Subgroups — #13018808

Order 1280.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10603845	C₂	1	6	yes	maximal
2	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
3	#65688	C₂²	1	4	yes	—
4	C₄×C₈×D₅	C₂²	1	3	yes	—
5	#64893	—	2	12	no	—
6	C₅×C₈²	C₄	1	1	yes	characteristic, complemented
7	C₈×#168	C₄	1	3	yes	complemented
8	#8817	—	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	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
13	C₈×Dic₅	Q₈	1	6	yes	—
14	C₄×#168	C₂×C₄	1	3	yes	—
15	#1834	—	5	6	no	—
16	C₂×C₈²	—	5	1	no	—
17	C₂²×AGL(1,𝔽₅)	—	4	4	no	—
18	C₂×C₄×D₅	C₂×D₄	1	3	yes	—
19	C₄⋊AGL(1,𝔽₅)	—	4	24	no	—
20	C₂×C₅×C₈	C₄⋊C₄	1	6	yes	—
21	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
22	C₄×Dic₅	C₂×Q₈	1	3	yes	—
23	C₂×#168	C₄⋊C₄	1	6	yes	—
24	C₈×D₅	—	2	12	no	—
25	C₂×C₄×C₈	—	5	3	no	—
26	#384	—	5	4	no	—
27	#340	—	10	12	no	—
28	C₈²	—	5	3	no	—
29	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
30	C₂²×D₅	#127	1	1	yes	characteristic
31	C₂×AGL(1,𝔽₅)	—	8	12	no	—
32	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
33	C₂×Dic₅	C₄⋊Q₈	1	3	yes	—
34	C₄×D₅	#135	1	12	yes	—
35	C₅×C₈	—	2	6	no	—
36	#168	—	2	6	no	—
37	C₂²×C₈	—	5	6	no	—
38	C₂×C₄⋊C₄	—	10	12	no	—
39	C₂×C₄²	—	5	1	no	—
40	C₄×C₈	—	5	6	no	—
41	C₄×C₈	—	5	3	no	—
42	C₄×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
43	C₂²×C₅	#384	1	1	yes	characteristic
44	C₂×D₅	#495	1	6	yes	—
45	AGL(1,𝔽₅)	—	16	4	no	—
46	C₄×C₅	#427	1	6	yes	—
47	Dic₅	#499	1	6	yes	—
48	C₂²×C₄	—	5	3	no	—
49	C₂²×C₄	—	20	4	no	—
50	C₂×C₈	—	5	6	no	—
51	C₂×C₈	—	10	12	no	—
52	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	6	yes	—
53	C₄⋊C₄	—	20	24	no	—
54	C₄²	—	5	3	no	—
55	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂×C₅	#1846	1	3	yes	—
57	D₅	#1711	1	4	yes	—
58	C₂³	—	5	1	no	—
59	C₂×C₄	—	5	3	no	—
60	C₂×C₄	—	5	12	no	—
61	C₂×C₄	—	40	12	no	—
62	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
63	C₈	—	2	6	no	—
64	C₈	—	10	6	no	—
65	C₅	#8817	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	5	6	no	—
67	C₂²	#65688	1	1	yes	characteristic
68	C₄	—	5	6	no	—
69	C₄	—	80	4	no	—
70	C₄	#65721	1	6	yes	—
71	C₂	—	5	4	no	—
72	C₂	#10603849	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	#8817
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
29. C₈²	AGL(1,𝔽₅)	semidirect
65. C₅	#8817	semidirect

Maximal subgroups

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

subgroup	index
#10603845	2
C₈²×D₅	2
#8817	5