Subgroups — #64970

Order 320.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4015	C₂	1	1	yes	maximal, characteristic
2	#4015	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	#3948	C₂	1	4	yes	maximal
5	C₂×Dic₁₀	—	2	1	no	—
6	C₂×Dic₁₀	C₂²	1	2	yes	—
7	C₂×C₅×C₈	C₂²	1	2	yes	—
8	C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₅	—	2	1	no	—
10	C₄⋊Dic₅	—	2	1	no	—
11	C₄⋊Dic₅	—	2	1	no	—
12	C₄⋊Dic₅	C₂²	1	2	yes	—
13	#487	—	5	1	no	maximal, Sylow
14	C₂×C₄×C₅	D₄	1	1	yes	characteristic
15	C₂×C₄×C₅	D₄	1	1	yes	characteristic
16	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂×Dic₅	—	2	2	no	—
18	C₂×Dic₅	—	2	2	no	—
19	Dic₁₀	—	2	4	no	—
20	Dic₁₀	—	4	2	no	—
21	C₅×C₈	—	2	2	no	—
22	C₄⋊Q₈	—	5	1	no	—
23	C₄⋊Q₈	—	5	1	no	—
24	#103	—	5	4	no	—
25	C₄×C₈	D₅	1	1	yes	characteristic
26	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
27	C₄×C₅	SD₁₆	1	2	yes	—
28	C₄×C₅	Q₁₆	1	2	yes	—
29	C₄×C₅	#41	1	2	yes	—
30	Dic₅	—	4	2	no	—
31	Dic₅	—	4	2	no	—
32	C₂×Q₈	—	5	2	no	—
33	C₂×Q₈	—	10	1	no	—
34	C₂×C₈	C₂×D₅	1	2	yes	—
35	C₄⋊C₄	—	5	2	no	—
36	C₄⋊C₄	—	10	1	no	—
37	C₄⋊C₄	—	10	1	no	—
38	C₄⋊C₄	—	10	1	no	—
39	C₄²	C₂×D₅	1	1	yes	characteristic
40	C₂×C₅	#124	1	1	yes	characteristic
41	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₅	C₂×Q₁₆	1	1	yes	characteristic
43	Q₈	—	10	4	no	—
44	Q₈	—	20	2	no	—
45	C₂×C₄	—	10	2	no	—
46	C₂×C₄	—	10	2	no	—
47	C₂×C₄	D₂₀	1	1	yes	characteristic
48	C₂×C₄	D₂₀	1	1	yes	characteristic
49	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₅	#487	1	1	yes	characteristic, complemented, Sylow
52	C₂²	C₂×D₂₀	1	1	yes	characteristic
53	C₄	—	20	2	no	—
54	C₄	—	20	2	no	—
55	C₄	#677	1	2	yes	—
56	C₄	Dic₂₀	1	2	yes	—
57	C₄	#709	1	2	yes	—
58	C₂	#4021	1	1	yes	characteristic
59	C₂	C₂×#677	1	1	yes	characteristic
60	C₂	C₂×Dic₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×C₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	C₄×C₈	D₅
hypocenter	C₅	#487
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
51. C₅	#487	semidirect

Maximal subgroups

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

subgroup	index
#4015	2
C₄×C₅×C₈	2
#3948	2
#487	5