Subgroups — #44548564

Order 32128.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₅₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#30631513	C₂	1	1	yes	maximal, characteristic
3	#366×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#22548649	—	2	1	no	—
5	C₂₅₁⋊C₃₂	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
7	#107×C₂₅₁	—	2	1	no	—
8	C₄⋊Dic₂₅₁	—	4	1	no	—
9	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
10	C₄⋊C₄×C₂₅₁	—	4	1	no	—
11	C₁₆×C₂₅₁	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₂₅₁	Q₈	1	1	yes	characteristic
13	C₂×Dic₂₅₁	—	8	1	no	—
14	C₂×C₄×C₂₅₁	—	8	1	no	—
15	C₂×C₄×C₂₅₁	D₈	1	1	yes	characteristic
16	C₈×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₂₅₁	Q₁₆	1	1	yes	characteristic
18	C₂²×C₂₅₁	D₁₆	1	1	yes	characteristic
19	C₄×C₂₅₁	—	16	1	no	—
20	C₄×C₂₅₁	#107	1	1	yes	characteristic
21	C₄×C₂₅₁	Q₃₂	1	1	yes	characteristic
22	Dic₂₅₁	—	16	1	no	—
23	C₂×C₂₅₁	#366	1	1	yes	characteristic
24	C₂×C₂₅₁	D₃₂	1	1	yes	characteristic
25	C₂×C₂₅₁	Q₆₄	1	1	yes	characteristic
26	C₂₅₁	#1423	1	1	yes	characteristic, complemented, Sylow
27	#1423	—	251	1	no	maximal, Sylow
28	C₂×C₃₂	—	251	1	no	—
29	#366	—	251	1	no	—
30	#366	D₂₅₁	1	1	yes	characteristic
31	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	#107	—	502	1	no	—
34	C₃₂	—	251	2	no	—
35	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
36	C₄⋊C₄	—	4	1	no	—
37	C₄⋊C₄	—	1004	1	no	—
38	C₁₆	C₄×D₂₅₁	1	1	yes	characteristic
39	C₁₆	Dic₅₀₂	1	1	yes	characteristic
40	C₂×C₄	—	8	1	no	—
41	C₂×C₄	—	2008	1	no	—
42	C₂×C₄	#16860795	1	1	yes	characteristic
43	C₈	Dic₂₅₁⋊C₄	1	1	yes	characteristic
44	C₈	#16860800	1	1	yes	characteristic
45	C₂²	#22548656	1	1	yes	characteristic
46	C₄	—	16	1	no	—
47	C₄	—	4016	1	no	—
48	C₄	#22548648	1	1	yes	characteristic
49	C₄	#22548661	1	1	yes	characteristic
50	C₂	#30631512	1	1	yes	characteristic
51	C₂	#30631522	1	1	yes	characteristic
52	C₂	#30631527	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22548656
commutator	C₁₆×C₂₅₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₅₁
Fitting	#366×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	D₁₆
cosocle	C₂×C₁₆×C₂₅₁	C₂²
hypercenter	#366	D₂₅₁
hypocenter	C₂₅₁	#1423
LCS	G ▷ C₁₆×C₂₅₁ ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
5. C₂₅₁⋊C₃₂	C₄	semidirect
26. C₂₅₁	#1423	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₅₁⋊C₃₂	2
#30631513	2
#366×C₂₅₁	2
#1423	251