Subgroups — C₂×C₅⋊(C₅⋊(D₃₁⋊C₉)) #42124525

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	C₅⋊(C₅⋊(D₃₁⋊C₉))	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₅⋊D₁₅₅	C₃	1	1	yes	maximal, characteristic
4	C₅²×C₃₁⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅⋊(D₃₁⋊C₉)	—	5	6	no	maximal
6	C₃×C₅⋊D₁₅₅	C₂×C₃	1	2	yes	—
7	C₂×C₃×C₅²×C₃₁	C₂×C₃	1	1	yes	characteristic
8	C₂×C₅⋊D₁₅₅	C₉	1	1	yes	characteristic, complemented, Hall
9	C₂×C₅×C₃₁⋊C₉	D₅	1	6	yes	complemented
10	C₅⋊(D₃₁⋊C₉)	—	5	12	no	—
11	C₃×C₅²×C₃₁	C₂²×C₃	1	1	yes	characteristic
12	C₂×C₃×D₁₅₅	—	5	6	no	—
13	C₂×C₅²×C₃₁	C₂×C₉	1	1	yes	characteristic, complemented
14	C₅⋊D₁₅₅	C₂×C₉	1	2	yes	complemented
15	C₅×C₃₁⋊C₉	C₂×D₅	1	6	yes	complemented
16	C₂×D₃₁⋊C₉	—	25	1	no	Hall
17	C₂×C₃×C₅×C₃₁	C₃×D₅	1	6	yes	—
18	C₃×D₁₅₅	—	5	12	no	—
19	C₂×C₉×C₅⋊D₅	—	31	1	no	maximal, Hall
20	C₅²×C₃₁	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
21	C₂×D₁₅₅	—	5	6	no	—
22	C₂×C₃₁⋊C₉	C₅⋊D₅	1	1	yes	characteristic, complemented
23	D₃₁⋊C₉	—	25	2	no	—
24	C₃×C₅×C₃₁	C₂×C₃×D₅	1	6	yes	—
25	C₂×C₅²×C₉	—	31	1	no	—
26	C₉×C₅⋊D₅	—	31	2	no	—
27	C₂×C₃×D₃₁	—	25	1	no	—
28	C₂×C₅×C₃₁	C₉×D₅	1	6	yes	complemented
29	D₁₅₅	—	5	12	no	—
30	C₂×C₃×C₅⋊D₅	—	31	1	no	—
31	C₃₁⋊C₉	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
32	C₅²×C₉	—	31	1	no	Hall
33	C₂×C₃×C₃₁	C₃×C₅⋊D₅	1	1	yes	characteristic
34	C₃×D₃₁	—	25	2	no	—
35	C₂×C₉×D₅	—	155	6	no	—
36	C₅×C₃₁	C₂×C₉×D₅	1	6	yes	complemented
37	C₂×C₃×C₅²	D₃₁⋊C₃	1	1	yes	characteristic
38	C₃×C₅⋊D₅	—	31	2	no	—
39	C₂×D₃₁	—	25	1	no	Hall
40	C₂×C₅⋊D₅	—	31	1	no	Hall
41	C₃×C₃₁	C₂×C₃×C₅⋊D₅	1	1	yes	characteristic
42	C₂×C₅×C₉	—	31	6	no	—
43	C₉×D₅	—	155	12	no	—
44	C₃×C₅²	C₂×D₃₁⋊C₃	1	1	yes	characteristic
45	C₂×C₃₁	C₉×C₅⋊D₅	1	1	yes	characteristic, complemented
46	D₃₁	—	25	2	no	—
47	C₂×C₃×D₅	—	155	6	no	—
48	C₂×C₅²	D₃₁⋊C₉	1	1	yes	characteristic, complemented
49	C₅⋊D₅	—	31	2	no	—
50	C₅×C₉	—	31	6	no	—
51	C₂²×C₉	—	775	1	no	Hall
52	C₃₁	C₂×C₉×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₅	C₅⋊(D₃₁⋊C₃)	1	6	yes	—
54	C₃×D₅	—	155	12	no	—
55	C₅²	C₂×D₃₁⋊C₉	1	1	yes	characteristic, complemented, Sylow
56	C₂×D₅	—	155	6	no	—
57	C₂×C₉	—	775	2	no	—
58	C₂×C₉	—	31	1	no	—
59	C₃×C₅	C₂×C₅⋊(D₃₁⋊C₃)	1	6	yes	—
60	C₂²×C₃	—	775	1	no	—
61	C₂×C₅	C₅⋊(D₃₁⋊C₉)	1	6	yes	complemented
62	D₅	—	155	12	no	—
63	C₉	—	31	1	no	Sylow
64	C₂×C₃	—	775	2	no	—
65	C₂×C₃	C₅⋊(C₅⋊(D₃₁⋊C₃))	1	1	yes	characteristic
66	C₅	C₂×C₅⋊(D₃₁⋊C₉)	1	6	yes	complemented
67	C₂²	—	775	1	no	Sylow
68	C₃	C₂×C₅⋊(C₅⋊(D₃₁⋊C₃))	1	1	yes	characteristic
69	C₂	—	775	2	no	—
70	C₂	C₅⋊(C₅⋊(D₃₁⋊C₉))	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	C₅⋊(C₅⋊(D₃₁⋊C₃))
commutator, hypocenter	C₅²×C₃₁	C₂²×C₉
Frattini	C₃	C₂×C₅⋊(C₅⋊(D₃₁⋊C₃))
Fitting, socle	C₂×C₃×C₅²×C₃₁	C₂×C₃
prime radical	C₂×C₃×C₅⋊D₁₅₅	C₃
cosocle	C₃×C₅²×C₃₁	C₂²×C₃
LCS	G ▷ C₅²×C₃₁
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×C₃₁⋊C₉	C₂	semidirect
2. C₅⋊(C₅⋊(D₃₁⋊C₉))	C₂	direct (N × H)
4. C₅²×C₃₁⋊C₉	C₂²	semidirect
8. C₂×C₅⋊D₁₅₅	C₉	semidirect
9. C₂×C₅×C₃₁⋊C₉	D₅	semidirect
13. C₂×C₅²×C₃₁	C₂×C₉	semidirect
14. C₅⋊D₁₅₅	C₂×C₉	semidirect
15. C₅×C₃₁⋊C₉	C₂×D₅	semidirect
20. C₅²×C₃₁	C₂²×C₉	semidirect
22. C₂×C₃₁⋊C₉	C₅⋊D₅	semidirect
28. C₂×C₅×C₃₁	C₉×D₅	semidirect
31. C₃₁⋊C₉	C₂×C₅⋊D₅	semidirect
36. C₅×C₃₁	C₂×C₉×D₅	semidirect
45. C₂×C₃₁	C₉×C₅⋊D₅	semidirect
48. C₂×C₅²	D₃₁⋊C₉	semidirect
52. C₃₁	C₂×C₉×C₅⋊D₅	semidirect
55. C₅²	C₂×D₃₁⋊C₉	semidirect
61. C₂×C₅	C₅⋊(D₃₁⋊C₉)	semidirect
66. C₅	C₂×C₅⋊(D₃₁⋊C₉)	semidirect
70. C₂	C₅⋊(C₅⋊(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
C₅⋊(C₅⋊(D₃₁⋊C₉))	2
C₂×C₃×C₅⋊D₁₅₅	3
C₂×C₅⋊(D₃₁⋊C₉)	5
C₂×C₉×C₅⋊D₅	31

Subgroups — C2×C5⋊(C5⋊(D31⋊C9)) #42124525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅⋊(C₅⋊(D₃₁⋊C₉)) #42124525