Subgroups — #44548559

Order 32128.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀₀₈	C₂	1	1	yes	maximal, characteristic
2	#30631517	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₀₄	—	2	1	no	—
5	Dic₂₀₀₈	—	2	1	no	—
6	Dic₂₀₀₈	C₄	1	2	yes	—
7	#22548651	—	2	1	no	—
8	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
9	C₃₂×C₂₅₁	—	2	1	no	—
10	C₂×Dic₅₀₂	—	4	1	no	—
11	Dic₁₀₀₄	—	2	2	no	—
12	Dic₁₀₀₄	—	4	1	no	—
13	#16860794	—	4	1	no	—
14	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
15	C₁₆×C₂₅₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₅₁	D₄	1	1	yes	characteristic
17	Dic₅₀₂	—	4	2	no	—
18	Dic₅₀₂	—	8	1	no	—
19	C₂×Dic₂₅₁	—	8	1	no	—
20	C₂₅₁⋊C₈	—	8	1	no	—
21	C₂×C₄×C₂₅₁	D₈	1	1	yes	characteristic
22	C₈×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₅₁	SD₃₂	1	1	yes	characteristic
25	C₄×C₂₅₁	#102	1	1	yes	characteristic
26	C₄×C₂₅₁	D₁₆	1	1	yes	characteristic
27	Dic₂₅₁	—	8	2	no	—
28	C₂×C₂₅₁	—	2	1	no	—
29	C₂×C₂₅₁	#357	1	1	yes	characteristic
30	C₂₅₁	#1420	1	1	yes	characteristic, complemented, Sylow
31	#1420	—	251	1	no	maximal, Sylow
32	C₂×Q₃₂	—	251	1	no	—
33	M₆₄	D₂₅₁	1	1	yes	characteristic
34	#368	—	251	1	no	—
35	C₂×Q₁₆	—	502	1	no	—
36	Q₃₂	—	502	1	no	—
37	Q₃₂	—	251	2	no	—
38	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
39	#108	—	502	1	no	—
40	C₃₂	—	2	1	no	—
41	C₂×Q₈	—	1004	1	no	—
42	Q₁₆	—	1004	1	no	—
43	Q₁₆	—	502	2	no	—
44	M₁₆	—	1004	1	no	—
45	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
46	C₁₆	C₄×D₂₅₁	1	1	yes	characteristic
47	C₁₆	C₂²⋊D₂₅₁	1	1	yes	characteristic
48	Q₈	—	2008	1	no	—
49	Q₈	—	1004	2	no	—
50	C₂×C₄	—	2008	1	no	—
51	C₂×C₄	D₂₀₀₈	1	1	yes	characteristic
52	C₈	—	2008	1	no	—
53	C₈	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
54	C₈	#16860799	1	1	yes	characteristic
55	C₂²	#22548660	1	1	yes	characteristic
56	C₄	—	2008	2	no	—
57	C₄	#22548637	1	1	yes	characteristic
58	C₄	D₄₀₁₆	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#30631489	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30631489
commutator	C₁₆×C₂₅₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₅₁
Fitting	M₆₄×C₂₅₁	C₂
socle	C₂×C₂₅₁	#357
prime radical	C₂²×C₂₅₁	SD₃₂
cosocle	C₂×C₁₆×C₂₅₁	C₂²
hypercenter	M₆₄	D₂₅₁
hypocenter	C₂₅₁	#1420
LCS	G ▷ C₁₆×C₂₅₁ ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
30. C₂₅₁	#1420	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀₀₈	2
#30631517	2
M₆₄×C₂₅₁	2
#1420	251