Subgroups — C₂×C₅×#226 #91940

Order 480.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₅×#226	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×Q₁₆	—	3	1	no	maximal, Hall
4	C₂×C₅×Dic₃	—	4	1	no	maximal
5	C₅×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂×#226	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₅×Q₈	—	3	1	no	—
8	C₂×C₅×Q₈	D₃	1	1	yes	characteristic
9	C₅×Q₁₆	—	3	2	no	—
10	C₅×Q₁₆	—	3	2	no	—
11	C₂×C₅×C₈	—	3	1	no	—
12	C₂²×C₃×C₅	—	4	1	no	—
13	C₅×Dic₃	—	4	2	no	—
14	C₂×SL(2,𝔽₃)	C₂×C₅	1	1	yes	characteristic
15	#226	C₂×C₅	1	2	yes	direct product factor
16	C₅×Q₈	—	3	1	no	—
17	C₅×Q₈	—	3	2	no	—
18	C₅×Q₈	—	6	1	no	—
19	C₅×Q₈	C₂×D₃	1	1	yes	characteristic
20	C₂×C₄×C₅	—	3	1	no	—
21	C₂×C₄×C₅	—	6	1	no	—
22	C₅×C₈	—	3	2	no	—
23	C₂×Q₁₆	—	3	1	no	Sylow
24	C₂×C₃×C₅	—	4	2	no	—
25	C₂×C₃×C₅	—	4	1	no	—
26	C₂×Dic₃	—	4	1	no	—
27	SL(2,𝔽₃)	C₂²×C₅	1	1	yes	characteristic
28	C₂²×C₅	S₄	1	1	yes	characteristic
29	C₄×C₅	—	3	1	no	—
30	C₄×C₅	—	3	1	no	—
31	C₄×C₅	—	6	2	no	—
32	C₂×Q₈	—	3	1	no	—
33	C₂×Q₈	C₅×D₃	1	1	yes	characteristic
34	Q₁₆	—	3	2	no	—
35	Q₁₆	—	3	2	no	—
36	C₂×C₈	—	3	1	no	—
37	C₃×C₅	—	4	1	no	Hall
38	C₂²×C₃	—	4	1	no	—
39	Dic₃	—	4	2	no	—
40	C₂×C₅	#226	1	2	yes	direct product factor
41	C₂×C₅	C₂×S₄	1	1	yes	characteristic
42	Q₈	—	3	1	no	—
43	Q₈	—	3	2	no	—
44	Q₈	—	6	1	no	—
45	Q₈	C₂×C₅×D₃	1	1	yes	characteristic
46	C₂×C₄	—	3	1	no	—
47	C₂×C₄	—	6	1	no	—
48	C₈	—	3	2	no	—
49	C₂×C₃	—	4	2	no	—
50	C₂×C₃	—	4	1	no	—
51	C₅	C₂×#226	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²	C₅×S₄	1	1	yes	characteristic
53	C₄	—	3	1	no	—
54	C₄	—	3	1	no	—
55	C₄	—	6	2	no	—
56	C₃	—	4	1	no	Sylow
57	C₂	C₅×#226	1	2	yes	direct product factor
58	C₂	C₂×C₅×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂²×C₅	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂²×C₅
Frattini	C₂	C₂×C₅×S₄
Fitting	C₂×C₅×Q₈	D₃
prime radical	C₂×C₅×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×#226	C₂	direct (N × H)
6. C₂×#226	C₅	direct (N × H)
15. #226	C₂×C₅	direct (N × H)
40. C₂×C₅	#226	direct (N × H)
51. C₅	C₂×#226	direct (N × H)
57. C₂	C₅×#226	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×C₅×SL(2,𝔽₃)	2
C₅×#226	2
C₂×C₅×Q₁₆	3
C₂×C₅×Dic₃	4
C₂×#226	5

Subgroups — C2×C5×#226 #91940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#226 #91940