Subgroups — C₃₁×D₄₅⋊D₅ #42124572

Order 27900.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃₁×D₄₅	C₂	1	2	yes	maximal, complemented
2	C₉×C₃₁×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×D₁₅⋊D₅	—	3	1	no	maximal
4	C₅²×C₉×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₅×D₉×C₃₁	—	5	2	no	maximal
6	C₅×D₁₅×C₃₁	—	3	2	no	—
7	C₃×C₃₁×C₅⋊D₅	D₃	1	1	yes	characteristic
8	D₅²×C₃₁	—	9	1	no	Hall
9	C₃₁×D₄₅	D₅	1	2	yes	complemented
10	C₅×D₉×C₃₁	—	5	2	no	—
11	C₉×D₅×C₃₁	—	5	2	no	—
12	C₉×D₅×C₃₁	—	10	2	no	—
13	C₃×C₅²×C₃₁	C₂×D₃	1	1	yes	characteristic
14	D₃×D₅×C₃₁	—	15	2	no	—
15	C₃₁×C₅⋊D₅	D₉	1	1	yes	characteristic, complemented
16	C₅×D₅×C₃₁	—	9	2	no	—
17	C₅×C₉×C₃₁	—	2	2	no	—
18	C₅×C₉×C₃₁	C₂×D₅	1	2	yes	complemented
19	C₂×D₉×C₃₁	—	25	1	no	Hall
20	D₁₅×C₃₁	—	3	2	no	—
21	C₅×D₃×C₃₁	—	15	2	no	—
22	C₃×D₅×C₃₁	—	5	2	no	—
23	C₃×D₅×C₃₁	—	10	2	no	—
24	D₄₅⋊D₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅²×C₃₁	C₂×D₉	1	1	yes	characteristic, complemented, Hall
26	C₂×D₅×C₃₁	—	45	2	no	—
27	C₂×C₉×C₃₁	—	25	1	no	—
28	D₉×C₃₁	—	5	2	no	—
29	C₃×C₅×C₃₁	—	2	2	no	—
30	C₃×C₅×C₃₁	D₃×D₅	1	2	yes	—
31	C₅×D₄₅	C₂×C₃₁	1	2	yes	complemented
32	C₉×C₅⋊D₅	C₂×C₃₁	1	1	yes	characteristic, complemented
33	C₂×D₃×C₃₁	—	75	1	no	—
34	C₂×C₅×C₃₁	—	45	2	no	—
35	D₅×C₃₁	—	5	2	no	—
36	D₅×C₃₁	—	9	2	no	—
37	D₅×C₃₁	—	10	2	no	—
38	D₁₅⋊D₅	—	3	1	no	—
39	C₉×C₃₁	D₅²	1	1	yes	characteristic, complemented, Hall
40	C₅²×C₉	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₃₁	—	25	1	no	—
42	D₃×C₃₁	—	15	2	no	—
43	D₅×D₉	—	5	2	no	—
44	C₅×C₃₁	—	2	2	no	—
45	C₅×C₃₁	D₅×D₉	1	2	yes	complemented
46	C₅×D₁₅	—	3	2	no	—
47	C₃×C₅⋊D₅	D₃×C₃₁	1	1	yes	characteristic
48	C₂²×C₃₁	—	225	1	no	Hall
49	D₅²	—	9	1	no	Hall
50	C₃×C₃₁	D₁₅⋊D₅	1	1	yes	characteristic
51	D₄₅	D₅×C₃₁	1	2	yes	complemented
52	C₉×D₅	—	5	2	no	—
53	C₉×D₅	—	10	2	no	—
54	C₅×D₉	—	5	2	no	—
55	C₃×C₅²	C₂×D₃×C₃₁	1	1	yes	characteristic
56	C₂×C₃₁	—	25	1	no	—
57	C₂×C₃₁	—	45	2	no	—
58	D₃×D₅	—	15	2	no	—
59	C₅⋊D₅	D₉×C₃₁	1	1	yes	characteristic, complemented
60	C₅×D₅	—	9	2	no	—
61	C₅×C₉	—	2	2	no	—
62	C₅×C₉	C₂×D₅×C₃₁	1	2	yes	complemented
63	C₂×D₉	—	25	1	no	Hall
64	C₃₁	D₄₅⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
65	D₁₅	—	3	2	no	—
66	C₃×D₅	—	5	2	no	—
67	C₃×D₅	—	10	2	no	—
68	C₅×D₃	—	15	2	no	—
69	C₅²	C₂×D₉×C₃₁	1	1	yes	characteristic, complemented, Sylow
70	C₂×D₅	—	45	2	no	—
71	C₂×C₉	—	25	1	no	—
72	D₉	—	5	2	no	—
73	C₃×C₅	—	2	2	no	—
74	C₃×C₅	D₃×D₅×C₃₁	1	2	yes	—
75	C₂×D₃	—	75	1	no	—
76	C₂×C₅	—	45	2	no	—
77	D₅	—	5	2	no	—
78	D₅	—	9	2	no	—
79	D₅	—	10	2	no	—
80	C₉	D₅²×C₃₁	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃	—	25	1	no	—
82	D₃	—	15	2	no	—
83	C₅	—	2	2	no	—
84	C₅	D₅×D₉×C₃₁	1	2	yes	complemented
85	C₂²	—	225	1	no	Sylow
86	C₃	C₃₁×D₁₅⋊D₅	1	1	yes	characteristic
87	C₂	—	25	1	no	—
88	C₂	—	45	2	no	—

	subgroup	G/H
center, hypercenter	C₃₁	D₄₅⋊D₅
commutator, cosocle, hypocenter	C₅²×C₉	C₂²×C₃₁
Frattini	C₃	C₃₁×D₁₅⋊D₅
Fitting	C₅²×C₉×C₃₁	C₂²
socle	C₃×C₅²×C₃₁	C₂×D₃
LCS	G ▷ C₅²×C₉
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
1. C₅×C₃₁×D₄₅	C₂	semidirect
2. C₉×C₃₁×C₅⋊D₅	C₂	semidirect
4. C₅²×C₉×C₃₁	C₂²	semidirect
9. C₃₁×D₄₅	D₅	semidirect
15. C₃₁×C₅⋊D₅	D₉	semidirect
18. C₅×C₉×C₃₁	C₂×D₅	semidirect
24. D₄₅⋊D₅	C₃₁	direct (N × H)
25. C₅²×C₃₁	C₂×D₉	semidirect
31. C₅×D₄₅	C₂×C₃₁	semidirect
32. C₉×C₅⋊D₅	C₂×C₃₁	semidirect
39. C₉×C₃₁	D₅²	semidirect
40. C₅²×C₉	C₂²×C₃₁	semidirect
45. C₅×C₃₁	D₅×D₉	semidirect
51. D₄₅	D₅×C₃₁	semidirect
59. C₅⋊D₅	D₉×C₃₁	semidirect
62. C₅×C₉	C₂×D₅×C₃₁	semidirect
64. C₃₁	D₄₅⋊D₅	direct (N × H)
69. C₅²	C₂×D₉×C₃₁	semidirect
80. C₉	D₅²×C₃₁	semidirect
84. C₅	D₅×D₉×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₃₁×D₄₅	2
C₉×C₃₁×C₅⋊D₅	2
C₃₁×D₁₅⋊D₅	3
D₅×D₉×C₃₁	5
D₄₅⋊D₅	31

Subgroups — C31×D45⋊D5 #42124572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×D₄₅⋊D₅ #42124572