Subgroups — C₄×C₃₁×C₅²⋊C₉ #42124366

Order 27900.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₅²⋊C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅²×C₃₁	C₃	1	1	yes	maximal, characteristic
3	C₃₁×C₅²⋊C₉	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₃×C₅²×C₃₁	C₂×C₃	1	1	yes	characteristic
5	C₄×C₅²×C₃₁	C₉	1	1	yes	characteristic, complemented, Hall
6	C₃×C₅²×C₃₁	C₃×C₄	1	1	yes	characteristic
7	C₃×C₄×C₅×C₃₁	—	3	2	no	—
8	C₂×C₅²×C₃₁	C₂×C₉	1	1	yes	characteristic
9	C₄×C₉×C₃₁	—	25	1	no	maximal, Hall
10	C₂×C₃×C₅×C₃₁	—	3	2	no	—
11	C₄×C₅²⋊C₉	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₅²×C₃₁	C₄×C₉	1	1	yes	characteristic, complemented, Hall
13	C₄×C₅×C₃₁	—	3	2	no	—
14	C₂×C₉×C₃₁	—	25	1	no	—
15	C₃×C₅×C₃₁	—	3	2	no	—
16	C₂×C₅²⋊C₉	C₂×C₃₁	1	1	yes	characteristic
17	C₃×C₄×C₃₁	C₅²⋊C₃	1	1	yes	characteristic
18	C₂×C₅×C₃₁	—	3	2	no	—
19	C₃×C₄×C₅²	C₃×C₃₁	1	1	yes	characteristic
20	C₉×C₃₁	—	25	1	no	Hall
21	C₅²⋊C₉	C₄×C₃₁	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₃×C₃₁	C₂×C₅²⋊C₃	1	1	yes	characteristic
23	C₅×C₃₁	—	3	2	no	—
24	C₂×C₃×C₅²	C₂×C₃×C₃₁	1	1	yes	characteristic
25	C₄×C₃₁	C₅²⋊C₉	1	1	yes	characteristic, direct product factor, Hall
26	C₄×C₅²	C₉×C₃₁	1	1	yes	characteristic, complemented, Hall
27	C₃×C₃₁	C₄×C₅²⋊C₃	1	1	yes	characteristic
28	C₃×C₅²	C₃×C₄×C₃₁	1	1	yes	characteristic
29	C₂×C₃₁	C₂×C₅²⋊C₉	1	1	yes	characteristic
30	C₃×C₄×C₅	—	3	2	no	—
31	C₂×C₅²	C₂×C₉×C₃₁	1	1	yes	characteristic
32	C₄×C₉	—	25	1	no	Hall
33	C₃₁	C₄×C₅²⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₃×C₅	—	3	2	no	—
35	C₅²	C₄×C₉×C₃₁	1	1	yes	characteristic, complemented, Sylow
36	C₄×C₅	—	3	2	no	—
37	C₂×C₉	—	25	1	no	—
38	C₃×C₅	—	3	2	no	—
39	C₃×C₄	C₃₁×C₅²⋊C₃	1	1	yes	characteristic
40	C₂×C₅	—	3	2	no	—
41	C₉	—	25	1	no	Sylow
42	C₂×C₃	C₂×C₃₁×C₅²⋊C₃	1	1	yes	characteristic
43	C₅	—	3	2	no	—
44	C₄	C₃₁×C₅²⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
45	C₃	C₄×C₃₁×C₅²⋊C₃	1	1	yes	characteristic
46	C₂	C₂×C₃₁×C₅²⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄×C₃₁	C₅²⋊C₃
commutator, hypocenter	C₅²	C₄×C₉×C₃₁
Frattini	C₂×C₃	C₂×C₃₁×C₅²⋊C₃
Fitting	C₃×C₄×C₅²×C₃₁	C₃
socle, prime radical	C₂×C₃×C₅²×C₃₁	C₂×C₃
cosocle	C₂×C₃×C₅²	C₂×C₃×C₃₁
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃×C₄×C₃₁

Complements

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

N	H ≅ G/N	type
3. C₃₁×C₅²⋊C₉	C₄	direct (N × H)
5. C₄×C₅²×C₃₁	C₉	semidirect
11. C₄×C₅²⋊C₉	C₃₁	direct (N × H)
12. C₅²×C₃₁	C₄×C₉	semidirect
21. C₅²⋊C₉	C₄×C₃₁	direct (N × H)
25. C₄×C₃₁	C₅²⋊C₉	direct (N × H)
26. C₄×C₅²	C₉×C₃₁	semidirect
33. C₃₁	C₄×C₅²⋊C₉	direct (N × H)
35. C₅²	C₄×C₉×C₃₁	semidirect
44. C₄	C₃₁×C₅²⋊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×C₃₁×C₅²⋊C₉	2
C₃×C₄×C₅²×C₃₁	3
C₄×C₉×C₃₁	25
C₄×C₅²⋊C₉	31

Subgroups — C4×C31×C52⋊C9 #42124366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×C₅²⋊C₉ #42124366