Subgroups — Q₈×SL(2,𝔽₅) #11751552

Order 960.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×SL(2,𝔽₅)	C₂	1	3	yes	maximal
2	C₂×SL(2,𝔽₅)	C₂²	1	1	yes	characteristic
3	Q₈×SL(2,𝔽₃)	—	5	1	no	maximal, Hall
4	Q₈×Dic₅	—	6	1	no	maximal
5	SL(2,𝔽₅)	Q₈	1	1	yes	characteristic, direct product factor
6	Q₈×Dic₃	—	10	1	no	maximal
7	C₄×SL(2,𝔽₃)	—	5	3	no	—
8	C₂×C₅×Q₈	—	6	1	no	—
9	C₄⋊Dic₅	—	6	3	no	—
10	C₄×Dic₅	—	6	3	no	—
11	Q₈²	—	5	1	no	Sylow
12	C₂×C₃×Q₈	—	10	1	no	—
13	C₂×SL(2,𝔽₃)	—	5	1	no	—
14	C₄⋊Dic₃	—	10	3	no	—
15	C₄×Dic₃	—	10	3	no	—
16	C₅×Q₈	—	6	4	no	—
17	C₂×C₄×C₅	—	6	3	no	—
18	C₂×Dic₅	—	6	1	no	—
19	C₂×Dic₅	—	6	3	no	—
20	C₄⋊Q₈	—	15	3	no	—
21	C₄×Q₈	—	5	3	no	—
22	C₄×Q₈	—	15	1	no	—
23	C₃×Q₈	—	10	4	no	—
24	C₂×C₃×C₄	—	10	3	no	—
25	C₂×Dic₃	—	10	1	no	—
26	C₂×Dic₃	—	10	3	no	—
27	SL(2,𝔽₃)	—	5	1	no	—
28	C₂²×C₅	—	6	1	no	—
29	C₄×C₅	—	6	6	no	—
30	Dic₅	—	6	1	no	—
31	Dic₅	—	6	1	no	—
32	Dic₅	—	12	3	no	—
33	C₂×Q₈	—	5	1	no	—
34	C₂×Q₈	—	15	2	no	—
35	C₂×Q₈	A₅	1	1	yes	maximal normal, characteristic
36	C₄⋊C₄	—	15	3	no	—
37	C₄⋊C₄	—	15	3	no	—
38	C₄²	—	15	3	no	—
39	C₂²×C₃	—	10	1	no	—
40	C₃×C₄	—	10	6	no	—
41	Dic₃	—	10	1	no	—
42	Dic₃	—	10	1	no	—
43	Dic₃	—	20	3	no	—
44	C₂×C₅	—	6	1	no	—
45	C₂×C₅	—	6	1	no	—
46	C₂×C₅	—	6	1	no	—
47	Q₈	—	5	1	no	—
48	Q₈	—	15	1	no	—
49	Q₈	—	60	2	no	—
50	Q₈	SL(2,𝔽₅)	1	4	yes	direct product factor
51	C₂×C₄	—	15	1	no	—
52	C₂×C₄	—	15	3	no	—
53	C₂×C₄	C₂×A₅	1	3	yes	—
54	C₂×C₃	—	10	1	no	—
55	C₂×C₃	—	10	1	no	—
56	C₂×C₃	—	10	1	no	—
57	C₅	—	6	1	no	Sylow
58	C₂²	C₂²×A₅	1	1	yes	characteristic
59	C₄	—	15	1	no	—
60	C₄	—	15	1	no	—
61	C₄	—	30	3	no	—
62	C₄	C₂×SL(2,𝔽₅)	1	6	yes	—
63	C₃	—	10	1	no	Sylow
64	C₂	Q₈×A₅	1	1	yes	characteristic
65	C₂	#91883	1	1	yes	characteristic
66	C₂	C₂²×SL(2,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²	C₂²×A₅
commutator, prime radical	C₂×SL(2,𝔽₅)	C₂²
Fitting, hypercenter	C₂×Q₈	A₅
hypocenter	SL(2,𝔽₅)	Q₈
LCS	G ▷ C₂×SL(2,𝔽₅) ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
5. SL(2,𝔽₅)	Q₈	direct (N × H)
50. Q₈	SL(2,𝔽₅)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×SL(2,𝔽₅)	2
Q₈×SL(2,𝔽₃)	5
Q₈×Dic₅	6
Q₈×Dic₃	10

Subgroups — Q8×SL(2,𝔽5) #11751552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×SL(2,𝔽₅) #11751552