Subgroups — C₅²×C₃₁×C₂²⋊C₉ #42124578

Order 27900.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅²×C₃₁	C₃	1	1	yes	maximal, characteristic
2	C₅²×C₉×C₃₁	—	4	1	no	maximal, Hall
3	C₅×C₃₁×C₂²⋊C₉	C₅	1	6	yes	maximal, direct product factor
4	C₂×C₃×C₅²×C₃₁	—	3	1	no	—
5	C₂²×C₅²×C₃₁	C₉	1	1	yes	characteristic, complemented, Hall
6	C₃×C₅²×C₃₁	A₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₃₁	C₃×C₅	1	6	yes	—
8	C₂×C₅²×C₃₁	—	3	1	no	—
9	C₅×C₉×C₃₁	—	4	6	no	—
10	C₃₁×C₂²⋊C₉	C₅²	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₃×C₅×C₃₁	—	3	6	no	—
12	C₅²×C₂²⋊C₉	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₅²×C₃₁	C₂²⋊C₉	1	1	yes	characteristic, direct product factor, Hall
14	C₂²×C₅×C₃₁	C₅×C₉	1	6	yes	complemented
15	C₃×C₅×C₃₁	C₅×A₄	1	6	yes	—
16	C₂²×C₃×C₃₁	C₃×C₅²	1	1	yes	characteristic
17	C₂×C₅×C₃₁	—	3	6	no	—
18	C₂²×C₃×C₅²	C₃×C₃₁	1	1	yes	characteristic
19	C₉×C₃₁	—	4	1	no	Hall
20	C₅²×C₉	—	4	1	no	Hall
21	C₂×C₃×C₃₁	—	3	1	no	—
22	C₅×C₂²⋊C₉	C₅×C₃₁	1	6	yes	direct product factor
23	C₅×C₃₁	C₅×C₂²⋊C₉	1	6	yes	direct product factor
24	C₂×C₃×C₅²	—	3	1	no	—
25	C₂²×C₃₁	C₅²×C₉	1	1	yes	characteristic, complemented, Hall
26	C₂²×C₅²	C₉×C₃₁	1	1	yes	characteristic, complemented, Hall
27	C₃×C₃₁	C₅²×A₄	1	1	yes	characteristic
28	C₃×C₅²	A₄×C₃₁	1	1	yes	characteristic
29	C₂×C₃₁	—	3	1	no	—
30	C₂²×C₃×C₅	C₃×C₅×C₃₁	1	6	yes	—
31	C₂×C₅²	—	3	1	no	—
32	C₅×C₉	—	4	6	no	—
33	C₂²⋊C₉	C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
34	C₃₁	C₅²×C₂²⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃×C₅	—	3	6	no	—
36	C₅²	C₃₁×C₂²⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
37	C₂²×C₅	C₅×C₉×C₃₁	1	6	yes	complemented
38	C₃×C₅	C₅×A₄×C₃₁	1	6	yes	—
39	C₂²×C₃	C₃×C₅²×C₃₁	1	1	yes	characteristic
40	C₂×C₅	—	3	6	no	—
41	C₉	—	4	1	no	Sylow
42	C₂×C₃	—	3	1	no	—
43	C₅	C₅×C₃₁×C₂²⋊C₉	1	6	yes	direct product factor
44	C₂²	C₅²×C₉×C₃₁	1	1	yes	characteristic, complemented, Sylow
45	C₃	C₅²×A₄×C₃₁	1	1	yes	characteristic
46	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅²×C₃₁	A₄
commutator, hypocenter	C₂²	C₅²×C₉×C₃₁
Frattini	C₃	C₅²×A₄×C₃₁
Fitting, socle, prime radical	C₂²×C₃×C₅²×C₃₁	C₃
cosocle	C₂²×C₃	C₃×C₅²×C₃₁
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₅²×C₃₁

Complements

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

N	H ≅ G/N	type
3. C₅×C₃₁×C₂²⋊C₉	C₅	direct (N × H)
5. C₂²×C₅²×C₃₁	C₉	semidirect
10. C₃₁×C₂²⋊C₉	C₅²	direct (N × H)
12. C₅²×C₂²⋊C₉	C₃₁	direct (N × H)
13. C₅²×C₃₁	C₂²⋊C₉	direct (N × H)
14. C₂²×C₅×C₃₁	C₅×C₉	semidirect
22. C₅×C₂²⋊C₉	C₅×C₃₁	direct (N × H)
23. C₅×C₃₁	C₅×C₂²⋊C₉	direct (N × H)
25. C₂²×C₃₁	C₅²×C₉	semidirect
26. C₂²×C₅²	C₉×C₃₁	semidirect
33. C₂²⋊C₉	C₅²×C₃₁	direct (N × H)
34. C₃₁	C₅²×C₂²⋊C₉	direct (N × H)
36. C₅²	C₃₁×C₂²⋊C₉	direct (N × H)
37. C₂²×C₅	C₅×C₉×C₃₁	semidirect
43. C₅	C₅×C₃₁×C₂²⋊C₉	direct (N × H)
44. C₂²	C₅²×C₉×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅²×C₃₁	3
C₅²×C₉×C₃₁	4
C₅×C₃₁×C₂²⋊C₉	5
C₅²×C₂²⋊C₉	31

Subgroups — C52×C31×C22⋊C9 #42124578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₃₁×C₂²⋊C₉ #42124578