Subgroups — #90986

Order 480.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
2	C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	#3940	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₅×Dic₃	—	2	1	no	—
7	C₂×C₃×Dic₅	—	2	1	no	—
8	C₃⋊#168	C₄	1	2	yes	complemented
9	#808	—	5	1	no	maximal, Hall
10	C₅×C₄⋊C₄	—	3	1	no	—
11	C₄⋊Dic₅	D₃	1	1	yes	characteristic
12	C₂×#168	—	3	1	no	—
13	C₂²×C₃×C₅	D₄	1	1	yes	characteristic
14	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄×C₅	Q₈	1	1	yes	characteristic
16	C₃×Dic₅	—	4	1	no	—
17	C₅×Dic₃	—	4	1	no	—
18	C₃×C₄⋊C₄	—	5	1	no	—
19	C₄⋊Dic₃	D₅	1	1	yes	characteristic
20	C₂×C₃⋊C₈	—	5	1	no	—
21	C₂×C₄×C₅	—	6	1	no	—
22	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
23	C₂×Dic₅	—	2	1	no	—
24	#168	—	3	2	no	—
25	#106	—	15	1	no	Sylow
26	C₂×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₃×C₅	SD₁₆	1	1	yes	characteristic
28	C₂×C₃×C₅	SD₁₆	1	1	yes	characteristic
29	C₂×C₃×C₄	—	10	1	no	—
30	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
31	C₂×Dic₃	—	2	1	no	—
32	C₃⋊C₈	—	5	2	no	—
33	C₂²×C₅	#67	1	1	yes	characteristic
34	C₄×C₅	—	12	1	no	—
35	C₄×C₅	Dic₆	1	1	yes	characteristic
36	C₄×C₅	C₄×D₃	1	1	yes	characteristic
37	Dic₅	—	4	1	no	—
38	C₂×C₈	—	15	1	no	—
39	C₄⋊C₄	—	3	1	no	—
40	C₄⋊C₄	—	5	1	no	—
41	C₃×C₅	#106	1	1	yes	characteristic, complemented, Hall
42	C₂²×C₃	C₂²⋊D₅	1	1	yes	characteristic
43	C₃×C₄	—	20	1	no	—
44	C₃×C₄	Dic₁₀	1	1	yes	characteristic
45	C₃×C₄	C₄×D₅	1	1	yes	characteristic
46	Dic₃	—	4	1	no	—
47	C₂×C₅	Dic₃⋊C₄	1	1	yes	characteristic
48	C₂×C₅	#214	1	1	yes	characteristic
49	C₂×C₅	#215	1	1	yes	characteristic
50	C₂×C₄	—	6	1	no	—
51	C₂×C₄	—	10	1	no	—
52	C₂×C₄	D₃×D₅	1	1	yes	characteristic
53	C₈	—	15	2	no	—
54	C₂×C₃	#683	1	1	yes	characteristic
55	C₂×C₃	#687	1	1	yes	characteristic
56	C₂×C₃	#688	1	1	yes	characteristic
57	C₅	#808	1	1	yes	characteristic, complemented, Sylow
58	C₂²	#1201	1	1	yes	characteristic
59	C₄	—	12	1	no	—
60	C₄	—	20	1	no	—
61	C₄	#1200	1	1	yes	characteristic
62	C₄	#1204	1	1	yes	characteristic
63	C₃	#3940	1	1	yes	characteristic, complemented, Sylow
64	C₂	#6655	1	1	yes	characteristic
65	C₂	#6656	1	1	yes	characteristic
66	C₂	#6669	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#1201
commutator	C₃×C₄×C₅	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃×D₅
Fitting, cosocle	C₂×C₃×C₄×C₅	C₂²
socle, prime radical	C₂²×C₃×C₅	D₄
hypocenter	C₃×C₅	#106
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
8. C₃⋊#168	C₄	semidirect
41. C₃×C₅	#106	semidirect
57. C₅	#808	semidirect
63. C₃	#3940	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊#168	2
C₅×C₄⋊Dic₃	2
C₃×C₄⋊Dic₅	2
#3940	3
#808	5