Subgroups — #13083703

Order 1440.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₁₈₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×D₄₀	C₂	1	1	yes	maximal, characteristic
3	C₅×C₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#90942	—	3	1	no	maximal
5	Dic₉₀	—	2	1	no	—
6	C₉×D₂₀	—	2	1	no	—
7	C₅×C₈×C₉	C₂²	1	1	yes	characteristic
8	#63386	—	5	1	no	maximal, Hall
9	Dic₆₀	—	3	1	no	—
10	C₃×D₄₀	D₃	1	1	yes	characteristic
11	C₅×C₃⋊C₁₆	—	3	1	no	—
12	C₂×C₉×D₅	—	4	1	no	—
13	C₄×C₅×C₉	D₄	1	1	yes	characteristic
14	Dic₄₅	—	4	1	no	—
15	#3932	—	9	1	no	Hall
16	C₉×D₈	—	5	1	no	—
17	Dic₃₆	—	5	1	no	—
18	C₉⋊C₁₆	D₅	1	1	yes	characteristic, complemented
19	Dic₃₀	—	6	1	no	—
20	C₃×D₂₀	—	2	1	no	—
21	C₃×C₅×C₈	C₂×D₃	1	1	yes	characteristic
22	#827	—	15	1	no	—
23	C₂×C₅×C₉	D₈	1	1	yes	characteristic
24	C₉×D₅	—	8	1	no	—
25	Dic₂₀	—	9	1	no	—
26	D₄₀	D₉	1	1	yes	characteristic
27	C₅×C₁₆	—	9	1	no	—
28	D₄×C₉	—	10	1	no	—
29	Dic₁₈	—	10	1	no	—
30	C₈×C₉	C₂×D₅	1	1	yes	characteristic
31	C₂×C₃×D₅	—	4	1	no	—
32	C₃×C₄×C₅	#67	1	1	yes	characteristic
33	Dic₁₅	—	12	1	no	—
34	C₃×D₈	—	5	1	no	—
35	Dic₁₂	—	15	1	no	—
36	C₃⋊C₁₆	—	3	1	no	—
37	C₅×C₉	SD₃₂	1	1	yes	characteristic, complemented, Hall
38	D₂₀	—	2	1	no	—
39	Dic₁₀	—	18	1	no	—
40	C₅×C₈	C₂×D₉	1	1	yes	characteristic
41	C₂²×C₉	—	20	1	no	—
42	C₄×C₉	D₂₀	1	1	yes	characteristic
43	Dic₉	—	20	1	no	—
44	SD₃₂	—	45	1	no	Sylow
45	C₂×C₃×C₅	#213	1	1	yes	characteristic
46	C₃×D₅	—	8	1	no	—
47	C₃×D₄	—	10	1	no	—
48	Dic₆	—	30	1	no	—
49	C₃×C₈	D₃×D₅	1	1	yes	characteristic
50	C₂×D₅	—	4	1	no	—
51	C₄×C₅	#611	1	1	yes	characteristic
52	Dic₅	—	36	1	no	—
53	C₂×C₉	—	40	1	no	—
54	C₂×C₉	D₄₀	1	1	yes	characteristic
55	Q₁₆	—	45	1	no	—
56	D₈	—	5	1	no	—
57	C₁₆	—	9	1	no	—
58	C₃×C₅	#827	1	1	yes	characteristic
59	C₂²×C₃	—	20	1	no	—
60	C₃×C₄	#1202	1	1	yes	characteristic
61	Dic₃	—	60	1	no	—
62	C₂×C₅	#3673	1	1	yes	characteristic
63	D₅	—	8	1	no	—
64	C₉	#3932	1	1	yes	characteristic, complemented, Sylow
65	Q₈	—	90	1	no	—
66	D₄	—	10	1	no	—
67	C₈	D₅×D₉	1	1	yes	characteristic
68	C₂×C₃	—	40	1	no	—
69	C₂×C₃	#6653	1	1	yes	characteristic
70	C₅	#63386	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	20	1	no	—
72	C₄	—	180	1	no	—
73	C₄	#67083	1	1	yes	characteristic
74	C₃	#90942	1	1	yes	characteristic
75	C₂	—	40	1	no	—
76	C₂	#10623549	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10623549
commutator, Fitting, cosocle	C₅×C₈×C₉	C₂²
Frattini	C₃×C₈	D₃×D₅
socle	C₂×C₃×C₅	#213
prime radical	C₃×D₄₀	D₃
hypercenter	C₈	D₅×D₉
hypocenter	C₅×C₉	SD₃₂
LCS	G ▷ C₅×C₈×C₉ ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. Dic₁₈₀	C₂	semidirect
3. C₅×C₉⋊C₁₆	C₂	semidirect
18. C₉⋊C₁₆	D₅	semidirect
37. C₅×C₉	SD₃₂	semidirect
64. C₉	#3932	semidirect
70. C₅	#63386	semidirect

Maximal subgroups

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

subgroup	index
Dic₁₈₀	2
C₉×D₄₀	2
C₅×C₉⋊C₁₆	2
#90942	3
#63386	5