Subgroups — #13018804

Order 1280.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10603847	C₂	1	6	yes	maximal
2	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#3993	—	2	12	no	—
4	#65688	C₂²	1	4	yes	—
5	C₄×C₈×D₅	C₂²	1	3	yes	—
6	C₅×C₈²	C₄	1	1	yes	characteristic, complemented
7	C₈×#168	C₄	1	3	yes	complemented
8	#8813	—	5	1	no	maximal, Sylow
9	C₂×C₄⋊AGL(1,𝔽₅)	—	2	12	no	—
10	C₂×C₈×D₅	D₄	1	6	yes	—
11	C₄²×D₅	C₂³	1	1	yes	characteristic
12	#3993	—	4	24	no	—
13	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
14	C₈×Dic₅	Q₈	1	6	yes	—
15	C₄×#168	C₂×C₄	1	3	yes	—
16	#1844	—	5	6	no	—
17	C₂×C₈²	—	5	1	no	—
18	C₂²×AGL(1,𝔽₅)	—	4	4	no	—
19	C₂×C₄×D₅	C₂×D₄	1	3	yes	—
20	C₄⋊AGL(1,𝔽₅)	—	4	24	no	—
21	C₂×C₅×C₈	C₄⋊C₄	1	6	yes	—
22	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
23	C₄×Dic₅	C₂×Q₈	1	3	yes	—
24	C₂×#168	C₄⋊C₄	1	6	yes	—
25	C₈×D₅	SD₁₆	1	24	yes	—
26	C₂×#106	—	10	12	no	—
27	C₂×C₄×C₈	—	5	3	no	—
28	#384	—	5	4	no	—
29	C₈²	—	5	3	no	—
30	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
31	C₂²×D₅	#127	1	1	yes	characteristic
32	C₂×AGL(1,𝔽₅)	—	8	12	no	—
33	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
34	C₂×Dic₅	C₄⋊Q₈	1	3	yes	—
35	C₄×D₅	C₂×SD₁₆	1	12	yes	—
36	C₅×C₈	#106	1	12	yes	complemented
37	#168	#106	1	12	yes	complemented
38	C₂²×C₈	—	5	6	no	—
39	C₂×C₄⋊C₄	—	10	12	no	—
40	C₂×C₄²	—	5	1	no	—
41	#106	—	20	24	no	—
42	C₄×C₈	—	5	6	no	—
43	C₄×C₈	—	5	3	no	—
44	C₄×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
45	C₂²×C₅	#384	1	1	yes	characteristic
46	C₂×D₅	#492	1	6	yes	—
47	AGL(1,𝔽₅)	—	16	4	no	—
48	C₄×C₅	C₂×#106	1	6	yes	—
49	Dic₅	#498	1	6	yes	—
50	C₂²×C₄	—	5	3	no	—
51	C₂²×C₄	—	20	4	no	—
52	C₂×C₈	—	5	6	no	—
53	C₂×C₈	—	5	24	no	—
54	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	6	yes	—
55	C₄⋊C₄	—	20	24	no	—
56	C₄²	—	5	3	no	—
57	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂×C₅	#1844	1	3	yes	—
59	D₅	#1705	1	4	yes	—
60	C₂³	—	5	1	no	—
61	C₂×C₄	—	5	3	no	—
62	C₂×C₄	—	5	12	no	—
63	C₂×C₄	—	40	12	no	—
64	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
65	C₈	—	5	12	no	—
66	C₈	#3993	1	12	yes	complemented
67	C₅	#8813	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	5	6	no	—
69	C₂²	#65688	1	1	yes	characteristic
70	C₄	—	5	6	no	—
71	C₄	—	80	4	no	—
72	C₄	C₂×#3993	1	6	yes	—
73	C₂	—	5	4	no	—
74	C₂	#10603847	1	3	yes	—

	subgroup	G/H
center	C₂²	#65688
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₈²	C₄
socle	C₂²×C₅	#384
prime radical	C₂²×D₅	#127
cosocle	C₄²×D₅	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#8813
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
6. C₅×C₈²	C₄	semidirect
7. C₈×#168	C₄	semidirect
30. C₈²	AGL(1,𝔽₅)	semidirect
36. C₅×C₈	#106	semidirect
37. #168	#106	semidirect
66. C₈	#3993	semidirect
67. C₅	#8813	semidirect

Maximal subgroups

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

subgroup	index
#10603847	2
C₈²×D₅	2
#8813	5