Subgroups — #91100

Order 480.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₃₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
4	#3948	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂×Dic₁₅	—	2	1	no	—
7	C₂×Dic₁₅	—	2	1	no	—
8	Dic₃₀	—	2	1	no	—
9	Dic₃₀	C₄	1	2	yes	complemented
10	C₃×C₅×C₈	—	2	1	no	—
11	#816	—	5	1	no	maximal, Hall
12	C₂×Dic₁₀	—	3	1	no	—
13	C₂×C₅×C₈	D₃	1	1	yes	characteristic
14	C₄⋊Dic₅	—	3	1	no	—
15	C₂²×C₃×C₅	D₄	1	1	yes	characteristic
16	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₃×C₄×C₅	D₄	1	1	yes	characteristic
18	Dic₁₅	—	2	2	no	—
19	Dic₁₅	—	4	1	no	—
20	C₂×Dic₆	—	5	1	no	—
21	C₂×C₃×C₈	D₅	1	1	yes	characteristic
22	C₄⋊Dic₃	—	5	1	no	—
23	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
24	C₂×Dic₅	—	6	1	no	—
25	C₂×Dic₅	—	6	1	no	—
26	Dic₁₀	—	3	2	no	—
27	Dic₁₀	—	6	1	no	—
28	C₅×C₈	—	2	1	no	—
29	#103	—	15	1	no	Sylow
30	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₃×C₅	SD₁₆	1	1	yes	characteristic
32	C₂×C₃×C₅	Q₁₆	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
34	C₂×Dic₃	—	10	1	no	—
35	C₂×Dic₃	—	10	1	no	—
36	Dic₆	—	5	2	no	—
37	Dic₆	—	10	1	no	—
38	C₃×C₈	—	2	1	no	—
39	C₂²×C₅	D₁₂	1	1	yes	characteristic
40	C₄×C₅	C₄×D₃	1	1	yes	characteristic
41	C₄×C₅	#67	1	1	yes	characteristic
42	Dic₅	—	6	2	no	—
43	Dic₅	—	12	1	no	—
44	C₂×Q₈	—	15	1	no	—
45	C₂×C₈	D₁₅	1	1	yes	characteristic
46	C₄⋊C₄	—	15	1	no	—
47	C₃×C₅	#103	1	1	yes	characteristic, complemented, Hall
48	C₂²×C₃	D₂₀	1	1	yes	characteristic
49	C₃×C₄	C₄×D₅	1	1	yes	characteristic
50	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
51	Dic₃	—	10	2	no	—
52	Dic₃	—	20	1	no	—
53	C₂×C₅	#204	1	1	yes	characteristic
54	C₂×C₅	Dic₁₂	1	1	yes	characteristic
55	C₂×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
56	Q₈	—	15	2	no	—
57	Q₈	—	30	1	no	—
58	C₂×C₄	—	30	1	no	—
59	C₂×C₄	—	30	1	no	—
60	C₂×C₄	C₂×D₁₅	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₂×C₃	#677	1	1	yes	characteristic
63	C₂×C₃	Dic₂₀	1	1	yes	characteristic
64	C₂×C₃	#685	1	1	yes	characteristic
65	C₅	#816	1	1	yes	characteristic, complemented, Sylow
66	C₂²	D₆₀	1	1	yes	characteristic
67	C₄	—	30	2	no	—
68	C₄	—	60	1	no	—
69	C₄	C₄×D₁₅	1	1	yes	characteristic
70	C₄	#1220	1	1	yes	characteristic
71	C₃	#3948	1	1	yes	characteristic, complemented, Sylow
72	C₂	#6706	1	1	yes	characteristic
73	C₂	Dic₆₀	1	1	yes	characteristic
74	C₂	#6714	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₆₀
commutator	C₃×C₄×C₅	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₅
Fitting	C₂×C₃×C₅×C₈	C₂
socle, prime radical	C₂²×C₃×C₅	D₄
cosocle	C₂×C₃×C₄×C₅	C₂²
hypercenter	C₂×C₈	D₁₅
hypocenter	C₃×C₅	#103
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
9. Dic₃₀	C₄	semidirect
47. C₃×C₅	#103	semidirect
65. C₅	#816	semidirect
71. C₃	#3948	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₃₀	2
C₂×C₃×C₅×C₈	2
C₄⋊Dic₁₅	2
#3948	3
#816	5