Subgroups — #44548432

Order 32128.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631508	C₂	1	1	yes	maximal, characteristic
2	#30631463	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#22548655	—	2	1	no	—
5	C₂×C₂₅₁⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₂₅₁⋊C₁₆	C₄	1	1	yes	characteristic
7	M₃₂×C₂₅₁	—	2	1	no	—
8	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
9	C₂₅₁⋊C₁₆	—	2	1	no	—
10	C₂₅₁⋊C₁₆	—	4	1	no	—
11	C₂₅₁⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
14	C₄²×C₂₅₁	Q₈	1	1	yes	characteristic
15	C₁₆×C₂₅₁	—	4	1	no	—
16	C₂×C₄×C₂₅₁	—	2	1	no	—
17	C₂×C₄×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₂₅₁	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₅₁	M₁₆	1	1	yes	characteristic
20	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
21	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
22	C₂²×C₂₅₁	#108	1	1	yes	characteristic
23	C₄×C₂₅₁	—	4	1	no	—
24	C₄×C₂₅₁	C₄⋊C₈	1	1	yes	characteristic
25	C₄×C₂₅₁	#106	1	1	yes	characteristic
26	C₂×C₂₅₁	—	2	1	no	—
27	C₂×C₂₅₁	#334	1	1	yes	characteristic
28	C₂₅₁	#1369	1	1	yes	characteristic, complemented, Sylow
29	#1369	—	251	1	no	maximal, Sylow
30	#364	—	251	1	no	—
31	#364	D₂₅₁	1	1	yes	characteristic
32	#346	—	251	1	no	—
33	M₃₂	—	2	1	no	—
34	M₃₂	—	502	1	no	—
35	C₂×C₁₆	—	251	1	no	—
36	C₂×C₁₆	—	251	1	no	—
37	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
38	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
40	C₄²	Dic₅₀₂	1	1	yes	characteristic
41	C₁₆	—	4	1	no	—
42	C₁₆	—	1004	1	no	—
43	C₁₆	—	502	1	no	—
44	C₁₆	—	251	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	Dic₂₅₁⋊C₄	1	1	yes	characteristic
47	C₈	C₈×D₂₅₁	1	1	yes	characteristic
48	C₈	#16860793	1	1	yes	characteristic
49	C₈	#16860797	1	1	yes	characteristic
50	C₈	#16860798	1	1	yes	characteristic
51	C₂²	#22548650	1	1	yes	characteristic
52	C₄	—	4	1	no	—
53	C₄	Dic₂₅₁⋊C₈	1	1	yes	characteristic
54	C₄	#22548646	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	#30631437	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#22548646
commutator	C₈×C₂₅₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₅₁
Fitting	#364×C₂₅₁	C₂
socle	C₂×C₂₅₁	#334
prime radical	C₂²×C₂₅₁	#108
cosocle	C₄×C₈×C₂₅₁	C₂²
hypercenter	#364	D₂₅₁
hypocenter	C₂₅₁	#1369
LCS	G ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
28. C₂₅₁	#1369	semidirect

Maximal subgroups

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

subgroup	index
#30631508	2
#30631463	2
#364×C₂₅₁	2
#1369	251