Subgroups — C₅²×C₃₁×C₃⋊Dic₃ #42124657

Order 27900.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₅²×Dic₃×C₃₁	—	3	4	no	maximal
3	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₅×C₃₁×C₃⋊Dic₃	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₃×C₅²×C₃₁	D₃	1	4	yes	—
6	C₄×C₅²×C₃₁	—	9	1	no	Hall
7	C₂×C₃²×C₅×C₃₁	C₂×C₅	1	6	yes	—
8	C₃×C₅²×C₃₁	Dic₃	1	4	yes	complemented
9	C₅×Dic₃×C₃₁	—	3	24	no	—
10	C₂×C₅²×C₃₁	C₃⋊D₃	1	1	yes	characteristic
11	C₃²×C₅×C₃₁	C₄×C₅	1	6	yes	complemented
12	C₃₁×C₃⋊Dic₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₃×C₅×C₃₁	C₅×D₃	1	24	yes	—
14	C₅²×C₃⋊Dic₃	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅²×C₃₁	C₃⋊Dic₃	1	1	yes	characteristic, direct product factor, Hall
16	C₄×C₅×C₃₁	—	9	6	no	—
17	C₂×C₃²×C₃₁	C₂×C₅²	1	1	yes	characteristic
18	C₃×C₅×C₃₁	C₅×Dic₃	1	24	yes	complemented
19	C₂×C₃²×C₅²	C₂×C₃₁	1	1	yes	characteristic
20	Dic₃×C₃₁	—	3	4	no	—
21	C₂×C₅×C₃₁	C₅×C₃⋊D₃	1	6	yes	—
22	C₅²×Dic₃	—	3	4	no	—
23	C₃²×C₃₁	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
24	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₃₁	C₅²×D₃	1	4	yes	—
26	C₅×C₃⋊Dic₃	C₅×C₃₁	1	6	yes	direct product factor
27	C₅×C₃₁	C₅×C₃⋊Dic₃	1	6	yes	direct product factor
28	C₂×C₃×C₅²	D₃×C₃₁	1	4	yes	—
29	C₄×C₃₁	—	9	1	no	Hall
30	C₄×C₅²	—	9	1	no	Hall
31	C₃×C₃₁	C₅²×Dic₃	1	4	yes	complemented
32	C₂×C₃²×C₅	C₂×C₅×C₃₁	1	6	yes	—
33	C₃×C₅²	Dic₃×C₃₁	1	4	yes	complemented
34	C₂×C₃₁	C₅²×C₃⋊D₃	1	1	yes	characteristic
35	C₅×Dic₃	—	3	24	no	—
36	C₂×C₅²	C₃⋊D₃×C₃₁	1	1	yes	characteristic
37	C₃²×C₅	C₄×C₅×C₃₁	1	6	yes	complemented
38	C₃⋊Dic₃	C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
39	C₃₁	C₅²×C₃⋊Dic₃	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₃×C₅	C₅×D₃×C₃₁	1	24	yes	—
41	C₅²	C₃₁×C₃⋊Dic₃	1	1	yes	characteristic, direct product factor, Sylow
42	C₄×C₅	—	9	6	no	—
43	C₂×C₃²	C₂×C₅²×C₃₁	1	1	yes	characteristic
44	C₃×C₅	C₅×Dic₃×C₃₁	1	24	yes	complemented
45	Dic₃	—	3	4	no	—
46	C₂×C₅	C₅×C₃⋊D₃×C₃₁	1	6	yes	—
47	C₃²	C₄×C₅²×C₃₁	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃	C₅²×D₃×C₃₁	1	4	yes	—
49	C₅	C₅×C₃₁×C₃⋊Dic₃	1	6	yes	direct product factor
50	C₄	—	9	1	no	Sylow
51	C₃	C₅²×Dic₃×C₃₁	1	4	yes	complemented
52	C₂	C₅²×C₃⋊D₃×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅²×C₃₁	C₃⋊D₃
commutator, hypocenter	C₃²	C₄×C₅²×C₃₁
Frattini	C₂	C₅²×C₃⋊D₃×C₃₁
Fitting, socle, prime radical	C₂×C₃²×C₅²×C₃₁	C₂
cosocle	C₂×C₃²	C₂×C₅²×C₃₁
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₅²×C₃₁

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅²×C₃₁	C₄	semidirect
4. C₅×C₃₁×C₃⋊Dic₃	C₅	direct (N × H)
8. C₃×C₅²×C₃₁	Dic₃	semidirect
11. C₃²×C₅×C₃₁	C₄×C₅	semidirect
12. C₃₁×C₃⋊Dic₃	C₅²	direct (N × H)
14. C₅²×C₃⋊Dic₃	C₃₁	direct (N × H)
15. C₅²×C₃₁	C₃⋊Dic₃	direct (N × H)
18. C₃×C₅×C₃₁	C₅×Dic₃	semidirect
23. C₃²×C₃₁	C₄×C₅²	semidirect
24. C₃²×C₅²	C₄×C₃₁	semidirect
26. C₅×C₃⋊Dic₃	C₅×C₃₁	direct (N × H)
27. C₅×C₃₁	C₅×C₃⋊Dic₃	direct (N × H)
31. C₃×C₃₁	C₅²×Dic₃	semidirect
33. C₃×C₅²	Dic₃×C₃₁	semidirect
37. C₃²×C₅	C₄×C₅×C₃₁	semidirect
38. C₃⋊Dic₃	C₅²×C₃₁	direct (N × H)
39. C₃₁	C₅²×C₃⋊Dic₃	direct (N × H)
41. C₅²	C₃₁×C₃⋊Dic₃	direct (N × H)
44. C₃×C₅	C₅×Dic₃×C₃₁	semidirect
47. C₃²	C₄×C₅²×C₃₁	semidirect
49. C₅	C₅×C₃₁×C₃⋊Dic₃	direct (N × H)
51. C₃	C₅²×Dic₃×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅²×C₃₁	2
C₅²×Dic₃×C₃₁	3
C₅×C₃₁×C₃⋊Dic₃	5
C₅²×C₃⋊Dic₃	31

Subgroups — C52×C31×C3⋊Dic3 #42124657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₃₁×C₃⋊Dic₃ #42124657