Subgroups — #91154

Order 480.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4009	—	3	1	no	maximal, Hall
5	C₂×ΓL(1,𝔽₁₆)	—	2	1	no	—
6	Dic₃₀	C₄	1	1	yes	characteristic, complemented
7	C₅×Dic₆	C₄	1	1	yes	characteristic, complemented
8	C₃×C₄×D₅	C₂²	1	1	yes	characteristic
9	#1204	—	2	1	no	—
10	D₅×Dic₃	—	2	1	no	—
11	C₃×#170	—	2	1	no	—
12	#816	—	5	1	no	maximal, Hall
13	Q₈×D₅	—	3	1	no	—
14	C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
15	D₅⋊C₈	D₃	1	1	yes	characteristic
16	C₂×C₃×D₅	D₄	1	1	yes	characteristic
17	ΓL(1,𝔽₁₆)	—	4	1	no	—
18	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	Dic₁₅	—	2	1	no	—
20	C₃×Dic₅	D₄	1	1	yes	characteristic
21	C₅×Dic₃	—	2	1	no	—
22	C₂×Dic₆	—	5	1	no	—
23	C₂×C₃×C₈	—	5	1	no	—
24	C₄⋊Dic₃	—	5	1	no	—
25	C₂×AGL(1,𝔽₅)	—	6	1	no	—
26	C₅×Q₈	—	3	1	no	—
27	C₄×D₅	—	6	1	no	—
28	C₄×D₅	C₂×D₃	1	1	yes	characteristic
29	Dic₁₀	—	3	1	no	—
30	Dic₁₀	—	6	1	no	—
31	#170	—	2	1	no	—
32	#103	—	15	1	no	Sylow
33	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
34	C₃×D₅	SD₁₆	1	1	yes	characteristic
35	C₃×D₅	Q₁₆	1	1	yes	characteristic
36	C₂×C₃×C₄	—	5	1	no	—
37	C₂×Dic₃	—	10	1	no	—
38	C₂×Dic₃	—	10	1	no	—
39	Dic₆	—	5	1	no	—
40	Dic₆	—	10	1	no	—
41	Dic₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
42	C₃×C₈	—	10	1	no	—
43	C₂×D₅	D₁₂	1	1	yes	characteristic
44	AGL(1,𝔽₅)	—	12	1	no	—
45	C₄×C₅	—	6	1	no	—
46	C₄×C₅	C₄×D₃	1	1	yes	characteristic
47	Dic₅	—	6	1	no	—
48	Dic₅	#67	1	1	yes	characteristic
49	C₂×Q₈	—	15	1	no	—
50	C₂×C₈	—	5	1	no	—
51	C₄⋊C₄	—	15	1	no	—
52	C₃×C₅	#103	1	1	yes	characteristic, complemented, Hall
53	C₂²×C₃	—	5	1	no	—
54	C₃×C₄	—	5	1	no	—
55	C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
56	Dic₃	—	2	1	no	—
57	Dic₃	—	10	1	no	—
58	Dic₃	—	20	1	no	—
59	C₂×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
60	D₅	#204	1	1	yes	characteristic
61	D₅	Dic₁₂	1	1	yes	characteristic
62	Q₈	—	3	1	no	—
63	Q₈	—	15	1	no	—
64	Q₈	—	30	1	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	30	1	no	—
67	C₂×C₄	—	30	1	no	—
68	C₈	—	10	1	no	—
69	C₂×C₃	—	5	1	no	—
70	C₂×C₃	—	5	1	no	—
71	C₂×C₃	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₅	#816	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	5	1	no	—
74	C₄	—	5	1	no	—
75	C₄	—	6	1	no	—
76	C₄	—	30	1	no	—
77	C₄	—	60	1	no	—
78	C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₃	#4009	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	5	1	no	—
81	C₂	—	5	1	no	—
82	C₂	(C₂×D₃)⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×D₃)⋊AGL(1,𝔽₅)
commutator, Fitting	C₃×C₄×C₅	C₂×C₄
Frattini, hypercenter	C₄	D₃×AGL(1,𝔽₅)
socle	C₂×C₃×C₅	C₂²⋊C₄
prime radical	C₂×C₃×D₅	D₄
cosocle	C₃×C₄×D₅	C₂²
hypocenter	C₃×C₅	#103
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
6. Dic₃₀	C₄	semidirect
7. C₅×Dic₆	C₄	semidirect
41. Dic₆	AGL(1,𝔽₅)	semidirect
52. C₃×C₅	#103	semidirect
72. C₅	#816	semidirect
79. C₃	#4009	semidirect

Maximal subgroups

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

subgroup	index
D₅×Dic₆	2
C₄⋊ΓL(1,𝔽₁₆)	2
C₃×D₅⋊C₈	2
#4009	3
#816	5