Subgroups — #10599859

Order 640.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²	C₂×D₄₀
commutator, cosocle	C₂×C₅×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₅
Fitting	C₄×C₅×C₁₆	C₂
socle, prime radical	C₂²×C₅	C₂×D₈
hypercenter	C₄×C₁₆	D₅
hypocenter	C₅	#2247
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
36. C₄×C₅	Q₃₂	semidirect
57. C₅	#2247	semidirect
60. C₄	Dic₄₀	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₄₀	2
#64988	2
C₄×C₅×C₁₆	2
#2247	5