Subgroups — C₃²×C₃²⋊#66524 #41007488

Order 26973.

2996 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#66524	C₃	1	3	yes	maximal, complemented
2	C₃×C₃²⋊#66524	C₃	1	36	yes	maximal, direct product factor
3	C₃⁵×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	—	3	27	no	—
5	C₃²×#66524	C₃²	1	36	yes	complemented
6	C₃²⋊#66524	C₃²	1	81	yes	direct product factor
7	C₃⁴×C₃₇	—	3	9	no	—
8	C₃⁴×C₃₇	—	3	18	no	—
9	C₃⁴×C₃₇	C₉	1	27	yes	complemented
10	C₃⁴×C₃₇	C₃²	1	12	yes	—
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
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×#66524	—	3	108	no	—
22	C₃×#66524	C₃³	1	27	yes	complemented
23	C₃²×C₃²⋊C₉	—	37	1	no	maximal, 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	complemented
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	#66524	—	3	27	no	—
34	C₃⁵	C₃₇⋊C₃	1	1	yes	characteristic
35	C₃³×C₉	—	37	3	no	—
36	C₃×C₃²⋊C₉	—	37	36	no	—
37	C₃×C₃₇	—	3	27	no	—
38	C₃×C₃₇	C₃×C₃²⋊C₉	1	36	yes	complemented
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	18	no	—
43	C₃⁴	—	3	9	no	—
44	C₃⁴	#66524	1	27	yes	complemented
45	C₃⁴	C₃×C₃₇⋊C₃	1	12	yes	—
46	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
47	C₃²×C₉	—	37	36	no	—
48	C₃²×C₉	—	111	27	no	—
49	C₃²⋊C₉	—	37	81	no	—
50	C₃₇	C₃²×C₃²⋊C₉	1	1	yes	characteristic, complemented, Sylow
51	C₃³	—	3	243	no	—
52	C₃³	—	3	36	no	—
53	C₃³	—	3	72	no	—
54	C₃³	C₃×#66524	1	9	yes	—
55	C₃³	C₃×#66524	1	108	yes	complemented
56	C₃³	#11758608	1	18	yes	—
57	C₃³	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
58	C₃³	C₃²×C₃₇⋊C₃	1	9	yes	—
59	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
60	C₃×C₉	—	37	27	no	—
61	C₃×C₉	—	111	108	no	—
62	C₃²	—	3	324	no	—
63	C₃²	—	3	9	no	—
64	C₃²	—	3	18	no	—
65	C₃²	C₃²⋊#66524	1	81	yes	direct product factor
66	C₃²	C₃²×#66524	1	12	yes	—
67	C₃²	C₃²×#66524	1	27	yes	complemented
68	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
69	C₃²	C₃×#11758608	1	24	yes	—
70	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
71	C₉	—	111	27	no	—
72	C₃	—	3	27	no	—
73	C₃	C₃×C₃²⋊#66524	1	36	yes	direct product factor
74	C₃	C₃³×#66524	1	1	yes	characteristic
75	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
76	C₃	C₃²×#11758608	1	2	yes	—

	subgroup	G/H
center	C₃⁴	C₃×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting, prime radical	C₃⁵×C₃₇	C₃
socle	C₃⁴×C₃₇	C₃²
cosocle	C₃²×C₃₇	C₃⁴
hypercenter	C₃⁵	C₃₇⋊C₃
hypocenter	C₃₇	C₃²×C₃²⋊C₉
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃⁴ ◁ C₃⁵

Complements

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

N	H ≅ G/N	type
1. C₃³×#66524	C₃	semidirect
2. C₃×C₃²⋊#66524	C₃	direct (N × H)
5. C₃²×#66524	C₃²	semidirect
6. C₃²⋊#66524	C₃²	direct (N × H)
9. C₃⁴×C₃₇	C₉	semidirect
16. C₃³×C₃₇	C₃×C₉	semidirect
22. C₃×#66524	C₃³	semidirect
27. C₃²×C₃₇	C₃²⋊C₉	semidirect
29. C₃²×C₃₇	C₃²×C₉	semidirect
38. C₃×C₃₇	C₃×C₃²⋊C₉	semidirect
44. C₃⁴	#66524	semidirect
50. C₃₇	C₃²×C₃²⋊C₉	semidirect
55. C₃³	C₃×#66524	semidirect
65. C₃²	C₃²⋊#66524	direct (N × H)
67. C₃²	C₃²×#66524	semidirect
73. C₃	C₃×C₃²⋊#66524	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×#66524	3
C₃×C₃²⋊#66524	3
C₃⁵×C₃₇	3
C₃²×C₃²⋊C₉	37

Subgroups — C32×C32⋊#66524 #41007488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃²⋊#66524 #41007488