Subgroups — C₃×C₅²×C₂²⋊(C₃₁⋊C₃) #42124976

Order 27900.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₂²⋊(C₃₁⋊C₃)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅²×C₃₁⋊C₃	—	4	1	no	maximal, Hall
4	C₃×C₅×C₂²⋊(C₃₁⋊C₃)	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₃×C₅²×C₃₁	—	3	1	no	—
6	C₂²×C₅²×C₃₁	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₅²×C₃₁⋊C₃	—	4	3	no	—
8	C₃×C₅²×C₃₁	A₄	1	1	yes	characteristic, complemented
9	C₂²×C₃×C₅×C₃₁	C₃×C₅	1	6	yes	complemented
10	C₅×C₂²⋊(C₃₁⋊C₃)	C₃×C₅	1	18	yes	direct product factor
11	C₂×C₅²×C₃₁	—	3	1	no	—
12	C₃×C₅×C₃₁⋊C₃	—	4	6	no	—
13	C₃×C₂²⋊(C₃₁⋊C₃)	C₅²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃×C₅×C₃₁	—	3	6	no	—
15	C₃×C₅²×A₄	—	31	1	no	maximal, Hall
16	C₅²×C₃₁	C₃×A₄	1	1	yes	characteristic, complemented, Hall
17	C₂²×C₅×C₃₁	C₃²×C₅	1	6	yes	complemented
18	C₃×C₅×C₃₁	C₅×A₄	1	6	yes	complemented
19	C₅×C₃₁⋊C₃	—	4	18	no	—
20	C₂²×C₃×C₃₁	C₃×C₅²	1	1	yes	characteristic, complemented
21	C₂²⋊(C₃₁⋊C₃)	C₃×C₅²	1	3	yes	direct product factor
22	C₂×C₅×C₃₁	—	3	6	no	—
23	C₂²×C₃×C₅²	C₃₁⋊C₃	1	1	yes	characteristic, complemented
24	C₅²×A₄	—	31	3	no	—
25	C₃×C₃₁⋊C₃	—	4	1	no	Hall
26	C₃²×C₅²	—	124	1	no	Hall
27	C₂×C₃×C₃₁	—	3	1	no	—
28	C₃×C₅×A₄	—	31	6	no	—
29	C₅×C₃₁	C₃×C₅×A₄	1	6	yes	complemented
30	C₂×C₃×C₅²	—	3	1	no	—
31	C₂²×C₃₁	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
32	C₂²×C₅²	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
33	C₃×C₃₁	C₅²×A₄	1	1	yes	characteristic, complemented
34	C₃₁⋊C₃	—	4	3	no	—
35	C₃×C₅²	—	124	3	no	—
36	C₃×C₅²	C₂²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, direct product factor
37	C₂×C₃₁	—	3	1	no	—
38	C₂²×C₃×C₅	C₅×C₃₁⋊C₃	1	6	yes	complemented
39	C₅×A₄	—	31	18	no	—
40	C₂×C₅²	—	3	1	no	—
41	C₃²×C₅	—	124	6	no	—
42	C₃×A₄	—	31	1	no	Hall
43	C₃₁	C₃×C₅²×A₄	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×C₅	—	3	6	no	—
45	C₅²	C₃×C₂²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²×C₅	C₃×C₅×C₃₁⋊C₃	1	6	yes	complemented
47	C₃×C₅	—	124	18	no	—
48	C₃×C₅	C₅×C₂²⋊(C₃₁⋊C₃)	1	6	yes	direct product factor
49	C₂²×C₃	C₅²×C₃₁⋊C₃	1	1	yes	characteristic, complemented
50	A₄	—	31	3	no	—
51	C₂×C₅	—	3	6	no	—
52	C₃²	—	124	1	no	Sylow
53	C₂×C₃	—	3	1	no	—
54	C₅	C₃×C₅×C₂²⋊(C₃₁⋊C₃)	1	6	yes	direct product factor
55	C₂²	C₃×C₅²×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	124	3	no	—
57	C₃	C₅²×C₂²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, direct product factor
58	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅²	C₂²⋊(C₃₁⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₃₁	C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₃₁	C₃
LCS	G ▷ C₂²×C₃₁
UCS	C₁ ◁ C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₅²×C₂²⋊(C₃₁⋊C₃)	C₃	direct (N × H)
2. C₂²×C₃×C₅²×C₃₁	C₃	semidirect
4. C₃×C₅×C₂²⋊(C₃₁⋊C₃)	C₅	direct (N × H)
6. C₂²×C₅²×C₃₁	C₃²	semidirect
8. C₃×C₅²×C₃₁	A₄	semidirect
9. C₂²×C₃×C₅×C₃₁	C₃×C₅	semidirect
10. C₅×C₂²⋊(C₃₁⋊C₃)	C₃×C₅	direct (N × H)
13. C₃×C₂²⋊(C₃₁⋊C₃)	C₅²	direct (N × H)
16. C₅²×C₃₁	C₃×A₄	semidirect
17. C₂²×C₅×C₃₁	C₃²×C₅	semidirect
18. C₃×C₅×C₃₁	C₅×A₄	semidirect
20. C₂²×C₃×C₃₁	C₃×C₅²	semidirect
21. C₂²⋊(C₃₁⋊C₃)	C₃×C₅²	direct (N × H)
23. C₂²×C₃×C₅²	C₃₁⋊C₃	semidirect
29. C₅×C₃₁	C₃×C₅×A₄	semidirect
31. C₂²×C₃₁	C₃²×C₅²	semidirect
32. C₂²×C₅²	C₃×C₃₁⋊C₃	semidirect
33. C₃×C₃₁	C₅²×A₄	semidirect
36. C₃×C₅²	C₂²⋊(C₃₁⋊C₃)	direct (N × H)
38. C₂²×C₃×C₅	C₅×C₃₁⋊C₃	semidirect
43. C₃₁	C₃×C₅²×A₄	semidirect
45. C₅²	C₃×C₂²⋊(C₃₁⋊C₃)	direct (N × H)
46. C₂²×C₅	C₃×C₅×C₃₁⋊C₃	semidirect
48. C₃×C₅	C₅×C₂²⋊(C₃₁⋊C₃)	direct (N × H)
49. C₂²×C₃	C₅²×C₃₁⋊C₃	semidirect
54. C₅	C₃×C₅×C₂²⋊(C₃₁⋊C₃)	direct (N × H)
55. C₂²	C₃×C₅²×C₃₁⋊C₃	semidirect
57. C₃	C₅²×C₂²⋊(C₃₁⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×C₂²⋊(C₃₁⋊C₃)	3
C₂²×C₃×C₅²×C₃₁	3
C₃×C₅²×C₃₁⋊C₃	4
C₃×C₅×C₂²⋊(C₃₁⋊C₃)	5
C₃×C₅²×A₄	31

Subgroups — C3×C52×C22⋊(C31⋊C3) #42124976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×C₂²⋊(C₃₁⋊C₃) #42124976