Subgroups — D₃₂₂₈₀ #72302001

Order 64560.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₆₁₄₀	C₂	1	2	yes	maximal, complemented
2	D₁₀₇₆₀	—	3	1	no	maximal, Hall
3	D₆₄₅₆	—	5	1	no	maximal, Hall
4	D₅₃₈₀	—	3	2	no	—
5	D₃₂₂₈	—	5	2	no	—
6	C₃×C₅×C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×C₈×C₂₆₉	D₃	1	1	yes	characteristic, complemented
8	C₃×C₈×C₂₆₉	D₅	1	1	yes	characteristic, complemented
9	C₂×D₄₀₃₅	—	2	2	no	—
10	C₃×C₄×C₅×C₂₆₉	C₂²	1	1	yes	characteristic
11	D₄₀₃₅	—	4	2	no	—
12	C₂×C₃×C₅×C₂₆₉	D₄	1	1	yes	characteristic
13	C₂×D₁₃₄₅	—	6	2	no	—
14	C₄×C₅×C₂₆₉	C₂×D₃	1	1	yes	characteristic
15	D₂₁₅₂	—	15	1	no	Hall
16	C₃×C₅×C₂₆₉	D₈	1	1	yes	characteristic, complemented, Hall
17	C₂×D₈₀₇	—	10	2	no	—
18	C₃×C₄×C₂₆₉	C₂×D₅	1	1	yes	characteristic
19	C₂×C₅×C₂₆₉	D₁₂	1	1	yes	characteristic
20	D₁₃₄₅	—	12	2	no	—
21	D₁₀₇₆	—	15	2	no	—
22	C₈×C₂₆₉	D₁₅	1	1	yes	characteristic, complemented
23	C₂×C₃×C₂₆₉	D₂₀	1	1	yes	characteristic
24	D₈₀₇	—	20	2	no	—
25	C₅×C₂₆₉	D₂₄	1	1	yes	characteristic, complemented, Hall
26	C₂×D₂₆₉	—	30	2	no	—
27	C₄×C₂₆₉	C₂×D₁₅	1	1	yes	characteristic
28	C₃×C₂₆₉	D₄₀	1	1	yes	characteristic, complemented, Hall
29	C₂×C₂₆₉	D₆₀	1	1	yes	characteristic
30	D₂₆₉	—	60	2	no	—
31	C₂₆₉	D₁₂₀	1	1	yes	characteristic, complemented, Sylow
32	D₁₂₀	—	269	1	no	maximal, Hall
33	D₆₀	—	269	2	no	—
34	C₃×C₅×C₈	D₂₆₉	1	1	yes	characteristic, complemented
35	D₄₀	—	807	1	no	Hall
36	C₂×D₁₅	—	538	2	no	—
37	C₃×C₄×C₅	C₂×D₂₆₉	1	1	yes	characteristic
38	D₂₄	—	1345	1	no	Hall
39	D₂₀	—	807	2	no	—
40	C₅×C₈	D₈₀₇	1	1	yes	characteristic, complemented
41	C₂×C₃×C₅	D₁₀₇₆	1	1	yes	characteristic
42	D₁₅	—	1076	2	no	—
43	D₁₂	—	1345	2	no	—
44	C₃×C₈	D₁₃₄₅	1	1	yes	characteristic, complemented
45	C₂×D₅	—	1614	2	no	—
46	C₄×C₅	C₂×D₈₀₇	1	1	yes	characteristic
47	D₈	—	4035	1	no	Sylow
48	C₃×C₅	D₂₁₅₂	1	1	yes	characteristic, complemented, Hall
49	C₂×D₃	—	2690	2	no	—
50	C₃×C₄	C₂×D₁₃₄₅	1	1	yes	characteristic
51	C₂×C₅	D₃₂₂₈	1	1	yes	characteristic
52	D₅	—	3228	2	no	—
53	D₄	—	4035	2	no	—
54	C₈	D₄₀₃₅	1	1	yes	characteristic, complemented
55	C₂×C₃	D₅₃₈₀	1	1	yes	characteristic
56	D₃	—	5380	2	no	—
57	C₅	D₆₄₅₆	1	1	yes	characteristic, complemented, Sylow
58	C₂²	—	8070	2	no	—
59	C₄	C₂×D₄₀₃₅	1	1	yes	characteristic
60	C₃	D₁₀₇₆₀	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	16140	2	no	—
62	C₂	D₁₆₁₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆₁₄₀
commutator, cosocle	C₃×C₄×C₅×C₂₆₉	C₂²
Frattini	C₄	C₂×D₄₀₃₅
Fitting	C₃×C₅×C₈×C₂₆₉	C₂
socle	C₂×C₃×C₅×C₂₆₉	D₄
hypercenter	C₈	D₄₀₃₅
hypocenter	C₃×C₅×C₂₆₉	D₈
LCS	G ▷ C₃×C₄×C₅×C₂₆₉ ▷ C₂×C₃×C₅×C₂₆₉ ▷ C₃×C₅×C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. D₁₆₁₄₀	C₂	semidirect
6. C₃×C₅×C₈×C₂₆₉	C₂	semidirect
7. C₅×C₈×C₂₆₉	D₃	semidirect
8. C₃×C₈×C₂₆₉	D₅	semidirect
16. C₃×C₅×C₂₆₉	D₈	semidirect
22. C₈×C₂₆₉	D₁₅	semidirect
25. C₅×C₂₆₉	D₂₄	semidirect
28. C₃×C₂₆₉	D₄₀	semidirect
31. C₂₆₉	D₁₂₀	semidirect
34. C₃×C₅×C₈	D₂₆₉	semidirect
40. C₅×C₈	D₈₀₇	semidirect
44. C₃×C₈	D₁₃₄₅	semidirect
48. C₃×C₅	D₂₁₅₂	semidirect
54. C₈	D₄₀₃₅	semidirect
57. C₅	D₆₄₅₆	semidirect
60. C₃	D₁₀₇₆₀	semidirect

Maximal subgroups

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

subgroup	index
D₁₆₁₄₀	2
D₁₀₇₆₀	3
D₆₄₅₆	5
C₃×C₅×C₈×C₂₆₉	2
D₁₂₀	269

Subgroups — D32280 #72302001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₂₂₈₀ #72302001