Subgroups — #11986970

Order 1280.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1398	C₂	1	1	yes	maximal, characteristic
2	#10598005	C₂	1	1	yes	maximal, characteristic
3	#10598004	C₂	1	1	yes	maximal, characteristic
4	C₅×M₆₄	C₄	1	1	yes	characteristic
5	C₅×M₆₄	C₄	1	1	yes	characteristic
6	C₅×#346	C₂²	1	1	yes	characteristic
7	#64683	—	2	1	no	—
8	#64677	—	2	1	no	—
9	#7457	—	5	1	no	maximal, Sylow
10	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅×C₁₆	Q₈	1	1	yes	characteristic
12	C₄×C₅×C₈	D₄	1	1	yes	characteristic
13	#3944	—	4	1	no	—
14	C₄⋊#168	—	4	1	no	—
15	C₅×C₃₂	—	2	1	no	—
16	C₅×C₃₂	C₈	1	2	yes	complemented
17	#1398	D₅	1	1	yes	characteristic
18	#1369	—	5	1	no	—
19	#1368	—	5	1	no	—
20	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
22	C₄²×C₅	SD₁₆	1	1	yes	characteristic
23	C₂×#168	—	8	1	no	—
24	C₅×C₁₆	—	2	1	no	—
25	C₅×C₁₆	C₂×C₈	1	1	yes	characteristic
26	C₅×C₁₆	M₁₆	1	1	yes	characteristic
27	#672	—	8	1	no	—
28	M₆₄	Dic₅	1	1	yes	characteristic
29	M₆₄	Dic₅	1	1	yes	characteristic
30	#364	—	10	1	no	—
31	#346	C₂×D₅	1	1	yes	characteristic
32	#334	—	10	1	no	—
33	C₂×C₄×C₅	—	2	1	no	—
34	C₂×C₄×C₅	#106	1	1	yes	characteristic
35	C₅×C₈	—	2	1	no	—
36	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
37	C₅×C₈	#108	1	1	yes	characteristic
38	#168	—	16	1	no	—
39	M₃₂	—	20	1	no	—
40	C₂×C₁₆	Dic₁₀	1	1	yes	characteristic
41	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
42	C₄⋊C₈	—	20	1	no	—
43	C₄×C₈	D₂₀	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	#168	1	2	yes	complemented
46	C₂²×C₅	#365	1	1	yes	characteristic
47	C₄×C₅	—	4	1	no	—
48	C₄×C₅	#334	1	1	yes	characteristic
49	C₄×C₅	#365	1	1	yes	characteristic
50	C₂×C₈	—	40	1	no	—
51	C₂×C₈	#677	1	1	yes	characteristic
52	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
53	C₄²	#677	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	40	1	no	—
56	C₁₆	C₂×#168	1	1	yes	characteristic
57	C₁₆	#681	1	1	yes	characteristic
58	C₂×C₅	—	2	1	no	—
59	C₂×C₅	#1368	1	1	yes	characteristic
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	#3949	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	—	80	1	no	—
64	C₈	C₄⋊#168	1	1	yes	characteristic
65	C₈	#3951	1	1	yes	characteristic
66	C₅	#7457	1	1	yes	characteristic, complemented, Sylow
67	C₂²	#64733	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#64677	1	1	yes	characteristic
70	C₄	#64733	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#10598004	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10598004
commutator	C₅×C₁₆	C₂×C₈
Frattini	#346	C₂×D₅
Fitting	C₅×#1398	C₂
socle	C₂×C₅	#1368
prime radical	C₂²×C₅	#365
cosocle	C₅×#346	C₂²
hypercenter	#1398	D₅
hypocenter	C₅	#7457
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #1398

Complements

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

N	H ≅ G/N	type
16. C₅×C₃₂	C₈	semidirect
45. C₃₂	#168	semidirect
66. C₅	#7457	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1398	2
#10598005	2
#10598004	2
#7457	5