Subgroups — #44550519

Order 32128.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₂₅₁⋊C₈)	C₂	1	6	yes	maximal
2	#376×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₅₁⋊C₈	C₂²	1	4	yes	—
4	C₄⋊(C₂₅₁⋊C₈)	—	2	12	no	—
5	C₂×C₄²×C₂₅₁	C₂²	1	3	yes	—
6	#95×C₂₅₁	C₄	1	4	yes	—
7	C₂×C₂₅₁⋊C₈	—	2	12	no	—
8	C₂²×C₄×C₂₅₁	C₂×C₄	1	6	yes	—
9	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
10	C₄²×C₂₅₁	D₄	1	6	yes	—
11	C₄²×C₂₅₁	Q₈	1	6	yes	—
12	C₂₅₁⋊C₈	—	4	4	no	—
13	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₅₁	—	2	12	no	—
15	C₂×C₄×C₂₅₁	C₄⋊C₄	1	12	yes	—
16	C₂×C₄×C₂₅₁	C₂×D₄	1	3	yes	—
17	C₂×C₄×C₂₅₁	C₂×Q₈	1	3	yes	—
18	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	3	yes	—
19	C₂²×C₂₅₁	#131	1	4	yes	—
20	C₄×C₂₅₁	—	2	12	no	—
21	C₄×C₂₅₁	#127	1	1	yes	characteristic
22	C₄×C₂₅₁	C₄⋊Q₈	1	3	yes	—
23	C₂×C₂₅₁	#384	1	1	yes	characteristic
24	C₂×C₂₅₁	#424	1	6	yes	—
25	C₂₅₁	#1843	1	1	yes	characteristic, complemented, Sylow
26	#1843	—	251	1	no	maximal, Sylow
27	C₂×C₄⋊C₈	—	251	6	no	—
28	#376	D₂₅₁	1	1	yes	characteristic
29	C₂²×C₈	—	251	4	no	—
30	C₂×C₄²	C₂×D₂₅₁	1	3	yes	—
31	C₄⋊C₈	—	502	12	no	—
32	#95	Dic₂₅₁	1	4	yes	—
33	C₂²×C₄	C₂×Dic₂₅₁	1	6	yes	—
34	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
35	C₂×C₈	—	502	12	no	—
36	C₄²	D₁₀₀₄	1	6	yes	—
37	C₄²	Dic₅₀₂	1	6	yes	—
38	C₂³	C₂²×Dic₂₅₁	1	1	yes	characteristic
39	C₂×C₄	—	2	12	no	—
40	C₂×C₄	C₄⋊Dic₂₅₁	1	12	yes	—
41	C₂×C₄	C₂×D₁₀₀₄	1	3	yes	—
42	C₂×C₄	C₂×Dic₅₀₂	1	3	yes	—
43	C₈	—	1004	4	no	—
44	C₂²	C₂×C₄⋊Dic₂₅₁	1	3	yes	—
45	C₂²	#22548722	1	4	yes	—
46	C₄	—	2	12	no	—
47	C₄	#22548712	1	1	yes	characteristic
48	C₄	#22548715	1	3	yes	—
49	C₂	#30631557	1	1	yes	characteristic
50	C₂	#30631714	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₅₁
commutator, socle, prime radical	C₂³×C₂₅₁	C₂²×C₄
Fitting	#376×C₂₅₁	C₂
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#376	D₂₅₁
hypocenter	C₂₅₁	#1843
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
25. C₂₅₁	#1843	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₂₅₁⋊C₈)	2
#376×C₂₅₁	2
#1843	251