Subgroups — C₂²×C₆₁×C₇⋊C₉ #29506752

Order 15372.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×C₇⋊C₉	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₇×C₆₁	C₃	1	1	yes	maximal, characteristic
3	C₆₁×C₇⋊C₉	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₃×C₇×C₆₁	C₂×C₃	1	3	yes	—
5	C₂²×C₉×C₆₁	—	7	1	no	maximal, Hall
6	C₂²×C₇×C₆₁	C₉	1	1	yes	characteristic, complemented, Hall
7	C₃×C₇×C₆₁	C₂²×C₃	1	1	yes	characteristic
8	C₂×C₉×C₆₁	—	7	3	no	—
9	C₂×C₇×C₆₁	C₂×C₉	1	3	yes	complemented
10	C₂²×C₃×C₆₁	ΓL(1,𝔽₈)	1	1	yes	characteristic
11	C₉×C₆₁	—	7	1	no	Hall
12	C₇×C₆₁	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₆₁	C₂×ΓL(1,𝔽₈)	1	3	yes	—
14	C₂²×C₇⋊C₉	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₆₁	C₇⋊C₉	1	1	yes	characteristic, direct product factor, Hall
16	C₃×C₆₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
17	C₂×C₇⋊C₉	C₂×C₆₁	1	3	yes	direct product factor
18	C₂×C₆₁	C₂×C₇⋊C₉	1	3	yes	direct product factor
19	C₂²×C₃×C₇	C₃×C₆₁	1	1	yes	characteristic
20	C₇⋊C₉	C₂²×C₆₁	1	1	yes	characteristic, direct product factor, Hall
21	C₆₁	C₂²×C₇⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×C₃×C₇	C₂×C₃×C₆₁	1	3	yes	—
23	C₂²×C₉	—	7	1	no	Hall
24	C₂²×C₇	C₉×C₆₁	1	1	yes	characteristic, complemented, Hall
25	C₃×C₇	C₂²×C₃×C₆₁	1	1	yes	characteristic
26	C₂×C₉	—	7	3	no	—
27	C₂×C₇	C₂×C₉×C₆₁	1	3	yes	complemented
28	C₂²×C₃	ΓL(1,𝔽₈)×C₆₁	1	1	yes	characteristic
29	C₉	—	7	1	no	Sylow
30	C₇	C₂²×C₉×C₆₁	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃	C₂×ΓL(1,𝔽₈)×C₆₁	1	3	yes	—
32	C₂²	C₆₁×C₇⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₂²×ΓL(1,𝔽₈)×C₆₁	1	1	yes	characteristic
34	C₂	C₂×C₆₁×C₇⋊C₉	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₆₁	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₂²×C₉×C₆₁
Frattini	C₃	C₂²×ΓL(1,𝔽₈)×C₆₁
Fitting, socle, prime radical	C₂²×C₃×C₇×C₆₁	C₃
cosocle	C₃×C₇	C₂²×C₃×C₆₁
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₃×C₆₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×C₇⋊C₉	C₂	direct (N × H)
3. C₆₁×C₇⋊C₉	C₂²	direct (N × H)
6. C₂²×C₇×C₆₁	C₉	semidirect
9. C₂×C₇×C₆₁	C₂×C₉	semidirect
12. C₇×C₆₁	C₂²×C₉	semidirect
14. C₂²×C₇⋊C₉	C₆₁	direct (N × H)
15. C₂²×C₆₁	C₇⋊C₉	direct (N × H)
17. C₂×C₇⋊C₉	C₂×C₆₁	direct (N × H)
18. C₂×C₆₁	C₂×C₇⋊C₉	direct (N × H)
20. C₇⋊C₉	C₂²×C₆₁	direct (N × H)
21. C₆₁	C₂²×C₇⋊C₉	direct (N × H)
24. C₂²×C₇	C₉×C₆₁	semidirect
27. C₂×C₇	C₂×C₉×C₆₁	semidirect
30. C₇	C₂²×C₉×C₆₁	semidirect
32. C₂²	C₆₁×C₇⋊C₉	direct (N × H)
34. C₂	C₂×C₆₁×C₇⋊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
C₂×C₆₁×C₇⋊C₉	2
C₂²×C₃×C₇×C₆₁	3
C₂²×C₉×C₆₁	7
C₂²×C₇⋊C₉	61

Subgroups — C22×C61×C7⋊C9 #29506752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×C₇⋊C₉ #29506752