Subgroups — #13013296

Order 1280.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10598513	C₂	1	2	yes	maximal
2	#10597989	C₂	1	1	yes	maximal, characteristic
3	C₄×#674	C₄	1	4	yes	—
4	C₄×C₅×C₁₆	C₄	1	1	yes	characteristic
5	C₄×#672	C₄	1	1	yes	characteristic
6	C₈×#168	C₂²	1	1	yes	characteristic
7	#7053	—	5	1	no	maximal, Sylow
8	C₂×#674	—	2	4	no	—
9	C₂×C₅×C₁₆	C₈	1	2	yes	—
10	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
11	C₂×#672	C₈	1	2	yes	—
12	C₄×#168	C₂×C₄	1	2	yes	—
13	#1312	—	5	2	no	—
14	#1311	—	5	1	no	—
15	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
16	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₅	C₄²	1	1	yes	characteristic
18	C₂×#168	M₁₆	1	4	yes	—
19	#674	—	4	4	no	—
20	C₅×C₁₆	C₁₆	1	4	yes	complemented
21	#672	—	2	2	no	—
22	C₄×C₁₆	—	5	4	no	—
23	C₄×C₁₆	—	5	1	no	—
24	C₄×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₈²	—	5	1	no	—
26	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
27	C₂×C₄×C₅	#97	1	2	yes	—
28	C₅×C₈	C₂×C₁₆	1	2	yes	—
29	C₅×C₈	M₃₂	1	2	yes	—
30	#168	—	2	4	no	—
31	C₂×C₁₆	—	5	2	no	—
32	C₂×C₁₆	—	10	4	no	—
33	C₂×C₁₆	#170	1	2	yes	—
34	C₄×C₈	—	5	2	no	—
35	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
36	C₂²×C₅	#322	1	1	yes	characteristic
37	C₄×C₅	C₄×C₁₆	1	1	yes	characteristic
38	C₄×C₅	#346	1	1	yes	characteristic
39	C₄×C₅	#347	1	4	yes	—
40	C₂×C₈	—	5	4	no	—
41	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₂×#170	1	1	yes	characteristic
43	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₁₆	—	10	2	no	—
45	C₁₆	—	20	4	no	—
46	C₁₆	#674	1	4	yes	complemented
47	C₂×C₅	#1312	1	1	yes	characteristic
48	C₂×C₅	#1313	1	2	yes	—
49	C₂×C₄	#3992	1	2	yes	—
50	C₂×C₄	C₄×#170	1	1	yes	characteristic
51	C₈	—	10	4	no	—
52	C₈	#3990	1	2	yes	—
53	C₈	C₂×#674	1	2	yes	—
54	C₅	#7053	1	1	yes	characteristic, complemented, Sylow
55	C₂²	#64879	1	1	yes	characteristic
56	C₄	#64846	1	4	yes	—
57	C₄	C₄×#674	1	1	yes	characteristic
58	C₄	#64886	1	1	yes	characteristic
59	C₂	#10598513	1	1	yes	characteristic
60	C₂	#10598638	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₄×C₁₆
Frattini	C₄×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₁₆	C₄
socle, prime radical	C₂²×C₅	#322
cosocle	C₈×#168	C₂²
hypercenter	C₄×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#7053
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
20. C₅×C₁₆	C₁₆	semidirect
46. C₁₆	#674	semidirect
54. C₅	#7053	semidirect

Maximal subgroups

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

subgroup	index
#10598513	2
#10597989	2
#7053	5