Subgroups — #64700

Order 320.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#168	C₂	1	1	yes	maximal, characteristic
3	C₄×#168	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₈	—	2	1	no	—
5	C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂×#168	—	2	1	no	—
7	C₂×#168	C₄	1	1	yes	characteristic
8	C₂×#168	C₄	1	1	yes	characteristic
9	#334	—	5	1	no	maximal, Sylow
10	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₅	D₄	1	1	yes	characteristic
12	C₂×C₄×C₅	Q₈	1	1	yes	characteristic
13	C₅×C₈	—	4	1	no	—
14	#168	—	2	1	no	—
15	#168	—	4	1	no	—
16	#168	C₈	1	2	yes	complemented
17	C₄⋊C₈	—	5	1	no	—
18	C₄⋊C₈	D₅	1	1	yes	characteristic
19	C₄×C₈	—	5	1	no	—
20	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
21	C₄×C₅	—	2	1	no	—
22	C₄×C₅	C₂×C₈	1	1	yes	characteristic
23	C₄×C₅	M₁₆	1	1	yes	characteristic
24	C₄×C₅	SD₁₆	1	1	yes	characteristic
25	C₄×C₅	SD₁₆	1	1	yes	characteristic
26	C₂×C₈	—	2	1	no	—
27	C₂×C₈	—	5	1	no	—
28	C₂×C₈	—	5	1	no	—
29	C₂×C₈	—	10	1	no	—
30	C₄²	C₂×D₅	1	1	yes	characteristic
31	C₂×C₅	C₄⋊C₈	1	1	yes	characteristic
32	C₂×C₅	#106	1	1	yes	characteristic
33	C₂×C₅	#108	1	1	yes	characteristic
34	C₂×C₄	Dic₁₀	1	1	yes	characteristic
35	C₂×C₄	C₄×D₅	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
37	C₈	—	4	1	no	—
38	C₈	—	5	2	no	—
39	C₈	—	10	1	no	—
40	C₈	—	20	1	no	—
41	C₅	#334	1	1	yes	characteristic, complemented, Sylow
42	C₂²	#683	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₅	1	1	yes	characteristic
45	C₄	#676	1	1	yes	characteristic
46	C₄	#687	1	1	yes	characteristic
47	C₄	#688	1	1	yes	characteristic
48	C₂	#3940	1	1	yes	characteristic
49	C₂	#3946	1	1	yes	characteristic
50	C₂	#3954	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₅
commutator	C₄×C₅	C₂×C₈
Frattini	C₄²	C₂×D₅
Fitting	C₅×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂²
hypercenter	C₄⋊C₈	D₅
hypocenter	C₅	#334
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
16. #168	C₈	semidirect
41. C₅	#334	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₄⋊C₈	2
C₄⋊#168	2
C₄×#168	2
#334	5