Subgroups — #13014079

Order 1280.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10598537	C₂	1	1	yes	maximal, characteristic
2	#10598537	C₂	1	1	yes	maximal, characteristic
3	#10597933	C₂	1	1	yes	maximal, characteristic
4	#64877	C₄	1	2	yes	—
5	#64877	C₄	1	2	yes	—
6	C₅×#322	C₄	1	1	yes	characteristic
7	#64681	C₄	1	1	yes	characteristic
8	#64681	C₂²	1	1	yes	characteristic
9	#7333	—	5	1	no	maximal, Sylow
10	C₄×C₅×C₈	—	2	1	no	—
11	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
12	C₂×#672	—	2	1	no	—
13	C₂×#672	C₂×C₄	1	1	yes	characteristic
14	C₂×#672	C₂×C₄	1	1	yes	characteristic
15	#3928	—	2	2	no	—
16	#3928	C₈	1	4	yes	complemented
17	#1398	—	5	1	no	—
18	#1398	—	5	1	no	—
19	#1313	—	5	1	no	—
20	C₂×C₅×C₈	—	4	1	no	—
21	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
22	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
23	C₄²×C₅	M₁₆	1	1	yes	characteristic
24	#672	—	4	1	no	—
25	#672	C₂×C₈	1	2	yes	—
26	#672	M₁₆	1	2	yes	—
27	M₆₄	—	5	2	no	—
28	M₆₄	—	5	2	no	—
29	#346	—	5	1	no	—
30	#346	—	5	1	no	—
31	#322	AGL(1,𝔽₅)	1	1	yes	characteristic
32	C₂×C₄×C₅	—	2	1	no	—
33	C₂×C₄×C₅	#97	1	1	yes	characteristic
34	C₅×C₈	—	2	1	no	—
35	C₅×C₈	—	8	1	no	—
36	C₅×C₈	C₄×C₈	1	1	yes	characteristic
37	C₅×C₈	#97	1	1	yes	characteristic
38	C₂×C₁₆	—	5	1	no	—
39	C₂×C₁₆	—	5	1	no	—
40	C₂×C₁₆	—	10	1	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
43	C₃₂	—	5	4	no	—
44	C₃₂	—	10	2	no	—
45	C₂²×C₅	#347	1	1	yes	characteristic
46	C₄×C₅	—	4	1	no	—
47	C₄×C₅	#322	1	1	yes	characteristic
48	C₄×C₅	#347	1	1	yes	characteristic
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	#700	1	1	yes	characteristic
51	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₄²	#704	1	1	yes	characteristic
53	C₁₆	—	5	2	no	—
54	C₁₆	—	5	2	no	—
55	C₁₆	—	20	1	no	—
56	C₂×C₅	—	2	1	no	—
57	C₂×C₅	#1313	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	#4002	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	—	8	1	no	—
62	C₈	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₈	#3992	1	1	yes	characteristic
64	C₅	#7333	1	1	yes	characteristic, complemented, Sylow
65	C₂²	#64860	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#170⋊C₈	1	1	yes	characteristic
68	C₄	#64890	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#10598515	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#170⋊C₈
commutator	C₅×C₈	C₄×C₈
Frattini	C₄×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₅×#322	C₄
socle	C₂×C₅	#1313
prime radical	C₂²×C₅	#347
cosocle	#64681	C₂²
hypercenter	#322	AGL(1,𝔽₅)
hypocenter	C₅	#7333
LCS	G ▷ C₅×C₈ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ #322

Complements

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

N	H ≅ G/N	type
16. #3928	C₈	semidirect
64. C₅	#7333	semidirect

Maximal subgroups

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

subgroup	index
#10598537	2
#10597933	2
#7333	5