Subgroups — #44552643

Order 32128.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632088	C₂	1	2	yes	maximal
2	#30631515	C₂	1	1	yes	maximal, characteristic
3	#30631512	C₂	1	2	yes	maximal
4	C₁₆×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
5	#367×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	#22548706	—	2	2	no	—
7	C₂×C₂₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#22548649	C₂²	1	2	yes	—
9	#22548646	—	2	2	no	—
10	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
12	#107×C₂₅₁	C₂²	1	2	yes	—
13	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
14	C₄⋊Dic₂₅₁	—	2	2	no	—
15	C₄⋊Dic₂₅₁	—	4	2	no	—
16	Dic₂₅₁⋊C₄	—	4	2	no	—
17	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
18	C₂₅₁⋊C₁₆	Q₈	1	2	yes	—
19	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₅₁	—	2	2	no	—
21	C₁₆×C₂₅₁	Q₈	1	2	yes	—
22	C₂×Dic₂₅₁	—	4	2	no	—
23	C₂×Dic₂₅₁	D₈	1	2	yes	—
24	C₂₅₁⋊C₈	Q₁₆	1	2	yes	—
25	C₂×C₄×C₂₅₁	—	4	2	no	—
26	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
27	C₈×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
28	C₈×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
30	C₄×C₂₅₁	—	8	2	no	—
31	C₄×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
32	C₄×C₂₅₁	C₂×Q₁₆	1	1	yes	characteristic
33	Dic₂₅₁	—	2	2	no	—
34	Dic₂₅₁	—	8	2	no	—
35	C₂×C₂₅₁	#500	1	1	yes	characteristic
36	C₂×C₂₅₁	#508	1	2	yes	—
37	C₂₅₁	#2253	1	1	yes	characteristic, complemented, Sylow
38	#2253	—	251	1	no	maximal, Sylow
39	#499	—	251	2	no	—
40	#367	—	251	1	no	—
41	#367	D₂₅₁	1	1	yes	characteristic
42	#366	—	251	2	no	—
43	C₄×C₁₆	—	251	1	no	—
44	#125	—	502	2	no	—
45	C₂×C₁₆	—	251	1	no	—
46	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
47	#107	—	251	2	no	—
48	#107	C₂×D₂₅₁	1	2	yes	—
49	#106	—	502	2	no	—
50	C₄×C₈	—	251	1	no	—
51	C₂×C₈	—	251	1	no	—
52	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	1004	2	no	—
55	C₄⋊C₄	—	1004	2	no	—
56	C₄⋊C₄	—	502	2	no	—
57	C₄²	—	251	1	no	—
58	C₁₆	—	251	2	no	—
59	C₁₆	Dic₅₀₂	1	2	yes	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	1004	2	no	—
62	C₂×C₄	—	251	2	no	—
63	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
64	C₈	—	251	2	no	—
65	C₈	Q₈×D₂₅₁	1	1	yes	characteristic
66	C₈	C₂×Dic₅₀₂	1	1	yes	characteristic
67	C₂²	D₈×D₂₅₁	1	1	yes	characteristic
68	C₄	—	8	2	no	—
69	C₄	—	2008	2	no	—
70	C₄	—	502	2	no	—
71	C₄	#22548714	1	1	yes	characteristic
72	C₄	Q₁₆×D₂₅₁	1	1	yes	characteristic
73	C₂	#30632091	1	1	yes	characteristic
74	C₂	#30632118	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₂₅₁
commutator, cosocle	C₂×C₈×C₂₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#367×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	C₂×D₈
hypercenter	#367	D₂₅₁
hypocenter	C₂₅₁	#2253
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
37. C₂₅₁	#2253	semidirect

Maximal subgroups

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

subgroup	index
#30632088	2
#30631515	2
#30631512	2
C₁₆×Dic₂₅₁	2
#367×C₂₅₁	2
#2253	251