Subgroups — #91177

Order 480.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊#170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#170	C₂	1	1	yes	maximal, characteristic
3	Dic₃×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#4004	—	3	1	no	maximal, Hall
5	C₂×Dic₁₅	C₄	1	1	yes	characteristic
6	C₂×C₅×Dic₃	C₄	1	1	yes	characteristic
7	C₂×C₃×Dic₅	C₂²	1	1	yes	characteristic
8	C₃⋊#170	—	2	1	no	—
9	C₃×#170	—	2	1	no	—
10	Dic₃⋊C₈	—	5	1	no	maximal, Hall
11	C₂×#170	—	3	1	no	—
12	C₂×#170	D₃	1	1	yes	characteristic
13	C₄×Dic₅	—	3	1	no	—
14	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
15	Dic₁₅	C₈	1	2	yes	complemented
16	C₃×Dic₅	D₄	1	1	yes	characteristic
17	C₃×Dic₅	Q₈	1	1	yes	characteristic
18	C₅×Dic₃	—	2	1	no	—
19	C₂×C₃×C₈	—	5	1	no	—
20	C₄×Dic₃	—	5	1	no	—
21	C₂×C₃⋊C₈	—	5	1	no	—
22	C₂×C₄×C₅	—	3	1	no	—
23	C₂×Dic₅	—	3	1	no	—
24	C₂×Dic₅	C₂×D₃	1	1	yes	characteristic
25	#170	—	2	1	no	—
26	#170	—	6	1	no	—
27	C₄⋊C₈	—	15	1	no	Sylow
28	C₂×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₃×C₅	C₂×C₈	1	1	yes	characteristic
30	C₂×C₃×C₅	M₁₆	1	1	yes	characteristic
31	C₂×C₃×C₄	—	5	1	no	—
32	C₂×Dic₃	—	5	1	no	—
33	C₂×Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic
34	C₃×C₈	—	10	1	no	—
35	C₃⋊C₈	—	10	1	no	—
36	C₂²×C₅	C₄×D₃	1	1	yes	characteristic
37	C₄×C₅	—	6	1	no	—
38	Dic₅	—	3	2	no	—
39	Dic₅	Dic₆	1	1	yes	characteristic
40	Dic₅	#67	1	1	yes	characteristic
41	C₂×C₈	—	5	1	no	—
42	C₂×C₈	—	15	1	no	—
43	C₄²	—	15	1	no	—
44	C₃×C₅	C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
45	C₂²×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₃×C₄	—	5	1	no	—
47	C₃×C₄	—	5	1	no	—
48	Dic₃	—	2	1	no	—
49	Dic₃	—	5	2	no	—
50	C₂×C₅	D₃×C₈	1	1	yes	characteristic
51	C₂×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
52	C₂×C₅	Dic₃⋊C₄	1	1	yes	characteristic
53	C₂×C₄	—	3	1	no	—
54	C₂×C₄	—	5	1	no	—
55	C₂×C₄	—	15	1	no	—
56	C₈	—	10	1	no	—
57	C₈	—	30	1	no	—
58	C₂×C₃	D₅⋊C₈	1	1	yes	characteristic
59	C₂×C₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₂×C₃	#704	1	1	yes	characteristic
61	C₅	Dic₃⋊C₈	1	1	yes	characteristic, complemented, Sylow
62	C₂²	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₄	—	5	1	no	—
64	C₄	—	5	1	no	—
65	C₄	—	6	1	no	—
66	C₄	—	15	2	no	—
67	C₃	#4004	1	1	yes	characteristic, complemented, Sylow
68	C₂	Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂	#6738	1	1	yes	characteristic
70	C₂	#6739	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂×C₈
Fitting, socle, prime radical	C₂²×C₃×C₅	C₂×C₄
cosocle	C₂×C₃×Dic₅	C₂²
hypocenter	C₃×C₅	C₄⋊C₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
15. Dic₁₅	C₈	semidirect
44. C₃×C₅	C₄⋊C₈	semidirect
61. C₅	Dic₃⋊C₈	semidirect
67. C₃	#4004	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊#170	2
C₂×C₃×#170	2
Dic₃×Dic₅	2
#4004	3
Dic₃⋊C₈	5