Subgroups — C₂²×(C₃⋊D₃)⋊D₅ #10624363

Order 720.

568 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₁₅	C₂	1	2	yes	maximal, complemented
3	C₂×(C₃⋊D₃)⋊D₅	C₂	1	12	yes	maximal, direct product factor
4	C₂²×D₃×D₅	—	3	2	no	maximal
5	C₂²×C₃²×C₅	C₂²	1	1	yes	characteristic, complemented
6	C₂×C₅×C₃⋊D₃	C₂²	1	6	yes	complemented
7	C₂×C₃×D₁₅	C₂²	1	12	yes	complemented
8	(C₃⋊D₃)⋊D₅	C₂²	1	16	yes	direct product factor
9	C₂²×D₃²	—	5	1	no	maximal, Hall
10	C₂²×D₁₅	D₃	1	2	yes	complemented
11	C₂²×C₅×D₃	—	3	2	no	—
12	C₂²×C₅×D₃	—	6	1	no	—
13	C₂²×C₃×D₅	—	3	2	no	—
14	C₂×D₃×D₅	—	3	24	no	—
15	C₂×C₃²×C₅	C₂³	1	3	yes	complemented
16	C₅×C₃⋊D₃	C₂³	1	4	yes	complemented
17	C₃×D₁₅	C₂³	1	8	yes	complemented
18	C₂³×D₅	—	9	1	no	Hall
19	C₂²×C₃⋊D₃	D₅	1	1	yes	characteristic, complemented
20	C₂²×C₃×D₃	—	5	2	no	—
21	C₂×D₃²	—	5	12	no	—
22	C₂²×C₃×C₅	—	2	1	no	—
23	C₂²×C₃×C₅	C₂×D₃	1	2	yes	complemented
24	C₂×D₁₅	C₂×D₃	1	12	yes	complemented
25	C₂×C₅×D₃	—	3	12	no	—
26	C₂×C₅×D₃	—	6	6	no	—
27	C₂×C₃×D₅	—	3	12	no	—
28	D₃×D₅	—	3	32	no	—
29	C₂³×D₃	—	15	2	no	—
30	C₃²×C₅	C₂⁴	1	1	yes	characteristic, complemented, Hall
31	C₂³×C₅	—	9	1	no	—
32	C₂²×D₅	—	3	2	no	—
33	C₂²×D₅	—	9	12	no	—
34	C₂²×C₃²	C₂×D₅	1	1	yes	characteristic, complemented
35	C₂×C₃⋊D₃	C₂×D₅	1	6	yes	complemented
36	C₂×C₃×D₃	—	5	12	no	—
37	D₃²	—	5	16	no	—
38	C₂×C₃×C₅	—	2	3	no	—
39	C₂×C₃×C₅	C₂²×D₃	1	6	yes	complemented
40	D₁₅	C₂²×D₃	1	8	yes	complemented
41	C₃×D₅	—	3	8	no	—
42	C₅×D₃	—	3	8	no	—
43	C₅×D₃	—	6	4	no	—
44	C₂³×C₃	—	15	2	no	—
45	C₂²×D₃	—	3	2	no	—
46	C₂²×D₃	—	5	2	no	—
47	C₂²×D₃	—	6	1	no	—
48	C₂²×D₃	—	15	24	no	—
49	C₂²×C₅	—	9	6	no	—
50	C₂²×C₅	D₃²	1	1	yes	characteristic, complemented
51	C₂×D₅	—	3	12	no	—
52	C₂×D₅	—	9	16	no	—
53	C₂×C₃²	C₂²×D₅	1	3	yes	complemented
54	C₃⋊D₃	C₂²×D₅	1	4	yes	complemented
55	C₃×D₃	—	5	8	no	—
56	C₂⁴	—	45	1	no	Sylow
57	C₃×C₅	—	2	1	no	—
58	C₃×C₅	C₂³×D₃	1	2	yes	complemented
59	C₂²×C₃	—	2	1	no	—
60	C₂²×C₃	—	15	12	no	—
61	C₂²×C₃	D₃×D₅	1	2	yes	complemented
62	C₂×D₃	—	3	12	no	—
63	C₂×D₃	—	5	12	no	—
64	C₂×D₃	—	6	6	no	—
65	C₂×D₃	—	15	32	no	—
66	C₂×C₅	—	9	4	no	—
67	C₂×C₅	C₂×D₃²	1	3	yes	complemented
68	D₅	—	3	8	no	—
69	C₃²	C₂³×D₅	1	1	yes	characteristic, complemented, Sylow
70	C₂³	—	9	1	no	—
71	C₂³	—	15	2	no	—
72	C₂³	—	45	12	no	—
73	C₂×C₃	—	2	3	no	—
74	C₂×C₃	—	15	8	no	—
75	C₂×C₃	C₂×D₃×D₅	1	6	yes	complemented
76	D₃	—	3	8	no	—
77	D₃	—	5	8	no	—
78	D₃	—	6	4	no	—
79	C₅	C₂²×D₃²	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	9	6	no	—
81	C₂²	—	15	12	no	—
82	C₂²	—	45	16	no	—
83	C₂²	(C₃⋊D₃)⋊D₅	1	1	yes	characteristic, direct product factor
84	C₃	—	2	1	no	—
85	C₃	C₂²×D₃×D₅	1	2	yes	complemented
86	C₂	—	9	4	no	—
87	C₂	—	15	8	no	—
88	C₂	C₂×(C₃⋊D₃)⋊D₅	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	(C₃⋊D₃)⋊D₅
commutator, cosocle, hypocenter	C₃²×C₅	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₅	C₂²
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₃⋊D₃	C₂	semidirect
2. C₂²×C₃×D₁₅	C₂	semidirect
3. C₂×(C₃⋊D₃)⋊D₅	C₂	direct (N × H)
5. C₂²×C₃²×C₅	C₂²	semidirect
6. C₂×C₅×C₃⋊D₃	C₂²	semidirect
7. C₂×C₃×D₁₅	C₂²	semidirect
8. (C₃⋊D₃)⋊D₅	C₂²	direct (N × H)
10. C₂²×D₁₅	D₃	semidirect
15. C₂×C₃²×C₅	C₂³	semidirect
16. C₅×C₃⋊D₃	C₂³	semidirect
17. C₃×D₁₅	C₂³	semidirect
19. C₂²×C₃⋊D₃	D₅	semidirect
23. C₂²×C₃×C₅	C₂×D₃	semidirect
24. C₂×D₁₅	C₂×D₃	semidirect
30. C₃²×C₅	C₂⁴	semidirect
34. C₂²×C₃²	C₂×D₅	semidirect
35. C₂×C₃⋊D₃	C₂×D₅	semidirect
39. C₂×C₃×C₅	C₂²×D₃	semidirect
40. D₁₅	C₂²×D₃	semidirect
50. C₂²×C₅	D₃²	semidirect
53. C₂×C₃²	C₂²×D₅	semidirect
54. C₃⋊D₃	C₂²×D₅	semidirect
58. C₃×C₅	C₂³×D₃	semidirect
61. C₂²×C₃	D₃×D₅	semidirect
67. C₂×C₅	C₂×D₃²	semidirect
69. C₃²	C₂³×D₅	semidirect
75. C₂×C₃	C₂×D₃×D₅	semidirect
79. C₅	C₂²×D₃²	semidirect
83. C₂²	(C₃⋊D₃)⋊D₅	direct (N × H)
85. C₃	C₂²×D₃×D₅	semidirect
88. C₂	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
C₂²×C₅×C₃⋊D₃	2
C₂²×C₃×D₁₅	2
C₂×(C₃⋊D₃)⋊D₅	2
C₂²×D₃×D₅	3
C₂²×D₃²	5

Subgroups — C22×(C3⋊D3)⋊D5 #10624363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×(C₃⋊D₃)⋊D₅ #10624363