Subgroups — #11986975

Order 1280.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#10598001	C₂	1	2	yes	maximal
3	C₄×C₅×C₁₆	C₄	1	1	yes	characteristic
4	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
5	C₅×C₈²	C₄	1	1	yes	characteristic
6	#64719	—	2	2	no	—
7	#7459	—	5	1	no	maximal, Sylow
8	C₂×C₅×C₁₆	C₈	1	1	yes	characteristic
9	C₂×C₅×C₁₆	C₈	1	1	yes	characteristic
10	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
11	C₂×C₅×C₁₆	Q₈	1	1	yes	characteristic
12	C₄×C₅×C₈	—	2	1	no	—
13	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	#3926	—	4	2	no	—
15	#1422	—	5	2	no	—
16	C₈×C₁₆	D₅	1	1	yes	characteristic
17	C₂×C₅×C₈	—	2	2	no	—
18	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
20	C₄²×C₅	M₁₆	1	1	yes	characteristic
21	C₅×C₁₆	—	2	1	no	—
22	C₅×C₁₆	—	2	1	no	—
23	C₅×C₁₆	C₁₆	1	2	yes	—
24	C₅×C₁₆	D₈	1	1	yes	characteristic
25	C₅×C₁₆	Q₁₆	1	1	yes	characteristic
26	M₆₄	—	10	2	no	—
27	C₄×C₁₆	Dic₅	1	1	yes	characteristic
28	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
29	C₈²	Dic₅	1	1	yes	characteristic
30	C₂×C₄×C₅	—	2	1	no	—
31	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
32	C₅×C₈	—	2	4	no	—
33	C₅×C₈	#107	1	1	yes	characteristic
34	C₅×C₈	#108	1	1	yes	characteristic
35	C₅×C₈	C₂×C₁₆	1	1	yes	characteristic
36	C₅×C₈	M₃₂	1	1	yes	characteristic
37	C₂×C₁₆	#168	1	1	yes	characteristic
38	C₂×C₁₆	#168	1	1	yes	characteristic
39	C₂×C₁₆	Dic₁₀	1	1	yes	characteristic
40	C₂×C₁₆	D₂₀	1	1	yes	characteristic
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
43	C₃₂	—	20	2	no	—
44	C₂²×C₅	#364	1	1	yes	characteristic
45	C₄×C₅	—	2	2	no	—
46	C₄×C₅	#335	1	1	yes	characteristic
47	C₄×C₅	C₄⋊C₁₆	1	1	yes	characteristic
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	C₂×#168	1	1	yes	characteristic
50	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
51	C₄²	#681	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	—	2	1	no	—
54	C₁₆	#672	1	2	yes	—
55	C₁₆	D₄₀	1	1	yes	characteristic
56	C₁₆	Dic₂₀	1	1	yes	characteristic
57	C₂×C₅	—	2	1	no	—
58	C₂×C₅	#1370	1	1	yes	characteristic
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
61	C₈	—	2	4	no	—
62	C₈	C₂×#672	1	1	yes	characteristic
63	C₈	#3944	1	1	yes	characteristic
64	C₈	#3950	1	1	yes	characteristic
65	C₈	#3951	1	1	yes	characteristic
66	C₅	#7459	1	1	yes	characteristic, complemented, Sylow
67	C₂²	#64683	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	#64678	1	1	yes	characteristic
70	C₄	C₄⋊#672	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#10597931	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₄₀
commutator	C₅×C₈	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₅
Fitting	C₅×C₈×C₁₆	C₂
socle	C₂×C₅	#1370
prime radical	C₂²×C₅	#364
cosocle	C₄×C₅×C₁₆	C₂²
hypercenter	C₈×C₁₆	D₅
hypocenter	C₅	#7459
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
66. C₅	#7459	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₈×C₁₆	2
#10598001	2
#7459	5