Subgroups — #13018809

Order 1280.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10603848	C₂	1	6	yes	maximal
2	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#3994	—	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	#8818	—	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	#3994	—	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	#1845	—	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	D₈	1	12	yes	—
26	C₈×D₅	Q₁₆	1	12	yes	—
27	C₂×#107	—	10	12	no	—
28	C₂×C₄×C₈	—	5	3	no	—
29	#384	—	5	4	no	—
30	C₈²	—	5	3	no	—
31	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
32	C₂²×D₅	#127	1	1	yes	characteristic
33	C₂×AGL(1,𝔽₅)	—	8	12	no	—
34	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
35	C₂×Dic₅	C₄⋊Q₈	1	3	yes	—
36	C₄×D₅	C₂×D₈	1	6	yes	—
37	C₄×D₅	C₂×Q₁₆	1	6	yes	—
38	C₅×C₈	#107	1	12	yes	complemented
39	#168	#107	1	12	yes	complemented
40	C₂²×C₈	—	5	6	no	—
41	C₂×C₄⋊C₄	—	10	12	no	—
42	C₂×C₄²	—	5	1	no	—
43	#107	—	20	24	no	—
44	C₄×C₈	—	5	6	no	—
45	C₄×C₈	—	5	3	no	—
46	C₄×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
47	C₂²×C₅	#384	1	1	yes	characteristic
48	C₂×D₅	#493	1	3	yes	—
49	C₂×D₅	#494	1	3	yes	—
50	AGL(1,𝔽₅)	—	16	4	no	—
51	C₄×C₅	C₂×#107	1	6	yes	—
52	Dic₅	#500	1	6	yes	—
53	C₂²×C₄	—	5	3	no	—
54	C₂²×C₄	—	20	4	no	—
55	C₂×C₈	—	5	6	no	—
56	C₂×C₈	—	5	12	no	—
57	C₂×C₈	—	5	12	no	—
58	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	6	yes	—
59	C₄⋊C₄	—	20	24	no	—
60	C₄²	—	5	3	no	—
61	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₂×C₅	#1845	1	3	yes	—
63	D₅	#1712	1	1	yes	characteristic
64	D₅	#1713	1	3	yes	—
65	C₂³	—	5	1	no	—
66	C₂×C₄	—	5	3	no	—
67	C₂×C₄	—	5	6	no	—
68	C₂×C₄	—	5	6	no	—
69	C₂×C₄	—	40	12	no	—
70	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
71	C₈	—	5	12	no	—
72	C₈	#3994	1	12	yes	complemented
73	C₅	#8818	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	5	3	no	—
75	C₂²	—	5	3	no	—
76	C₂²	#65688	1	1	yes	characteristic
77	C₄	—	5	6	no	—
78	C₄	—	80	4	no	—
79	C₄	C₂×#3994	1	6	yes	—
80	C₂	—	5	1	no	—
81	C₂	—	5	3	no	—
82	C₂	#10603848	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	#8818
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
31. C₈²	AGL(1,𝔽₅)	semidirect
38. C₅×C₈	#107	semidirect
39. #168	#107	semidirect
72. C₈	#3994	semidirect
73. C₅	#8818	semidirect

Maximal subgroups

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

subgroup	index
#10603848	2
C₈²×D₅	2
#8818	5