇	C₃²×#731	1	1	yes	characteristic, direct product factor, Sylow
62	C₃³	—	3	9	no	—
63	C₃³	—	3	12	no	—
64	C₃³	—	9	9	no	—
65	C₃³	H(3,𝔽₃)×C₃₇	1	8	yes	—
66	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
67	C₃³	C₃³×C₃₇	1	4	yes	—
68	3₋¹⁺²	—	3	48	no	—
69	3₋¹⁺²	C₃³×C₃₇	1	18	yes	complemented
70	H(3,𝔽₃)	—	3	24	no	—
71	H(3,𝔽₃)	C₃³×C₃₇	1	9	yes	—
72	C₃×C₉	—	3	24	no	—
73	C₃×C₉	—	3	12	no	—
74	C₃×C₉	—	3	24	no	—
75	C₃×C₉	C₃³×C₃₇	1	9	yes	complemented
76	C₃²	—	3	24	no	—
77	C₃²	—	3	12	no	—
78	C₃²	—	3	9	no	—
79	C₃²	—	9	36	no	—
80	C₃²	C₃₇×#731	1	9	yes	direct product factor
81	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	8	yes	—
82	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	4	yes	—
83	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
84	C₉	—	3	9	no	—
85	C₉	—	3	18	no	—
86	C₃	—	3	8	no	—
87	C₃	—	3	1	no	—
88	C₃	—	9	9	no	—
89	C₃	C₃×C₃₇×#731	1	12	yes	direct product factor
90	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	C₃²×H(3,𝔽₃)×C₃₇	C₃
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. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
3. C₃³×C₉×C₃₇	C₃	semidirect
4. C₃×C₃₇×#731	C₃	direct (N × H)
8. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
14. C₃²×C₉×C₃₇	C₃²	semidirect
15. C₃₇×#731	C₃²	direct (N × H)
23. 3₋¹⁺²×C₃₇	C₃³	semidirect
29. C₃×C₉×C₃₇	C₃³	semidirect
30. C₃²×#731	C₃₇	direct (N × H)
35. C₃²×C₃₇	#731	direct (N × H)
41. C₃²×3₋¹⁺²	C₃×C₃₇	semidirect
43. C₃³×C₉	C₃×C₃₇	semidirect
44. C₃×#731	C₃×C₃₇	direct (N × H)
48. C₃×C₃₇	C₃×#731	direct (N × H)
53. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
59. C₃²×C₉	C₃²×C₃₇	semidirect
60. #731	C₃²×C₃₇	direct (N × H)
61. C₃₇	C₃²×#731	direct (N × H)
69. 3₋¹⁺²	C₃³×C₃₇	semidirect
75. C₃×C₉	C₃³×C₃₇	semidirect
80. C₃²	C₃₇×#731	direct (N × H)
89. C₃	C₃×C₃₇×#731	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
C₃²×H(3,𝔽₃)×C₃₇	3
C₃³×C₉×C₃₇	3
C₃×C₃₇×#731	3
C₃²×#731	37

Subgroups — C32×C37×#731 #41004709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₇×#731 #41004709