Subgroups — C₃²×C₃₇×C₃²⋊C₉ #41004645

Order 26973.

2996 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁵×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃³×C₉×C₃₇	C₃	1	3	yes	maximal, complemented
3	C₃×C₃₇×C₃²⋊C₉	C₃	1	36	yes	maximal, direct product factor
4	C₃⁴×C₃₇	—	3	9	no	—
5	C₃⁴×C₃₇	—	3	18	no	—
6	C₃⁴×C₃₇	C₉	1	27	yes	complemented
7	C₃⁴×C₃₇	C₃²	1	12	yes	—
8	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
9	C₃²×C₉×C₃₇	—	3	27	no	—
10	C₃²×C₉×C₃₇	C₃²	1	36	yes	complemented
11	C₃₇×C₃²⋊C₉	C₃²	1	81	yes	direct product factor
12	C₃³×C₃₇	—	3	243	no	—
13	C₃³×C₃₇	—	3	36	no	—
14	C₃³×C₃₇	—	3	72	no	—
15	C₃³×C₃₇	C₃×C₉	1	9	yes	—
16	C₃³×C₃₇	C₃×C₉	1	108	yes	complemented
17	C₃³×C₃₇	H(3,𝔽₃)	1	9	yes	—
18	C₃³×C₃₇	3₋¹⁺²	1	18	yes	—
19	C₃³×C₃₇	C₃³	1	9	yes	—
20	C₃³×C₃₇	C₃³	1	4	yes	—
21	C₃×C₉×C₃₇	—	3	108	no	—
22	C₃×C₉×C₃₇	C₃³	1	27	yes	complemented
23	C₃²×C₃²⋊C₉	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₃²×C₃₇	—	3	9	no	—
25	C₃²×C₃₇	—	3	18	no	—
26	C₃²×C₃₇	—	3	324	no	—
27	C₃²×C₃₇	C₃²⋊C₉	1	81	yes	direct product factor
28	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
29	C₃²×C₃₇	C₃²×C₉	1	27	yes	complemented
30	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
31	C₃²×C₃₇	C₃×3₋¹⁺²	1	24	yes	—
32	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
33	C₉×C₃₇	—	3	27	no	—
34	C₃⁵	C₃×C₃₇	1	1	yes	characteristic
35	C₃³×C₉	C₃×C₃₇	1	3	yes	complemented
36	C₃×C₃²⋊C₉	C₃×C₃₇	1	36	yes	direct product factor
37	C₃×C₃₇	—	3	27	no	—
38	C₃×C₃₇	C₃×C₃²⋊C₉	1	36	yes	direct product factor
39	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
40	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃×C₃₇	C₃²×3₋¹⁺²	1	2	yes	—
42	C₃⁴	—	3	9	no	—
43	C₃⁴	—	3	18	no	—
44	C₃⁴	C₉×C₃₇	1	27	yes	complemented
45	C₃⁴	C₃²×C₃₇	1	12	yes	—
46	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
47	C₃²×C₉	—	3	27	no	—
48	C₃²×C₉	C₃²×C₃₇	1	36	yes	complemented
49	C₃²⋊C₉	C₃²×C₃₇	1	81	yes	direct product factor
50	C₃₇	C₃²×C₃²⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
51	C₃³	—	3	243	no	—
52	C₃³	—	3	36	no	—
53	C₃³	—	3	72	no	—
54	C₃³	C₃×C₉×C₃₇	1	9	yes	—
55	C₃³	C₃×C₉×C₃₇	1	108	yes	complemented
56	C₃³	H(3,𝔽₃)×C₃₇	1	9	yes	—
57	C₃³	3₋¹⁺²×C₃₇	1	18	yes	—
58	C₃³	C₃³×C₃₇	1	9	yes	—
59	C₃³	C₃³×C₃₇	1	4	yes	—
60	C₃×C₉	—	3	108	no	—
61	C₃×C₉	C₃³×C₃₇	1	27	yes	complemented
62	C₃²	—	3	324	no	—
63	C₃²	—	3	9	no	—
64	C₃²	—	3	18	no	—
65	C₃²	C₃₇×C₃²⋊C₉	1	81	yes	direct product factor
66	C₃²	C₃²×C₉×C₃₇	1	12	yes	—
67	C₃²	C₃²×C₉×C₃₇	1	27	yes	complemented
68	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	12	yes	—
69	C₃²	C₃×3₋¹⁺²×C₃₇	1	24	yes	—
70	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
71	C₉	—	3	27	no	—
72	C₃	—	3	27	no	—
73	C₃	C₃×C₃₇×C₃²⋊C₉	1	36	yes	direct product factor
74	C₃	C₃³×C₉×C₃₇	1	1	yes	characteristic
75	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
76	C₃	C₃²×3₋¹⁺²×C₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₃⁴×C₃₇	C₃²
commutator	C₃	C₃³×C₉×C₃₇
Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃⁵×C₃₇	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃⁴×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. C₃×C₃₇×C₃²⋊C₉	C₃	direct (N × H)
6. C₃⁴×C₃₇	C₉	semidirect
10. C₃²×C₉×C₃₇	C₃²	semidirect
11. C₃₇×C₃²⋊C₉	C₃²	direct (N × H)
16. C₃³×C₃₇	C₃×C₉	semidirect
22. C₃×C₉×C₃₇	C₃³	semidirect
23. C₃²×C₃²⋊C₉	C₃₇	direct (N × H)
27. C₃²×C₃₇	C₃²⋊C₉	direct (N × H)
29. C₃²×C₃₇	C₃²×C₉	semidirect
35. C₃³×C₉	C₃×C₃₇	semidirect
36. C₃×C₃²⋊C₉	C₃×C₃₇	direct (N × H)
38. C₃×C₃₇	C₃×C₃²⋊C₉	direct (N × H)
44. C₃⁴	C₉×C₃₇	semidirect
48. C₃²×C₉	C₃²×C₃₇	semidirect
49. C₃²⋊C₉	C₃²×C₃₇	direct (N × H)
50. C₃₇	C₃²×C₃²⋊C₉	direct (N × H)
55. C₃³	C₃×C₉×C₃₇	semidirect
61. C₃×C₉	C₃³×C₃₇	semidirect
65. C₃²	C₃₇×C₃²⋊C₉	direct (N × H)
67. C₃²	C₃²×C₉×C₃₇	semidirect
73. C₃	C₃×C₃₇×C₃²⋊C₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃⁵×C₃₇	3
C₃³×C₉×C₃₇	3
C₃×C₃₇×C₃²⋊C₉	3
C₃²×C₃²⋊C₉	37

Subgroups — C32×C37×C32⋊C9 #41004645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₇×C₃²⋊C₉ #41004645