Subgroups — #11987195

Order 1280.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₆₄	C₂	1	1	yes	maximal, characteristic
2	Dic₅×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#64663	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₃₂	C₂²	1	1	yes	characteristic
5	C₁₆×Dic₅	C₄	1	1	yes	characteristic
6	C₂×#3926	C₄	1	1	yes	characteristic
7	C₅×C₆₄	—	2	1	no	—
8	#64663	—	2	1	no	—
9	C₄⋊C₆₄	—	5	1	no	maximal, Sylow
10	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
11	C₈×Dic₅	C₈	1	1	yes	characteristic
12	C₂×#672	C₈	1	1	yes	characteristic
13	C₅×C₃₂	D₄	1	1	yes	characteristic
14	C₅×C₃₂	Q₈	1	1	yes	characteristic
15	#3926	—	2	1	no	—
16	C₂×C₆₄	—	5	1	no	—
17	C₂×C₆₄	D₅	1	1	yes	characteristic
18	C₄×C₃₂	—	5	1	no	—
19	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
20	C₄×Dic₅	C₁₆	1	1	yes	characteristic
21	C₂×#168	C₁₆	1	1	yes	characteristic
22	C₅×C₁₆	C₄⋊C₄	1	1	yes	characteristic
23	C₅×C₁₆	M₁₆	1	1	yes	characteristic
24	#672	—	2	1	no	—
25	C₂×C₃₂	—	5	1	no	—
26	C₂×C₃₂	C₂×D₅	1	1	yes	characteristic
27	C₄×C₁₆	—	5	1	no	—
28	C₆₄	—	2	1	no	—
29	C₆₄	—	10	1	no	—
30	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
31	C₂×Dic₅	C₃₂	1	1	yes	characteristic
32	C₂×Dic₅	C₃₂	1	1	yes	characteristic
33	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
34	C₅×C₈	M₃₂	1	1	yes	characteristic
35	#168	—	2	1	no	—
36	C₂×C₁₆	—	5	1	no	—
37	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
38	C₄×C₈	—	5	1	no	—
39	C₃₂	—	10	1	no	—
40	C₃₂	Dic₁₀	1	1	yes	characteristic
41	C₃₂	C₂²⋊D₅	1	1	yes	characteristic
42	C₂²×C₅	C₂×C₃₂	1	1	yes	characteristic
43	C₄×C₅	C₄⋊C₁₆	1	1	yes	characteristic
44	C₄×C₅	M₆₄	1	1	yes	characteristic
45	Dic₅	—	2	1	no	—
46	Dic₅	C₆₄	1	2	yes	complemented
47	C₂×C₈	—	5	1	no	—
48	C₂×C₈	C₈×D₅	1	1	yes	characteristic
49	C₄²	—	5	1	no	—
50	C₁₆	—	10	1	no	—
51	C₁₆	#676	1	1	yes	characteristic
52	C₁₆	#683	1	1	yes	characteristic
53	C₂×C₅	C₄⋊C₃₂	1	1	yes	characteristic
54	C₂×C₅	C₂×C₆₄	1	1	yes	characteristic
55	C₂×C₅	M₁₂₈	1	1	yes	characteristic
56	C₂×C₄	—	5	1	no	—
57	C₂×C₄	—	5	1	no	—
58	C₂×C₄	D₅×C₁₆	1	1	yes	characteristic
59	C₈	—	10	1	no	—
60	C₈	#3930	1	1	yes	characteristic
61	C₈	#3946	1	1	yes	characteristic
62	C₅	C₄⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
63	C₂²	D₅×C₃₂	1	1	yes	characteristic
64	C₄	—	5	2	no	—
65	C₄	—	10	1	no	—
66	C₄	#64667	1	1	yes	characteristic
67	C₄	#64721	1	1	yes	characteristic
68	C₂	D₅×C₆₄	1	1	yes	characteristic
69	C₂	#10597923	1	1	yes	characteristic
70	C₂	#10598081	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₅
commutator	C₂×C₅	C₂×C₆₄
Fitting	C₂×C₅×C₆₄	C₂
socle, prime radical	C₂²×C₅	C₂×C₃₂
cosocle	C₂×C₅×C₃₂	C₂²
hypercenter	C₂×C₆₄	D₅
hypocenter	C₅	C₄⋊C₆₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
46. Dic₅	C₆₄	semidirect
62. C₅	C₄⋊C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₆₄	2
Dic₅×C₃₂	2
C₂×#64663	2
C₄⋊C₆₄	5