Subgroups — #12014044

Order 1280.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10599839	C₂	1	4	yes	maximal
2	#10598858	C₂	1	2	yes	maximal
3	C₅×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#64988	—	2	2	no	—
5	#64971	—	2	2	no	—
6	#64971	C₂²	1	4	yes	—
7	C₄×C₅×C₁₆	C₂²	1	2	yes	—
8	C₅×C₈²	C₂²	1	1	yes	characteristic
9	#64725	—	2	8	no	—
10	#12523	—	5	1	no	maximal, Sylow
11	C₂×Dic₂₀	—	4	4	no	—
12	#4015	—	2	4	no	—
13	C₂×C₅×C₁₆	D₄	1	4	yes	—
14	C₄×C₅×C₈	D₄	1	2	yes	—
15	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
16	#3950	—	2	8	no	—
17	#3950	—	4	4	no	—
18	#2254	—	5	4	no	—
19	#1713	—	5	2	no	—
20	C₈×C₁₆	D₅	1	1	yes	characteristic
21	C₂×Dic₁₀	—	4	4	no	—
22	C₂×C₅×C₈	D₈	1	4	yes	—
23	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
24	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
25	C₄⋊Dic₅	—	4	4	no	—
26	C₄⋊Dic₅	—	4	4	no	—
27	Dic₂₀	—	8	8	no	—
28	C₅×C₁₆	Q₁₆	1	8	yes	complemented
29	#500	—	5	4	no	—
30	#500	—	10	2	no	—
31	#494	—	10	2	no	—
32	#367	—	10	8	no	—
33	C₄×C₁₆	C₂×D₅	1	2	yes	—
34	C₈²	C₂×D₅	1	1	yes	characteristic
35	C₂×C₄×C₅	#127	1	1	yes	characteristic
36	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
37	C₂×Dic₅	—	8	4	no	—
38	Dic₁₀	—	8	8	no	—
39	C₅×C₈	SD₃₂	1	8	yes	—
40	C₅×C₈	C₂×Q₁₆	1	4	yes	—
41	C₂×Q₁₆	—	20	4	no	—
42	C₄⋊Q₈	—	10	4	no	—
43	C₂×C₁₆	D₂₀	1	4	yes	—
44	#107	—	10	8	no	—
45	#107	—	20	4	no	—
46	C₄×C₈	D₂₀	1	2	yes	—
47	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
48	C₂²×C₅	#493	1	1	yes	characteristic
49	C₄×C₅	#494	1	2	yes	—
50	C₄×C₅	C₂×SD₃₂	1	4	yes	—
51	Dic₅	—	16	4	no	—
52	C₂×Q₈	—	20	4	no	—
53	Q₁₆	—	40	8	no	—
54	C₂×C₈	D₄₀	1	4	yes	—
55	C₂×C₈	C₂×D₂₀	1	2	yes	—
56	C₄⋊C₄	—	20	4	no	—
57	C₄⋊C₄	—	20	4	no	—
58	C₄²	C₂×D₂₀	1	1	yes	characteristic
59	C₁₆	Dic₂₀	1	8	yes	complemented
60	C₂×C₅	#1713	1	1	yes	characteristic
61	C₂×C₅	#2248	1	2	yes	—
62	Q₈	—	40	8	no	—
63	C₂×C₄	—	40	4	no	—
64	C₂×C₄	#4020	1	1	yes	characteristic
65	C₂×C₄	C₂×D₄₀	1	2	yes	—
66	C₈	#3932	1	8	yes	—
67	C₈	C₂×Dic₂₀	1	4	yes	—
68	C₅	#12523	1	1	yes	characteristic, complemented, Sylow
69	C₂²	#64984	1	1	yes	characteristic
70	C₄	—	80	4	no	—
71	C₄	#64988	1	2	yes	—
72	C₄	C₂×#3932	1	4	yes	—
73	C₂	#10598858	1	1	yes	characteristic
74	C₂	#10599854	1	2	yes	—

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×C₈×C₁₆	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	C₈×C₁₆	D₅
hypocenter	C₅	#12523
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
28. C₅×C₁₆	Q₁₆	semidirect
59. C₁₆	Dic₂₀	semidirect
68. C₅	#12523	semidirect

Maximal subgroups

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

subgroup	index
#10599839	2
#10598858	2
C₅×C₈×C₁₆	2
#12523	5