Subgroups — #64896

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#700	C₂	1	2	yes	maximal
2	C₂×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
4	C₂×#170	—	2	2	no	—
5	#700	—	2	2	no	—
6	#700	C₄	1	4	yes	—
7	C₂×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂×#168	C₄	1	1	yes	characteristic
9	C₈×D₅	—	2	2	no	—
10	#341	—	5	1	no	maximal, Sylow
11	C₂²×D₅	Q₈	1	1	yes	characteristic
12	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂×Dic₅	D₄	1	1	yes	characteristic
14	C₄×D₅	C₂×C₄	1	2	yes	—
15	C₄×D₅	D₄	1	2	yes	—
16	#170	—	2	4	no	—
17	C₅×C₈	—	2	1	no	—
18	#168	—	2	1	no	—
19	C₂×M₁₆	—	5	2	no	—
20	C₂²×C₈	—	5	1	no	—
21	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
22	C₂×D₅	—	2	1	no	—
23	C₂×D₅	—	2	1	no	—
24	C₂×D₅	C₄⋊C₄	1	2	yes	—
25	C₄×C₅	C₄²	1	1	yes	characteristic
26	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	Dic₅	C₂²⋊C₄	1	2	yes	—
28	C₂²×C₄	—	5	1	no	—
29	M₁₆	—	5	4	no	—
30	M₁₆	—	10	2	no	—
31	C₂×C₈	—	5	1	no	—
32	C₂×C₈	—	10	2	no	—
33	C₂×C₈	—	10	2	no	—
34	C₂×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₂×C₅	#95	1	1	yes	characteristic
36	C₂×C₅	#108	1	1	yes	characteristic
37	C₂×C₅	#108	1	1	yes	characteristic
38	D₅	—	2	2	no	—
39	C₂³	—	5	1	no	—
40	C₂×C₄	—	5	1	no	—
41	C₂×C₄	—	5	2	no	—
42	C₂×C₄	—	5	2	no	—
43	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	—	10	4	no	—
46	C₈	—	10	1	no	—
47	C₅	#341	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	5	2	no	—
49	C₂²	—	10	1	no	—
50	C₂²	—	10	1	no	—
51	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₄	—	5	2	no	—
53	C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₂	—	10	2	no	—
56	C₂	#3995	1	1	yes	characteristic
57	C₂	#3996	1	1	yes	characteristic
58	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₄²
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂²×C₅	C₄⋊C₄
prime radical	C₂²×D₅	Q₈
cosocle	C₂×C₄×D₅	C₂²
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#341
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
47. C₅	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×#700	2
C₂×C₈×D₅	2
#341	5