Subgroups — #12014015

Order 1280.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10599859	C₂	1	4	yes	maximal
2	#10598858	C₂	1	2	yes	maximal
3	C₅×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₀	—	2	8	no	—
5	#64988	C₂²	1	4	yes	—
6	#64971	—	2	4	no	—
7	C₄×C₅×C₁₆	C₂²	1	2	yes	—
8	C₅×C₈²	C₂²	1	1	yes	characteristic
9	#12516	—	5	1	no	maximal, Sylow
10	C₂×Dic₂₀	—	2	8	no	—
11	#4015	—	2	4	no	—
12	C₂×C₅×C₁₆	D₄	1	4	yes	—
13	C₄×C₅×C₈	D₄	1	2	yes	—
14	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
15	#3950	—	4	8	no	—
16	Dic₄₀	—	4	16	no	—
17	#2247	—	5	4	no	—
18	#1713	—	5	2	no	—
19	C₈×C₁₆	D₅	1	1	yes	characteristic
20	C₂×Dic₁₀	—	4	4	no	—
21	C₂×C₅×C₈	D₈	1	4	yes	—
22	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
23	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
24	C₄⋊Dic₅	—	4	8	no	—
25	Dic₂₀	—	4	16	no	—
26	C₅×C₁₆	D₈	1	8	yes	—
27	C₂×Q₃₂	—	10	8	no	—
28	#500	—	10	4	no	—
29	#494	—	5	4	no	—
30	C₄×C₁₆	C₂×D₅	1	2	yes	—
31	C₈²	C₂×D₅	1	1	yes	characteristic
32	C₂×C₄×C₅	#127	1	1	yes	characteristic
33	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
34	C₂×Dic₅	—	8	4	no	—
35	Dic₁₀	—	8	8	no	—
36	C₅×C₈	Q₃₂	1	8	yes	complemented
37	C₅×C₈	C₂×D₈	1	4	yes	—
38	C₂×Q₁₆	—	10	8	no	—
39	C₄⋊Q₈	—	10	4	no	—
40	Q₃₂	—	20	16	no	—
41	C₂×C₁₆	D₂₀	1	4	yes	—
42	#107	—	20	8	no	—
43	C₄×C₈	D₂₀	1	2	yes	—
44	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
45	C₂²×C₅	#493	1	1	yes	characteristic
46	C₄×C₅	#493	1	2	yes	—
47	C₄×C₅	C₂×Q₃₂	1	4	yes	—
48	Dic₅	—	16	4	no	—
49	C₂×Q₈	—	20	4	no	—
50	Q₁₆	—	20	16	no	—
51	C₂×C₈	D₄₀	1	4	yes	—
52	C₂×C₈	C₂×D₂₀	1	2	yes	—
53	C₄⋊C₄	—	20	8	no	—
54	C₄²	C₂×D₂₀	1	1	yes	characteristic
55	C₁₆	D₄₀	1	8	yes	—
56	C₂×C₅	#1712	1	1	yes	characteristic
57	C₂×C₅	#2247	1	2	yes	—
58	Q₈	—	40	8	no	—
59	C₂×C₄	—	40	4	no	—
60	C₂×C₄	#4020	1	1	yes	characteristic
61	C₂×C₄	C₂×D₄₀	1	2	yes	—
62	C₈	Dic₄₀	1	8	yes	complemented
63	C₈	C₂×D₄₀	1	4	yes	—
64	C₅	#12516	1	1	yes	characteristic, complemented, Sylow
65	C₂²	#64984	1	1	yes	characteristic
66	C₄	—	80	4	no	—
67	C₄	#64984	1	2	yes	—
68	C₄	C₂×Dic₄₀	1	4	yes	—
69	C₂	#10598855	1	1	yes	characteristic
70	C₂	#10599859	1	2	yes	—

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×C₈×C₁₆	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	C₈×C₁₆	D₅
hypocenter	C₅	#12516
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
36. C₅×C₈	Q₃₂	semidirect
62. C₈	Dic₄₀	semidirect
64. C₅	#12516	semidirect

Maximal subgroups

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

subgroup	index
#10599859	2
#10598858	2
C₅×C₈×C₁₆	2
#12516	5