Subgroups — C₂×Dic₁₈₀ #13084547

Order 1440.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₉₀	C₂	1	2	yes	maximal
2	C₂×C₅×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	Dic₁₈₀	C₂	1	4	yes	maximal, direct product factor
4	C₂×Dic₆₀	—	3	1	no	maximal
5	C₂×C₄×C₅×C₉	C₂²	1	1	yes	characteristic
6	C₂×Dic₄₅	—	2	2	no	—
7	Dic₉₀	—	2	2	no	—
8	Dic₉₀	C₂²	1	4	yes	—
9	C₅×C₈×C₉	C₂²	1	2	yes	—
10	C₂×Dic₃₆	—	5	1	no	maximal, Hall
11	C₂×Dic₃₀	—	3	2	no	—
12	C₂×C₃×C₅×C₈	D₃	1	1	yes	characteristic
13	Dic₆₀	—	3	4	no	—
14	C₂²×C₅×C₉	D₄	1	1	yes	characteristic
15	C₄×C₅×C₉	D₄	1	1	yes	characteristic
16	C₄×C₅×C₉	C₂³	1	1	yes	characteristic
17	Dic₄₅	—	2	4	no	—
18	C₂×Dic₂₀	—	9	1	no	Hall
19	C₂×Dic₁₈	—	5	2	no	—
20	C₂×C₈×C₉	D₅	1	1	yes	characteristic
21	Dic₃₆	—	5	4	no	—
22	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
23	C₂×Dic₁₅	—	6	2	no	—
24	Dic₃₀	—	3	4	no	—
25	Dic₃₀	—	6	2	no	—
26	C₃×C₅×C₈	C₂×D₃	1	2	yes	—
27	C₂×Dic₁₂	—	15	1	no	—
28	C₂×C₅×C₉	Q₁₆	1	2	yes	complemented
29	C₂×C₅×C₉	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₁₀	—	9	2	no	—
31	C₂×C₅×C₈	D₉	1	1	yes	characteristic
32	Dic₂₀	—	9	4	no	—
33	C₂×C₄×C₉	C₂×D₅	1	1	yes	characteristic
34	C₂×Dic₉	—	10	2	no	—
35	Dic₁₈	—	5	4	no	—
36	Dic₁₈	—	10	2	no	—
37	C₈×C₉	C₂×D₅	1	2	yes	—
38	C₂²×C₃×C₅	D₁₂	1	1	yes	characteristic
39	C₃×C₄×C₅	D₁₂	1	1	yes	characteristic
40	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
41	Dic₁₅	—	6	4	no	—
42	C₂×Dic₆	—	15	2	no	—
43	C₂×C₃×C₈	D₁₅	1	1	yes	characteristic
44	Dic₁₂	—	15	4	no	—
45	C₅×C₉	C₂×Q₁₆	1	1	yes	characteristic, complemented, Hall
46	C₂×C₄×C₅	C₂×D₉	1	1	yes	characteristic
47	C₂×Dic₅	—	18	2	no	—
48	Dic₁₀	—	9	4	no	—
49	Dic₁₀	—	18	2	no	—
50	C₅×C₈	C₂×D₉	1	2	yes	—
51	C₂²×C₉	D₂₀	1	1	yes	characteristic
52	C₄×C₉	D₂₀	1	1	yes	characteristic
53	C₄×C₉	C₂²×D₅	1	1	yes	characteristic
54	Dic₉	—	10	4	no	—
55	C₂×Q₁₆	—	45	1	no	Sylow
56	C₂×C₃×C₅	Dic₁₂	1	2	yes	—
57	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
58	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
59	C₂×Dic₃	—	30	2	no	—
60	Dic₆	—	15	4	no	—
61	Dic₆	—	30	2	no	—
62	C₃×C₈	C₂×D₁₅	1	2	yes	—
63	C₂²×C₅	D₃₆	1	1	yes	characteristic
64	C₄×C₅	D₃₆	1	1	yes	characteristic
65	C₄×C₅	C₂²×D₉	1	1	yes	characteristic
66	Dic₅	—	18	4	no	—
67	C₂×C₉	Dic₂₀	1	2	yes	complemented
68	C₂×C₉	C₂×D₂₀	1	1	yes	characteristic
69	C₂×Q₈	—	45	2	no	—
70	Q₁₆	—	45	4	no	—
71	C₂×C₈	D₄₅	1	1	yes	characteristic
72	C₃×C₅	C₂×Dic₁₂	1	1	yes	characteristic
73	C₂²×C₃	D₆₀	1	1	yes	characteristic
74	C₃×C₄	D₆₀	1	1	yes	characteristic
75	C₃×C₄	C₂²×D₁₅	1	1	yes	characteristic
76	Dic₃	—	30	4	no	—
77	C₂×C₅	Dic₃₆	1	2	yes	complemented
78	C₂×C₅	C₂×D₃₆	1	1	yes	characteristic
79	C₉	C₂×Dic₂₀	1	1	yes	characteristic, complemented, Sylow
80	Q₈	—	45	4	no	—
81	Q₈	—	90	2	no	—
82	C₂×C₄	—	90	2	no	—
83	C₂×C₄	C₂×D₄₅	1	1	yes	characteristic
84	C₈	C₂×D₄₅	1	2	yes	—
85	C₂×C₃	Dic₆₀	1	2	yes	—
86	C₂×C₃	C₂×D₆₀	1	1	yes	characteristic
87	C₅	C₂×Dic₃₆	1	1	yes	characteristic, complemented, Sylow
88	C₂²	D₁₈₀	1	1	yes	characteristic
89	C₄	—	90	4	no	—
90	C₄	D₁₈₀	1	1	yes	characteristic
91	C₄	C₂²×D₄₅	1	1	yes	characteristic
92	C₃	C₂×Dic₆₀	1	1	yes	characteristic
93	C₂	Dic₁₈₀	1	2	yes	direct product factor
94	C₂	C₂×D₁₈₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₈₀
commutator, cosocle	C₄×C₅×C₉	C₂³
Frattini	C₃×C₄	C₂²×D₁₅
Fitting	C₂×C₅×C₈×C₉	C₂
socle, prime radical	C₂²×C₃×C₅	D₁₂
hypercenter	C₂×C₈	D₄₅
hypocenter	C₅×C₉	C₂×Q₁₆
LCS	G ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
3. Dic₁₈₀	C₂	direct (N × H)
28. C₂×C₅×C₉	Q₁₆	semidirect
45. C₅×C₉	C₂×Q₁₆	semidirect
67. C₂×C₉	Dic₂₀	semidirect
77. C₂×C₅	Dic₃₆	semidirect
79. C₉	C₂×Dic₂₀	semidirect
87. C₅	C₂×Dic₃₆	semidirect
93. C₂	Dic₁₈₀	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×Dic₉₀	2
C₂×C₅×C₈×C₉	2
Dic₁₈₀	2
C₂×Dic₆₀	3
C₂×Dic₃₆	5

Subgroups — C2×Dic180 #13084547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₁₈₀ #13084547