Subgroups — #13015263

Order 1280.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10603839	C₂	1	3	yes	maximal
2	#10603806	C₂	1	3	yes	maximal
3	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
4	C₄²×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
5	C₄×D₅⋊C₈	C₂²	1	3	yes	—
6	C₄×C₈×D₅	C₂²	1	3	yes	—
7	(C₂×AGL(1,𝔽₅))⋊C₈	C₄	1	12	yes	—
8	#170⋊C₈	C₄	1	12	yes	complemented
9	C₅×C₈²	C₄	1	1	yes	characteristic, complemented
10	C₈×#168	C₄	1	3	yes	complemented
11	#7845	—	5	1	no	maximal, Sylow
12	C₂×C₄×AGL(1,𝔽₅)	C₂×C₄	1	6	yes	—
13	C₂×D₅⋊C₈	C₂×C₄	1	6	yes	—
14	C₂×C₈×D₅	C₂×C₄	1	6	yes	—
15	C₄²×D₅	C₂³	1	1	yes	characteristic
16	C₄×#170	C₂×C₄	1	12	yes	—
17	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
18	C₈×Dic₅	C₂×C₄	1	6	yes	—
19	C₄×#168	C₂×C₄	1	3	yes	—
20	#1744	—	5	3	no	—
21	C₂×#322	—	5	3	no	—
22	C₂×C₈²	—	5	1	no	—
23	C₂²×AGL(1,𝔽₅)	C₄²	1	4	yes	—
24	C₂×C₄×D₅	C₂²×C₄	1	3	yes	—
25	C₂×#170	C₄²	1	12	yes	—
26	C₄×AGL(1,𝔽₅)	—	2	12	no	—
27	D₅⋊C₈	—	2	12	no	—
28	C₂×C₅×C₈	C₄²	1	6	yes	—
29	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₄×Dic₅	C₂²×C₄	1	3	yes	—
31	C₂×#168	C₄²	1	6	yes	—
32	C₈×D₅	—	2	12	no	—
33	C₂×C₄×C₈	—	5	3	no	—
34	C₂×C₄×C₈	—	5	3	no	—
35	C₄³	—	5	1	no	—
36	#336	—	5	12	no	—
37	#322	—	5	12	no	—
38	C₈²	—	5	3	no	—
39	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
40	C₂²×D₅	C₂×C₄²	1	1	yes	characteristic
41	C₂×AGL(1,𝔽₅)	—	2	12	no	—
42	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
43	C₂×Dic₅	C₂×C₄²	1	3	yes	—
44	C₄×D₅	#131	1	12	yes	—
45	#170	—	2	12	no	—
46	C₅×C₈	—	2	6	no	—
47	#168	—	2	6	no	—
48	C₂²×C₈	—	5	6	no	—
49	C₂²×C₈	—	5	6	no	—
50	C₂×C₄²	—	5	1	no	—
51	C₂×C₄²	—	5	6	no	—
52	C₄×C₈	—	5	12	no	—
53	C₄×C₈	—	5	6	no	—
54	C₄×C₈	—	5	3	no	—
55	C₄×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
56	C₂²×C₅	C₄³	1	1	yes	characteristic
57	C₂×D₅	#405	1	6	yes	—
58	AGL(1,𝔽₅)	—	4	4	no	—
59	C₄×C₅	#405	1	6	yes	—
60	Dic₅	#405	1	6	yes	—
61	C₂²×C₄	—	5	3	no	—
62	C₂²×C₄	—	5	4	no	—
63	C₂×C₈	—	5	12	no	—
64	C₂×C₈	—	5	6	no	—
65	C₂×C₈	—	10	12	no	—
66	C₂×C₈	—	10	12	no	—
67	C₂×C₈	C₄×AGL(1,𝔽₅)	1	6	yes	—
68	C₄²	—	5	3	no	—
69	C₄²	—	10	12	no	—
70	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
71	C₂×C₅	#1726	1	3	yes	—
72	D₅	#1454	1	4	yes	—
73	C₂³	—	5	1	no	—
74	C₂×C₄	—	5	3	no	—
75	C₂×C₄	—	5	12	no	—
76	C₂×C₄	—	10	12	no	—
77	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	3	yes	—
78	C₈	—	2	6	no	—
79	C₈	—	10	12	no	—
80	C₈	—	10	6	no	—
81	C₅	#7845	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	5	6	no	—
83	C₂²	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₄	—	5	6	no	—
85	C₄	—	20	4	no	—
86	C₄	#65718	1	6	yes	—
87	C₂	—	5	4	no	—
88	C₂	#10603838	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₄³
Fitting	C₅×C₈²	C₄
prime radical	C₂²×D₅	C₂×C₄²
cosocle	C₄²×D₅	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#7845
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
8. #170⋊C₈	C₄	semidirect
9. C₅×C₈²	C₄	semidirect
10. C₈×#168	C₄	semidirect
39. C₈²	AGL(1,𝔽₅)	semidirect
81. C₅	#7845	semidirect

Maximal subgroups

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

subgroup	index
#10603839	2
#10603806	2
C₈²×D₅	2
#7845	5