Subgroups — #64887

Order 320.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#674	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₅	C₄	1	1	yes	characteristic
4	C₂×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂×#168	C₂²	1	1	yes	characteristic
6	C₈×D₅	—	2	2	no	—
7	#674	—	2	2	no	—
8	C₂²⋊C₁₆	—	5	1	no	maximal, Sylow
9	C₂²×D₅	C₈	1	1	yes	characteristic
10	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
11	C₂×Dic₅	C₈	1	1	yes	characteristic
12	C₄×D₅	—	2	1	no	—
13	C₄×D₅	—	2	1	no	—
14	C₅×C₈	—	2	1	no	—
15	#168	D₄	1	2	yes	—
16	C₂²×C₈	—	5	1	no	—
17	C₂×C₁₆	—	5	2	no	—
18	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
19	C₂×D₅	—	2	1	no	—
20	C₂×D₅	—	2	1	no	—
21	C₂×D₅	C₁₆	1	2	yes	complemented
22	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₅	M₁₆	1	1	yes	characteristic
24	Dic₅	—	2	1	no	—
25	C₂²×C₄	—	5	1	no	—
26	C₂×C₈	—	5	1	no	—
27	C₂×C₈	—	10	2	no	—
28	C₂×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
29	C₁₆	—	10	2	no	—
30	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
31	C₂×C₅	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₅	M₃₂	1	1	yes	characteristic
33	D₅	—	2	2	no	—
34	C₂³	—	5	1	no	—
35	C₂×C₄	—	5	1	no	—
36	C₂×C₄	—	10	1	no	—
37	C₂×C₄	—	10	1	no	—
38	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₈	—	2	1	no	—
40	C₈	—	5	2	no	—
41	C₅	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	5	2	no	—
43	C₂²	—	10	1	no	—
44	C₂²	—	10	1	no	—
45	C₂²	D₅⋊C₈	1	1	yes	characteristic
46	C₄	—	10	1	no	—
47	C₄	#700	1	1	yes	characteristic
48	C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₂	—	10	2	no	—
50	C₂	D₅⋊C₁₆	1	1	yes	characteristic
51	C₂	#3990	1	1	yes	characteristic
52	C₂	#4003	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₁₆
Fitting	C₂×C₅×C₈	C₄
socle	C₂²×C₅	C₂×C₈
prime radical	C₂²×D₅	C₈
cosocle	C₂×#168	C₂²
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂²⋊C₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#674	C₂	semidirect
21. C₂×D₅	C₁₆	semidirect
41. C₅	C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₅	2
C₂×#674	2
C₂²⋊C₁₆	5