Subgroups — #44552639

Order 32128.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632091	C₂	1	2	yes	maximal
2	#30631513	C₂	1	1	yes	maximal, characteristic
3	#30631512	C₂	1	2	yes	maximal
4	C₁₆×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
5	#366×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	#22548714	—	2	2	no	—
7	C₂×C₂₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#22548649	C₂²	1	2	yes	—
9	#22548648	—	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×Dic₅₀₂	—	4	2	no	—
14	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₂₅₁	—	2	2	no	—
16	Dic₂₅₁⋊C₄	—	4	2	no	—
17	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
18	C₂₅₁⋊C₁₆	Q₈	1	2	yes	complemented
19	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₅₁	—	2	2	no	—
21	C₁₆×C₂₅₁	Q₈	1	2	yes	complemented
22	Dic₅₀₂	—	8	4	no	—
23	C₂×Dic₂₅₁	—	4	2	no	—
24	C₂×Dic₂₅₁	D₈	1	1	yes	characteristic
25	C₂×Dic₂₅₁	D₈	1	1	yes	characteristic
26	C₂₅₁⋊C₈	Q₁₆	1	2	yes	—
27	C₂×C₄×C₂₅₁	—	4	2	no	—
28	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
29	C₈×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
32	C₄×C₂₅₁	—	8	2	no	—
33	C₄×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
34	C₄×C₂₅₁	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₂₅₁	—	8	2	no	—
36	Dic₂₅₁	D₁₆	1	2	yes	—
37	Dic₂₅₁	Q₃₂	1	2	yes	—
38	C₂×C₂₅₁	#500	1	1	yes	characteristic
39	C₂×C₂₅₁	C₂×D₁₆	1	1	yes	characteristic
40	C₂×C₂₅₁	C₂×Q₃₂	1	1	yes	characteristic
41	C₂₅₁	#2252	1	1	yes	characteristic, complemented, Sylow
42	#2252	—	251	1	no	maximal, Sylow
43	#500	—	251	2	no	—
44	#366	—	251	1	no	—
45	#366	—	251	2	no	—
46	#366	D₂₅₁	1	1	yes	characteristic
47	C₄×C₁₆	—	251	1	no	—
48	C₄⋊Q₈	—	502	2	no	—
49	C₂×C₁₆	—	251	1	no	—
50	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
51	#107	—	502	2	no	—
52	#107	—	251	2	no	—
53	#107	C₂×D₂₅₁	1	2	yes	—
54	C₄×C₈	—	251	1	no	—
55	C₂×Q₈	—	1004	2	no	—
56	C₂×C₈	—	251	1	no	—
57	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	1004	2	no	—
60	C₄⋊C₄	—	502	2	no	—
61	C₄²	—	251	1	no	—
62	C₁₆	—	251	2	no	—
63	C₁₆	Dic₅₀₂	1	2	yes	complemented
64	Q₈	—	2008	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	1004	2	no	—
67	C₂×C₄	—	251	1	no	—
68	C₂×C₄	—	251	1	no	—
69	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
70	C₈	—	251	2	no	—
71	C₈	Q₈×D₂₅₁	1	1	yes	characteristic
72	C₈	C₂×Dic₅₀₂	1	1	yes	characteristic
73	C₂²	D₈×D₂₅₁	1	1	yes	characteristic
74	C₄	—	8	2	no	—
75	C₄	—	2008	2	no	—
76	C₄	—	251	2	no	—
77	C₄	—	251	2	no	—
78	C₄	#22548714	1	1	yes	characteristic
79	C₄	Q₁₆×D₂₅₁	1	1	yes	characteristic
80	C₂	#30632091	1	1	yes	characteristic
81	C₂	D₁₆×D₂₅₁	1	1	yes	characteristic
82	C₂	Q₃₂×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₅₁
commutator, cosocle	C₂×C₈×C₂₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#366×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	C₂×D₈
hypercenter	#366	D₂₅₁
hypocenter	C₂₅₁	#2252
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
18. C₂₅₁⋊C₁₆	Q₈	semidirect
21. C₁₆×C₂₅₁	Q₈	semidirect
41. C₂₅₁	#2252	semidirect
63. C₁₆	Dic₅₀₂	semidirect

Maximal subgroups

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

subgroup	index
#30632091	2
#30631513	2
#30631512	2
C₁₆×Dic₂₅₁	2
#366×C₂₅₁	2
#2252	251