Subgroups — #44548572

Order 32128.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631521	C₂	1	1	yes	maximal, characteristic
2	#30631515	C₂	1	1	yes	maximal, characteristic
3	#368×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	#108×C₂₅₁	—	2	1	no	—
8	C₄⋊Dic₂₅₁	—	4	1	no	—
9	M₁₆×C₂₅₁	—	4	1	no	—
10	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
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	D₈	1	1	yes	characteristic
15	C₈×C₂₅₁	—	8	1	no	—
16	C₈×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₂₅₁	Q₁₆	1	1	yes	characteristic
18	C₂²×C₂₅₁	SD₃₂	1	1	yes	characteristic
19	C₄×C₂₅₁	#107	1	1	yes	characteristic
20	C₄×C₂₅₁	SD₃₂	1	1	yes	characteristic
21	Dic₂₅₁	—	16	1	no	—
22	C₂×C₂₅₁	—	2	1	no	—
23	C₂×C₂₅₁	#367	1	1	yes	characteristic
24	C₂₅₁	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	251	1	no	maximal, Sylow
26	M₆₄	—	251	1	no	—
27	#368	D₂₅₁	1	1	yes	characteristic
28	#367	—	251	1	no	—
29	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
30	#108	—	2	1	no	—
31	#107	—	502	1	no	—
32	C₃₂	—	251	2	no	—
33	M₁₆	—	4	1	no	—
34	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
35	C₄⋊C₄	—	1004	1	no	—
36	C₁₆	C₄×D₂₅₁	1	1	yes	characteristic
37	C₁₆	Dic₅₀₂	1	1	yes	characteristic
38	C₂×C₄	—	2008	1	no	—
39	C₂×C₄	#16860795	1	1	yes	characteristic
40	C₈	—	8	1	no	—
41	C₈	Dic₂₅₁⋊C₄	1	1	yes	characteristic
42	C₈	#16860800	1	1	yes	characteristic
43	C₂²	#22548658	1	1	yes	characteristic
44	C₄	—	4016	1	no	—
45	C₄	#22548648	1	1	yes	characteristic
46	C₄	#22548659	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#30631514	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30631514
commutator	C₁₆×C₂₅₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₅₁
Fitting	#368×C₂₅₁	C₂
socle	C₂×C₂₅₁	#367
prime radical	C₂²×C₂₅₁	SD₃₂
cosocle	C₂×C₁₆×C₂₅₁	C₂²
hypercenter	#368	D₂₅₁
hypocenter	C₂₅₁	#1426
LCS	G ▷ C₁₆×C₂₅₁ ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
5. C₂₅₁⋊C₃₂	C₄	semidirect
24. C₂₅₁	#1426	semidirect

Maximal subgroups

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

subgroup	index
#30631521	2
#30631515	2
#368×C₂₅₁	2
#1426	251