Subgroups — C₆₁×#6891 #28414992

Order 14823.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₆₁	C₃	1	1	yes	maximal, characteristic
2	C₃×3₋¹⁺²×C₆₁	C₃	1	3	yes	maximal, complemented
3	C₆₁×C₃²⋊C₉	C₃	1	9	yes	maximal, complemented
4	C₃³×C₆₁	—	3	3	no	—
5	C₃³×C₆₁	—	3	9	no	—
6	C₃³×C₆₁	C₃²	1	1	yes	characteristic
7	C₃³×C₆₁	C₃²	1	3	yes	—
8	3₋¹⁺²×C₆₁	—	3	9	no	—
9	C₃×C₉×C₆₁	C₃²	1	9	yes	complemented
10	C₃²×C₆₁	—	3	27	no	—
11	C₃²×C₆₁	—	3	3	no	—
12	C₃²×C₆₁	—	3	3	no	—
13	C₃²×C₆₁	—	3	6	no	—
14	C₃²×C₆₁	H(3,𝔽₃)	1	3	yes	—
15	C₃²×C₆₁	3₋¹⁺²	1	9	yes	complemented
16	C₃²×C₆₁	C₃³	1	1	yes	characteristic
17	C₉×C₆₁	—	3	9	no	—
18	#6891	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₃×C₆₁	—	3	3	no	—
20	C₃×C₆₁	—	3	9	no	—
21	C₃×C₆₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×C₆₁	C₃×3₋¹⁺²	1	3	yes	—
23	C₃⁴	C₃×C₆₁	1	1	yes	characteristic
24	C₃×3₋¹⁺²	C₃×C₆₁	1	3	yes	complemented
25	C₃²⋊C₉	C₃×C₆₁	1	9	yes	complemented
26	C₆₁	#6891	1	1	yes	characteristic, direct product factor, Sylow
27	C₃³	—	3	3	no	—
28	C₃³	—	3	9	no	—
29	C₃³	C₃²×C₆₁	1	1	yes	characteristic
30	C₃³	C₃²×C₆₁	1	3	yes	—
31	3₋¹⁺²	—	3	9	no	—
32	C₃×C₉	C₃²×C₆₁	1	9	yes	complemented
33	C₃²	—	3	27	no	—
34	C₃²	—	3	3	no	—
35	C₃²	—	3	3	no	—
36	C₃²	—	3	6	no	—
37	C₃²	H(3,𝔽₃)×C₆₁	1	3	yes	—
38	C₃²	3₋¹⁺²×C₆₁	1	9	yes	complemented
39	C₃²	C₃³×C₆₁	1	1	yes	characteristic
40	C₉	—	3	9	no	—
41	C₃	—	3	3	no	—
42	C₃	—	3	9	no	—
43	C₃	C₃×H(3,𝔽₃)×C₆₁	1	1	yes	characteristic
44	C₃	C₃×3₋¹⁺²×C₆₁	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₆₁	C₃³
commutator, Frattini, cosocle	C₃²	C₃³×C₆₁
prime radical	C₃⁴×C₆₁	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×3₋¹⁺²×C₆₁	C₃	semidirect
3. C₆₁×C₃²⋊C₉	C₃	semidirect
9. C₃×C₉×C₆₁	C₃²	semidirect
15. C₃²×C₆₁	3₋¹⁺²	semidirect
18. #6891	C₆₁	direct (N × H)
24. C₃×3₋¹⁺²	C₃×C₆₁	semidirect
25. C₃²⋊C₉	C₃×C₆₁	semidirect
26. C₆₁	#6891	direct (N × H)
32. C₃×C₉	C₃²×C₆₁	semidirect
38. C₃²	3₋¹⁺²×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₃⁴×C₆₁	3
C₃×3₋¹⁺²×C₆₁	3
C₆₁×C₃²⋊C₉	3
#6891	61

Subgroups — C61×#6891 #28414992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#6891 #28414992