Subgroups — C₂×C₅²×D₃₁⋊C₉ #42124522

Order 27900.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₃₁⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×D₃₁⋊C₉	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₅²×D₃₁	C₃	1	1	yes	maximal, characteristic
4	C₅²×C₃₁⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×D₃₁⋊C₉	C₅	1	6	yes	maximal, direct product factor
6	C₃×C₅²×D₃₁	C₂×C₃	1	2	yes	—
7	C₂×C₃×C₅²×C₃₁	C₂×C₃	1	1	yes	characteristic
8	C₂×C₅²×D₃₁	C₉	1	1	yes	characteristic, complemented, Hall
9	C₂×C₅×C₃₁⋊C₉	C₂×C₅	1	6	yes	complemented
10	C₅×D₃₁⋊C₉	C₂×C₅	1	12	yes	direct product factor
11	C₃×C₅²×C₃₁	C₂²×C₃	1	1	yes	characteristic
12	C₂×C₃×C₅×D₃₁	C₃×C₅	1	6	yes	—
13	C₂×C₅²×C₃₁	C₂×C₉	1	1	yes	characteristic, complemented
14	C₅²×D₃₁	C₂×C₉	1	2	yes	complemented
15	C₅×C₃₁⋊C₉	C₂²×C₅	1	6	yes	complemented
16	C₂×D₃₁⋊C₉	C₅²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃×C₅×C₃₁	C₂×C₃×C₅	1	6	yes	—
18	C₃×C₅×D₃₁	C₂×C₃×C₅	1	12	yes	—
19	C₂²×C₅²×C₉	—	31	1	no	maximal, Hall
20	C₅²×C₃₁	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
21	C₂×C₅×D₃₁	C₅×C₉	1	6	yes	complemented
22	C₂×C₃₁⋊C₉	C₂×C₅²	1	1	yes	characteristic, complemented
23	D₃₁⋊C₉	C₂×C₅²	1	2	yes	direct product factor
24	C₃×C₅×C₃₁	C₂²×C₃×C₅	1	6	yes	—
25	C₂×C₅²×C₉	—	31	2	no	—
26	C₂×C₅²×C₉	—	31	1	no	—
27	C₂×C₃×D₃₁	C₃×C₅²	1	1	yes	characteristic
28	C₂×C₅×C₃₁	C₂×C₅×C₉	1	6	yes	complemented
29	C₅×D₃₁	C₂×C₅×C₉	1	12	yes	complemented
30	C₂²×C₃×C₅²	—	31	1	no	—
31	C₃₁⋊C₉	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
32	C₅²×C₉	—	31	1	no	Hall
33	C₂×C₃×C₃₁	C₂×C₃×C₅²	1	1	yes	characteristic
34	C₃×D₃₁	C₂×C₃×C₅²	1	2	yes	—
35	C₂²×C₅×C₉	—	31	6	no	—
36	C₅×C₃₁	C₂²×C₅×C₉	1	6	yes	complemented
37	C₂×C₃×C₅²	—	31	2	no	—
38	C₂×C₃×C₅²	D₃₁⋊C₃	1	1	yes	characteristic
39	C₂×D₃₁	C₅²×C₉	1	1	yes	characteristic, complemented, Hall
40	C₂²×C₅²	—	31	1	no	Hall
41	C₃×C₃₁	C₂²×C₃×C₅²	1	1	yes	characteristic
42	C₂×C₅×C₉	—	31	12	no	—
43	C₂×C₅×C₉	—	31	6	no	—
44	C₃×C₅²	C₂×D₃₁⋊C₃	1	1	yes	characteristic
45	C₂×C₃₁	C₂×C₅²×C₉	1	1	yes	characteristic, complemented
46	D₃₁	C₂×C₅²×C₉	1	2	yes	complemented
47	C₂²×C₃×C₅	—	31	6	no	—
48	C₂×C₅²	—	31	2	no	—
49	C₂×C₅²	D₃₁⋊C₉	1	1	yes	characteristic, direct product factor
50	C₅×C₉	—	31	6	no	—
51	C₂²×C₉	—	31	1	no	Hall
52	C₃₁	C₂²×C₅²×C₉	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₅	—	31	12	no	—
54	C₂×C₃×C₅	C₅×D₃₁⋊C₃	1	6	yes	—
55	C₅²	C₂×D₃₁⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₅	—	31	6	no	—
57	C₂×C₉	—	31	2	no	—
58	C₂×C₉	—	31	1	no	—
59	C₃×C₅	C₂×C₅×D₃₁⋊C₃	1	6	yes	—
60	C₂²×C₃	—	31	1	no	—
61	C₂×C₅	—	31	12	no	—
62	C₂×C₅	C₅×D₃₁⋊C₉	1	6	yes	direct product factor
63	C₉	—	31	1	no	Sylow
64	C₂×C₃	—	31	2	no	—
65	C₂×C₃	C₅²×D₃₁⋊C₃	1	1	yes	characteristic
66	C₅	C₂×C₅×D₃₁⋊C₉	1	6	yes	direct product factor
67	C₂²	—	31	1	no	Sylow
68	C₃	C₂×C₅²×D₃₁⋊C₃	1	1	yes	characteristic
69	C₂	—	31	2	no	—
70	C₂	C₅²×D₃₁⋊C₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅²	D₃₁⋊C₃
commutator, hypocenter	C₃₁	C₂²×C₅²×C₉
Frattini	C₃	C₂×C₅²×D₃₁⋊C₃
Fitting, socle	C₂×C₃×C₅²×C₃₁	C₂×C₃
prime radical	C₂×C₃×C₅²×D₃₁	C₃
cosocle	C₃×C₃₁	C₂²×C₃×C₅²
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂×C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×C₃₁⋊C₉	C₂	semidirect
2. C₅²×D₃₁⋊C₉	C₂	direct (N × H)
4. C₅²×C₃₁⋊C₉	C₂²	semidirect
5. C₂×C₅×D₃₁⋊C₉	C₅	direct (N × H)
8. C₂×C₅²×D₃₁	C₉	semidirect
9. C₂×C₅×C₃₁⋊C₉	C₂×C₅	semidirect
10. C₅×D₃₁⋊C₉	C₂×C₅	direct (N × H)
13. C₂×C₅²×C₃₁	C₂×C₉	semidirect
14. C₅²×D₃₁	C₂×C₉	semidirect
15. C₅×C₃₁⋊C₉	C₂²×C₅	semidirect
16. C₂×D₃₁⋊C₉	C₅²	direct (N × H)
20. C₅²×C₃₁	C₂²×C₉	semidirect
21. C₂×C₅×D₃₁	C₅×C₉	semidirect
22. C₂×C₃₁⋊C₉	C₂×C₅²	semidirect
23. D₃₁⋊C₉	C₂×C₅²	direct (N × H)
28. C₂×C₅×C₃₁	C₂×C₅×C₉	semidirect
29. C₅×D₃₁	C₂×C₅×C₉	semidirect
31. C₃₁⋊C₉	C₂²×C₅²	semidirect
36. C₅×C₃₁	C₂²×C₅×C₉	semidirect
39. C₂×D₃₁	C₅²×C₉	semidirect
45. C₂×C₃₁	C₂×C₅²×C₉	semidirect
46. D₃₁	C₂×C₅²×C₉	semidirect
49. C₂×C₅²	D₃₁⋊C₉	direct (N × H)
52. C₃₁	C₂²×C₅²×C₉	semidirect
55. C₅²	C₂×D₃₁⋊C₉	direct (N × H)
62. C₂×C₅	C₅×D₃₁⋊C₉	direct (N × H)
66. C₅	C₂×C₅×D₃₁⋊C₉	direct (N × H)
70. C₂	C₅²×D₃₁⋊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×C₅²×C₃₁⋊C₉	2
C₅²×D₃₁⋊C₉	2
C₂×C₃×C₅²×D₃₁	3
C₂×C₅×D₃₁⋊C₉	5
C₂²×C₅²×C₉	31

Subgroups — C2×C52×D31⋊C9 #42124522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×D₃₁⋊C₉ #42124522