Subgroups — #44548287

Order 32128.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631463	C₂	1	1	yes	maximal, characteristic
2	#30631408	C₂	1	1	yes	maximal, characteristic
3	#346×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₅₁⋊C₁₆	C₄	1	2	yes	—
5	C₄×C₂₅₁⋊C₈	C₄	1	2	yes	—
6	C₂×C₁₆×C₂₅₁	C₄	1	2	yes	—
7	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
8	C₂×C₂₅₁⋊C₈	—	2	2	no	—
9	C₂₅₁⋊C₁₆	C₈	1	4	yes	complemented
10	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₂₅₁	—	2	2	no	—
14	C₂₅₁⋊C₈	—	4	2	no	—
15	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₂₅₁	M₁₆	1	2	yes	—
17	C₈×C₂₅₁	C₂×C₈	1	2	yes	—
18	C₈×C₂₅₁	M₁₆	1	2	yes	—
19	C₂²×C₂₅₁	#97	1	1	yes	characteristic
20	C₄×C₂₅₁	—	2	2	no	—
21	C₄×C₂₅₁	C₄×C₈	1	1	yes	characteristic
22	C₄×C₂₅₁	#97	1	1	yes	characteristic
23	C₂×C₂₅₁	#322	1	1	yes	characteristic
24	C₂×C₂₅₁	#347	1	2	yes	—
25	C₂₅₁	#1313	1	1	yes	characteristic, complemented, Sylow
26	#1313	—	251	1	no	maximal, Sylow
27	#346	—	251	1	no	—
28	#346	D₂₅₁	1	1	yes	characteristic
29	#322	—	251	1	no	—
30	C₂×C₁₆	—	251	2	no	—
31	C₂×C₁₆	Dic₂₅₁	1	2	yes	—
32	C₄×C₈	—	251	2	no	—
33	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
34	C₂×C₈	—	502	2	no	—
35	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₂₅₁	1	1	yes	characteristic
37	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
38	C₁₆	—	2	2	no	—
39	C₁₆	—	251	4	no	—
40	C₂×C₄	C₄×Dic₂₅₁	1	1	yes	characteristic
41	C₂×C₄	#16860793	1	2	yes	—
42	C₈	—	1004	2	no	—
43	C₈	C₈×D₂₅₁	1	2	yes	—
44	C₈	#16860794	1	2	yes	—
45	C₂²	#22548625	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₈×Dic₂₅₁	1	1	yes	characteristic
48	C₄	#22548626	1	1	yes	characteristic
49	C₂	#30631409	1	1	yes	characteristic
50	C₂	#30631464	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₂₅₁
commutator	C₄×C₂₅₁	C₄×C₈
Frattini	C₄×C₈	C₂×D₂₅₁
Fitting	#346×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#97
cosocle	C₄×C₈×C₂₅₁	C₂²
hypercenter	#346	D₂₅₁
hypocenter	C₂₅₁	#1313
LCS	G ▷ C₄×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #346

Complements

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

N	H ≅ G/N	type
9. C₂₅₁⋊C₁₆	C₈	semidirect
25. C₂₅₁	#1313	semidirect

Maximal subgroups

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

subgroup	index
#30631463	2
#30631408	2
#346×C₂₅₁	2
#1313	251