Subgroups — #66262

Order 320.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₅	C₂	1	1	yes	maximal, characteristic
2	Q₈×AGL(1,𝔽₅)	C₂	1	8	yes	maximal, complemented
3	#4130	C₂	1	6	yes	maximal
4	C₂×C₅×Q₈	C₄	1	1	yes	characteristic, complemented
5	Q₈×D₅	C₄	1	4	yes	complemented
6	Q₈×D₅	C₂²	1	4	yes	—
7	C₂×C₄×D₅	C₂²	1	3	yes	—
8	C₂×Dic₁₀	C₄	1	3	yes	complemented
9	C₂²⋊AGL(1,𝔽₅)	C₂²	1	4	yes	—
10	C₄⋊AGL(1,𝔽₅)	C₂²	1	12	yes	—
11	C₄×AGL(1,𝔽₅)	C₂²	1	12	yes	—
12	#519	—	5	1	no	maximal, Sylow
13	C₂²×D₅	C₂³	1	1	yes	characteristic
14	C₂×AGL(1,𝔽₅)	C₂³	1	8	yes	—
15	C₅×Q₈	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₅	C₂×C₄	1	3	yes	—
17	C₂×Dic₅	C₂×C₄	1	3	yes	—
18	C₄×D₅	C₂×C₄	1	6	yes	—
19	C₄×D₅	C₂³	1	6	yes	—
20	Dic₁₀	C₂×C₄	1	12	yes	—
21	C₂²×Q₈	—	5	1	no	—
22	C₄×Q₈	—	5	8	no	—
23	#117	—	5	6	no	—
24	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
25	C₂×D₅	—	2	2	no	—
26	C₂×D₅	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₅	C₂⁴	1	1	yes	characteristic
28	AGL(1,𝔽₅)	—	2	8	no	—
29	C₄×C₅	C₂²×C₄	1	6	yes	—
30	Dic₅	C₂²×C₄	1	6	yes	—
31	C₂×Q₈	—	5	4	no	—
32	C₂×Q₈	—	5	4	no	—
33	C₂×Q₈	—	5	3	no	—
34	C₂×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
35	C₂²×C₄	—	5	3	no	—
36	C₄⋊C₄	—	5	12	no	—
37	C₂²⋊C₄	—	5	4	no	—
38	C₄²	—	5	12	no	—
39	C₂×C₅	—	2	1	no	—
40	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
41	D₅	—	2	1	no	—
42	D₅	2₋¹⁺⁴	1	2	yes	—
43	C₂³	—	5	1	no	—
44	Q₈	—	5	12	no	—
45	Q₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
46	C₂×C₄	—	5	3	no	—
47	C₂×C₄	—	5	6	no	—
48	C₂×C₄	—	5	6	no	—
49	C₂×C₄	—	5	8	no	—
50	C₂×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
51	C₅	#519	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	5	1	no	—
53	C₂²	—	5	1	no	—
54	C₂²	—	10	2	no	—
55	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₄	—	5	6	no	—
57	C₄	—	10	8	no	—
58	C₄	C₂²×AGL(1,𝔽₅)	1	6	yes	—
59	C₂	—	2	1	no	—
60	C₂	—	5	2	no	—
61	C₂	—	10	1	no	—
62	C₂	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂³×C₄
Fitting	C₂×C₅×Q₈	C₄
prime radical	C₂²×D₅	C₂³
cosocle	C₂×D₅	C₂⁴
hypercenter	C₂×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#519
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×AGL(1,𝔽₅)	C₂	semidirect
4. C₂×C₅×Q₈	C₄	semidirect
5. Q₈×D₅	C₄	semidirect
8. C₂×Dic₁₀	C₄	semidirect
34. C₂×Q₈	AGL(1,𝔽₅)	semidirect
51. C₅	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₅	2
Q₈×AGL(1,𝔽₅)	2
#4130	2
#519	5