Subgroups — D₅×C₅⋊(D₆₁⋊C₃) #32913041

Order 18300.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊(C₅⋊(D₆₁⋊C₃))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₅×C₆₁⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₅⋊(D₆₁⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×D₃₀₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₅²×C₆₁⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₅⋊(D₆₁⋊C₃)	—	5	1	no	maximal
7	D₅×D₆₁⋊C₃	—	5	1	no	maximal
8	C₅⋊D₃₀₅	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×D₃₀₅	C₂×C₃	1	1	yes	characteristic, complemented
10	C₅×D₅×C₆₁	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₅×C₆₁⋊C₃	—	5	1	no	—
12	C₅⋊(D₆₁⋊C₃)	—	5	1	no	—
13	C₅⋊(D₆₁⋊C₃)	—	5	1	no	—
14	C₅⋊(D₆₁⋊C₃)	—	10	2	no	—
15	C₅⋊(D₆₁⋊C₃)	D₅	1	1	yes	characteristic, direct product factor
16	D₅×C₆₁⋊C₃	D₅	1	1	yes	characteristic, complemented
17	C₅×D₆₁⋊C₃	—	5	1	no	—
18	C₅²×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
19	C₂×D₃₀₅	—	5	1	no	—
20	D₅×D₆₁	—	5	1	no	—
21	C₅×C₆₁⋊C₃	—	2	2	no	—
22	C₅×C₆₁⋊C₃	C₂×D₅	1	1	yes	characteristic, complemented
23	C₅×C₆₁⋊C₃	C₂×D₅	1	1	yes	characteristic, complemented
24	C₂×D₆₁⋊C₃	—	25	1	no	Hall
25	C₂×C₅×C₆₁	—	5	1	no	—
26	D₃₀₅	—	5	1	no	—
27	D₃₀₅	—	5	1	no	—
28	D₃₀₅	—	10	2	no	—
29	D₃₀₅	C₃×D₅	1	1	yes	characteristic, complemented
30	C₅×D₆₁	—	5	1	no	—
31	D₅×C₆₁	C₃×D₅	1	1	yes	characteristic, complemented
32	C₂×C₆₁⋊C₃	—	5	1	no	—
33	D₆₁⋊C₃	—	5	1	no	—
34	D₆₁⋊C₃	—	25	1	no	—
35	C₅×C₆₁	—	2	2	no	—
36	C₅×C₆₁	C₂×C₃×D₅	1	1	yes	characteristic, complemented
37	C₅×C₆₁	C₂×C₃×D₅	1	1	yes	characteristic, complemented
38	C₃×D₅²	—	61	1	no	maximal, Hall
39	C₂×D₆₁	—	25	1	no	Hall
40	C₆₁⋊C₃	D₅²	1	1	yes	characteristic, complemented, Hall
41	C₃×C₅⋊D₅	—	61	1	no	—
42	C₃×C₅×D₅	—	61	1	no	—
43	C₃×C₅×D₅	—	122	1	no	—
44	C₂×C₆₁	—	5	1	no	—
45	D₆₁	—	5	1	no	—
46	D₆₁	—	25	1	no	—
47	D₅²	—	61	1	no	Hall
48	C₃×C₅²	—	122	1	no	Hall
49	C₆₁	C₃×D₅²	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃×D₅	—	305	1	no	—
51	C₂×C₃×D₅	—	305	1	no	—
52	C₅⋊D₅	—	61	1	no	—
53	C₅×D₅	—	61	1	no	—
54	C₅×D₅	D₆₁⋊C₃	1	1	yes	characteristic, complemented
55	C₂×C₃×C₅	—	305	1	no	—
56	C₂×C₃×C₅	—	610	1	no	—
57	C₃×D₅	—	305	1	no	—
58	C₃×D₅	—	305	1	no	—
59	C₃×D₅	—	61	1	no	—
60	C₃×D₅	—	610	2	no	—
61	C₃×D₅	—	122	1	no	—
62	C₅²	C₂×D₆₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₅	—	305	1	no	—
64	C₂×D₅	—	305	1	no	—
65	C₃×C₅	—	122	1	no	—
66	C₃×C₅	—	122	1	no	—
67	C₃×C₅	—	244	2	no	—
68	C₂²×C₃	—	1525	1	no	Hall
69	C₂×C₅	—	5	1	no	—
70	C₂×C₅	—	305	1	no	—
71	D₅	—	305	1	no	—
72	D₅	—	305	1	no	—
73	D₅	—	61	1	no	—
74	D₅	—	610	2	no	—
75	D₅	C₅⋊(D₆₁⋊C₃)	1	1	yes	characteristic, direct product factor
76	C₂×C₃	—	305	1	no	—
77	C₂×C₃	—	610	1	no	—
78	C₂×C₃	—	1525	1	no	—
79	C₅	—	2	2	no	—
80	C₅	D₅×D₆₁⋊C₃	1	1	yes	characteristic, complemented
81	C₅	C₂×C₅⋊(D₆₁⋊C₃)	1	1	yes	characteristic, complemented
82	C₂²	—	1525	1	no	Sylow
83	C₃	—	122	1	no	Sylow
84	C₂	—	5	1	no	—
85	C₂	—	305	1	no	—
86	C₂	—	1525	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅²×C₆₁	C₂²×C₃
LCS	G ▷ C₅²×C₆₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅⋊(C₅⋊(D₆₁⋊C₃))	C₂	semidirect
2. C₅×D₅×C₆₁⋊C₃	C₂	semidirect
3. C₅×C₅⋊(D₆₁⋊C₃)	C₂	semidirect
4. D₅×D₃₀₅	C₃	semidirect
5. C₅²×C₆₁⋊C₃	C₂²	semidirect
8. C₅⋊D₃₀₅	C₂×C₃	semidirect
9. C₅×D₃₀₅	C₂×C₃	semidirect
10. C₅×D₅×C₆₁	C₂×C₃	semidirect
15. C₅⋊(D₆₁⋊C₃)	D₅	direct (N × H)
16. D₅×C₆₁⋊C₃	D₅	semidirect
18. C₅²×C₆₁	C₂²×C₃	semidirect
22. C₅×C₆₁⋊C₃	C₂×D₅	semidirect
23. C₅×C₆₁⋊C₃	C₂×D₅	semidirect
29. D₃₀₅	C₃×D₅	semidirect
31. D₅×C₆₁	C₃×D₅	semidirect
36. C₅×C₆₁	C₂×C₃×D₅	semidirect
37. C₅×C₆₁	C₂×C₃×D₅	semidirect
40. C₆₁⋊C₃	D₅²	semidirect
49. C₆₁	C₃×D₅²	semidirect
54. C₅×D₅	D₆₁⋊C₃	semidirect
62. C₅²	C₂×D₆₁⋊C₃	semidirect
75. D₅	C₅⋊(D₆₁⋊C₃)	direct (N × H)
80. C₅	D₅×D₆₁⋊C₃	semidirect
81. C₅	C₂×C₅⋊(D₆₁⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊(C₅⋊(D₆₁⋊C₃))	2
C₅×D₅×C₆₁⋊C₃	2
C₅×C₅⋊(D₆₁⋊C₃)	2
D₅×D₃₀₅	3
C₂×C₅⋊(D₆₁⋊C₃)	5
D₅×D₆₁⋊C₃	5
C₃×D₅²	61

Subgroups — D5×C5⋊(D61⋊C3) #32913041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×C₅⋊(D₆₁⋊C₃) #32913041