Subgroups — C₂×(C₃⋊D₃)⋊D₅ #67225

Order 360.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₁₅	C₂	1	2	yes	maximal, complemented
3	(C₃⋊D₃)⋊D₅	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₃×D₅	—	3	2	no	maximal
5	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic, complemented
6	C₅×C₃⋊D₃	C₂²	1	2	yes	complemented
7	C₃×D₁₅	C₂²	1	4	yes	complemented
8	C₂×D₃²	—	5	1	no	maximal, Hall
9	C₂×D₁₅	D₃	1	2	yes	complemented
10	C₂×C₅×D₃	—	3	2	no	—
11	C₂×C₅×D₃	—	6	1	no	—
12	C₂×C₃×D₅	—	3	2	no	—
13	D₃×D₅	—	3	8	no	—
14	C₃²×C₅	C₂³	1	1	yes	characteristic, complemented, Hall
15	C₂²×D₅	—	9	1	no	Hall
16	C₂×C₃⋊D₃	D₅	1	1	yes	characteristic, complemented
17	C₂×C₃×D₃	—	5	2	no	—
18	D₃²	—	5	4	no	—
19	C₂×C₃×C₅	—	2	1	no	—
20	C₂×C₃×C₅	C₂×D₃	1	2	yes	complemented
21	D₁₅	C₂×D₃	1	4	yes	complemented
22	C₃×D₅	—	3	4	no	—
23	C₅×D₃	—	3	4	no	—
24	C₅×D₃	—	6	2	no	—
25	C₂²×D₃	—	15	2	no	—
26	C₂²×C₅	—	9	1	no	—
27	C₂×D₅	—	3	2	no	—
28	C₂×D₅	—	9	4	no	—
29	C₂×C₃²	C₂×D₅	1	1	yes	characteristic, complemented
30	C₃⋊D₃	C₂×D₅	1	2	yes	complemented
31	C₃×D₃	—	5	4	no	—
32	C₃×C₅	—	2	1	no	—
33	C₃×C₅	C₂²×D₃	1	2	yes	complemented
34	C₂²×C₃	—	15	2	no	—
35	C₂×D₃	—	3	2	no	—
36	C₂×D₃	—	5	2	no	—
37	C₂×D₃	—	6	1	no	—
38	C₂×D₃	—	15	8	no	—
39	C₂×C₅	—	9	2	no	—
40	C₂×C₅	D₃²	1	1	yes	characteristic, complemented
41	D₅	—	3	4	no	—
42	C₃²	C₂²×D₅	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	45	1	no	Sylow
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	—	15	4	no	—
46	C₂×C₃	D₃×D₅	1	2	yes	complemented
47	D₃	—	3	4	no	—
48	D₃	—	5	4	no	—
49	D₃	—	6	2	no	—
50	C₅	C₂×D₃²	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	9	1	no	—
52	C₂²	—	15	2	no	—
53	C₂²	—	45	4	no	—
54	C₃	—	2	1	no	—
55	C₃	C₂×D₃×D₅	1	2	yes	complemented
56	C₂	—	9	2	no	—
57	C₂	—	15	4	no	—
58	C₂	(C₃⋊D₃)⋊D₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	(C₃⋊D₃)⋊D₅
commutator, cosocle, hypocenter	C₃²×C₅	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅	C₂²
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₃⋊D₃	C₂	semidirect
2. C₂×C₃×D₁₅	C₂	semidirect
3. (C₃⋊D₃)⋊D₅	C₂	direct (N × H)
5. C₂×C₃²×C₅	C₂²	semidirect
6. C₅×C₃⋊D₃	C₂²	semidirect
7. C₃×D₁₅	C₂²	semidirect
9. C₂×D₁₅	D₃	semidirect
14. C₃²×C₅	C₂³	semidirect
16. C₂×C₃⋊D₃	D₅	semidirect
20. C₂×C₃×C₅	C₂×D₃	semidirect
21. D₁₅	C₂×D₃	semidirect
29. C₂×C₃²	C₂×D₅	semidirect
30. C₃⋊D₃	C₂×D₅	semidirect
33. C₃×C₅	C₂²×D₃	semidirect
40. C₂×C₅	D₃²	semidirect
42. C₃²	C₂²×D₅	semidirect
46. C₂×C₃	D₃×D₅	semidirect
50. C₅	C₂×D₃²	semidirect
55. C₃	C₂×D₃×D₅	semidirect
58. C₂	(C₃⋊D₃)⋊D₅	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×C₅×C₃⋊D₃	2
C₂×C₃×D₁₅	2
(C₃⋊D₃)⋊D₅	2
C₂×D₃×D₅	3
C₂×D₃²	5

Subgroups — C2×(C3⋊D3)⋊D5 #67225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₃⋊D₃)⋊D₅ #67225