Subgroups — C₂²×C₅×Dic₁₅ #11925713

Order 1200.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₁₅	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×Dic₅	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅²	C₄	1	4	yes	complemented
5	C₂²×C₃×C₅²	C₂²	1	3	yes	—
6	C₅×Dic₁₅	C₂²	1	4	yes	direct product factor
7	C₂²×Dic₁₅	C₅	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₅×Dic₃	—	5	1	no	maximal
9	C₂³×C₅²	D₃	1	1	yes	characteristic
10	C₂×C₅×Dic₅	—	3	6	no	—
11	C₂×C₃×C₅²	C₂×C₄	1	6	yes	complemented
12	C₂×C₃×C₅²	C₂³	1	1	yes	characteristic
13	C₂³×C₃×C₅	—	2	2	no	—
14	C₂³×C₃×C₅	D₅	1	1	yes	characteristic
15	C₂³×C₃×C₅	C₂×C₅	1	1	yes	characteristic
16	C₂×Dic₁₅	C₂×C₅	1	6	yes	direct product factor
17	C₂×C₅×Dic₃	—	5	6	no	—
18	C₂²×C₅²	Dic₃	1	4	yes	complemented
19	C₂²×C₅²	C₂×D₃	1	3	yes	—
20	C₅×Dic₅	—	3	4	no	—
21	C₂²×C₄×C₅	—	15	1	no	—
22	C₂²×Dic₅	—	3	1	no	—
23	C₃×C₅²	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
24	C₂²×C₃×C₅	—	2	8	no	—
25	C₂²×C₃×C₅	—	2	6	no	—
26	C₂²×C₃×C₅	Dic₅	1	4	yes	complemented
27	C₂²×C₃×C₅	C₄×C₅	1	4	yes	complemented
28	C₂²×C₃×C₅	C₂×D₅	1	3	yes	—
29	C₂²×C₃×C₅	C₂²×C₅	1	3	yes	—
30	Dic₁₅	C₂²×C₅	1	4	yes	direct product factor
31	C₅×Dic₃	—	5	4	no	—
32	C₂×C₅²	C₂×Dic₃	1	6	yes	complemented
33	C₂×C₅²	C₂²×D₃	1	1	yes	characteristic
34	C₂²×Dic₃	—	5	1	no	Hall
35	C₂³×C₅	—	2	2	no	—
36	C₂³×C₅	C₅×D₃	1	1	yes	characteristic
37	C₂³×C₅	D₁₅	1	1	yes	characteristic
38	C₂×C₄×C₅	—	15	6	no	—
39	C₂×Dic₅	—	3	6	no	—
40	C₂×C₃×C₅	—	2	12	no	—
41	C₂×C₃×C₅	—	2	2	no	—
42	C₂×C₃×C₅	C₂×Dic₅	1	6	yes	complemented
43	C₂×C₃×C₅	C₂×C₄×C₅	1	6	yes	complemented
44	C₂×C₃×C₅	C₂²×D₅	1	1	yes	characteristic
45	C₂×C₃×C₅	C₂³×C₅	1	1	yes	characteristic
46	C₅²	C₂²×Dic₃	1	1	yes	characteristic, complemented, Sylow
47	C₂³×C₃	C₅×D₅	1	1	yes	characteristic
48	C₂×Dic₃	—	5	6	no	—
49	C₂²×C₅	—	2	8	no	—
50	C₂²×C₅	—	2	6	no	—
51	C₂²×C₅	C₅×Dic₃	1	4	yes	complemented
52	C₂²×C₅	Dic₁₅	1	4	yes	direct product factor
53	C₂²×C₅	C₂×C₅×D₃	1	3	yes	—
54	C₂²×C₅	C₂×D₁₅	1	3	yes	—
55	C₄×C₅	—	15	4	no	—
56	Dic₅	—	3	4	no	—
57	C₂²×C₄	—	15	1	no	Sylow
58	C₃×C₅	—	2	2	no	—
59	C₃×C₅	C₂²×Dic₅	1	1	yes	characteristic, complemented
60	C₃×C₅	C₂²×C₄×C₅	1	1	yes	characteristic, complemented
61	C₂²×C₃	C₅×Dic₅	1	4	yes	complemented
62	C₂²×C₃	C₂×C₅×D₅	1	3	yes	—
63	Dic₃	—	5	4	no	—
64	C₂×C₅	—	2	12	no	—
65	C₂×C₅	—	2	2	no	—
66	C₂×C₅	C₂×C₅×Dic₃	1	6	yes	complemented
67	C₂×C₅	C₂×Dic₁₅	1	6	yes	direct product factor
68	C₂×C₅	C₂²×C₅×D₃	1	1	yes	characteristic
69	C₂×C₅	C₂²×D₁₅	1	1	yes	characteristic
70	C₂³	C₅×D₁₅	1	1	yes	characteristic
71	C₂×C₄	—	15	6	no	—
72	C₂×C₃	C₂×C₅×Dic₅	1	6	yes	complemented
73	C₂×C₃	C₂²×C₅×D₅	1	1	yes	characteristic
74	C₅	—	2	2	no	—
75	C₅	C₂²×C₅×Dic₃	1	1	yes	characteristic, complemented
76	C₅	C₂²×Dic₁₅	1	1	yes	characteristic, direct product factor
77	C₂²	C₅×Dic₁₅	1	4	yes	direct product factor
78	C₂²	C₂×C₅×D₁₅	1	3	yes	—
79	C₄	—	15	4	no	—
80	C₃	C₂²×C₅×Dic₅	1	1	yes	characteristic, complemented, Sylow
81	C₂	C₂×C₅×Dic₁₅	1	6	yes	direct product factor
82	C₂	C₂²×C₅×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₅	D₁₅
commutator, hypocenter	C₃×C₅	C₂²×C₄×C₅
Frattini	C₂	C₂²×C₅×D₁₅
Fitting, socle, prime radical	C₂³×C₃×C₅²	C₂
cosocle	C₂×C₃×C₅	C₂³×C₅
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂³×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×Dic₁₅	C₂	direct (N × H)
4. C₂²×C₃×C₅²	C₄	semidirect
6. C₅×Dic₁₅	C₂²	direct (N × H)
7. C₂²×Dic₁₅	C₅	direct (N × H)
11. C₂×C₃×C₅²	C₂×C₄	semidirect
16. C₂×Dic₁₅	C₂×C₅	direct (N × H)
18. C₂²×C₅²	Dic₃	semidirect
23. C₃×C₅²	C₂²×C₄	semidirect
26. C₂²×C₃×C₅	Dic₅	semidirect
27. C₂²×C₃×C₅	C₄×C₅	semidirect
30. Dic₁₅	C₂²×C₅	direct (N × H)
32. C₂×C₅²	C₂×Dic₃	semidirect
42. C₂×C₃×C₅	C₂×Dic₅	semidirect
43. C₂×C₃×C₅	C₂×C₄×C₅	semidirect
46. C₅²	C₂²×Dic₃	semidirect
51. C₂²×C₅	C₅×Dic₃	semidirect
52. C₂²×C₅	Dic₁₅	direct (N × H)
59. C₃×C₅	C₂²×Dic₅	semidirect
60. C₃×C₅	C₂²×C₄×C₅	semidirect
61. C₂²×C₃	C₅×Dic₅	semidirect
66. C₂×C₅	C₂×C₅×Dic₃	semidirect
67. C₂×C₅	C₂×Dic₁₅	direct (N × H)
72. C₂×C₃	C₂×C₅×Dic₅	semidirect
75. C₅	C₂²×C₅×Dic₃	semidirect
76. C₅	C₂²×Dic₁₅	direct (N × H)
77. C₂²	C₅×Dic₁₅	direct (N × H)
80. C₃	C₂²×C₅×Dic₅	semidirect
81. C₂	C₂×C₅×Dic₁₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅²	2
C₂×C₅×Dic₁₅	2
C₂²×C₅×Dic₅	3
C₂²×Dic₁₅	5
C₂²×C₅×Dic₃	5

Subgroups — C22×C5×Dic15 #11925713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×Dic₁₅ #11925713