Subgroups — #4064

Order 160.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#713	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#689	C₂	1	1	yes	maximal, characteristic, complemented
5	#688	C₂	1	1	yes	maximal, characteristic, complemented
6	#677	C₂	1	1	yes	maximal, characteristic, complemented
7	#676	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₅×Q₈	C₂²	1	1	yes	characteristic
10	D₂₀	—	2	1	no	—
11	D₂₀	C₂²	1	1	yes	characteristic
12	C₄×D₅	—	2	1	no	—
13	C₄×D₅	—	2	1	no	—
14	C₄×D₅	C₂²	1	1	yes	characteristic
15	Dic₁₀	—	2	1	no	—
16	Dic₁₀	C₂²	1	1	yes	characteristic
17	C₅×C₈	C₂²	1	1	yes	characteristic
18	#168	C₂²	1	1	yes	characteristic
19	#137	—	5	1	no	maximal, Sylow
20	C₂×D₅	—	2	1	no	—
21	C₂×D₅	D₄	1	1	yes	characteristic
22	C₄×C₅	—	2	1	no	—
23	C₄×C₅	—	2	1	no	—
24	C₄×C₅	C₂³	1	1	yes	characteristic
25	Dic₅	—	2	1	no	—
26	Dic₅	D₄	1	1	yes	characteristic
27	#41	—	5	1	no	—
28	C₂×Q₈	—	5	1	no	—
29	Q₁₆	—	5	1	no	—
30	Q₁₆	D₅	1	1	yes	characteristic, complemented
31	SD₁₆	—	5	1	no	—
32	SD₁₆	—	5	1	no	—
33	M₁₆	—	5	1	no	—
34	C₂×C₅	C₂×D₄	1	1	yes	characteristic
35	D₅	—	2	1	no	—
36	D₅	—	4	1	no	—
37	Q₈	—	5	1	no	—
38	Q₈	—	10	1	no	—
39	Q₈	C₂×D₅	1	1	yes	characteristic
40	Q₈	C₂×D₅	1	1	yes	characteristic
41	D₄	—	5	1	no	—
42	D₄	—	10	1	no	—
43	C₂×C₄	—	5	1	no	—
44	C₂×C₄	—	10	1	no	—
45	C₂×C₄	—	10	1	no	—
46	C₈	—	5	1	no	—
47	C₈	C₂×D₅	1	1	yes	characteristic
48	C₅	#137	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	5	1	no	—
50	C₂²	—	10	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	5	1	no	—
54	C₄	—	10	1	no	—
55	C₄	C₂²×D₅	1	1	yes	characteristic
56	C₂	—	10	1	no	—
57	C₂	—	20	1	no	—
58	C₂	D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₅
commutator, cosocle	C₄×C₅	C₂³
Frattini	C₄	C₂²×D₅
Fitting	C₅×Q₁₆	C₂
socle	C₂×C₅	C₂×D₄
prime radical	#713	C₂
hypercenter	Q₁₆	D₅
hypocenter	C₅	#137
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₅	C₂	semidirect
3. C₅×Q₁₆	C₂	semidirect
4. #689	C₂	semidirect
5. #688	C₂	semidirect
6. #677	C₂	semidirect
7. #676	C₂	semidirect
30. Q₁₆	D₅	semidirect
48. C₅	#137	semidirect

Maximal subgroups

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

subgroup	index
#713	2
Q₈×D₅	2
C₅×Q₁₆	2
#689	2
#688	2
#677	2
#676	2
#137	5