Subgroups — #44548430

Order 32128.

68 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	#30631438	C₂	1	1	yes	maximal, characteristic
3	#334×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₅₁⋊C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₂₅₁⋊C₁₆	C₄	1	1	yes	characteristic
6	C₄⋊(C₂₅₁⋊C₈)	—	2	1	no	—
7	C₄⋊C₈×C₂₅₁	—	2	1	no	—
8	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
9	C₂×C₂₅₁⋊C₈	—	4	1	no	—
10	C₂₅₁⋊C₁₆	—	2	1	no	—
11	C₂₅₁⋊C₁₆	C₈	1	2	yes	complemented
12	C₂×C₈×C₂₅₁	—	4	1	no	—
13	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₅₁	Q₈	1	1	yes	characteristic
15	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
16	C₂₅₁⋊C₈	—	8	1	no	—
17	C₂×C₄×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₅₁	SD₁₆	1	1	yes	characteristic
19	C₂×C₄×C₂₅₁	SD₁₆	1	1	yes	characteristic
20	C₈×C₂₅₁	—	2	1	no	—
21	C₈×C₂₅₁	—	8	1	no	—
22	C₈×C₂₅₁	C₂×C₈	1	1	yes	characteristic
23	C₈×C₂₅₁	M₁₆	1	1	yes	characteristic
24	C₂²×C₂₅₁	#106	1	1	yes	characteristic
25	C₄×C₂₅₁	—	2	1	no	—
26	C₄×C₂₅₁	—	2	1	no	—
27	C₄×C₂₅₁	C₄⋊C₈	1	1	yes	characteristic
28	C₄×C₂₅₁	#108	1	1	yes	characteristic
29	C₂×C₂₅₁	#334	1	1	yes	characteristic
30	C₂×C₂₅₁	#365	1	1	yes	characteristic
31	C₂×C₂₅₁	#365	1	1	yes	characteristic
32	C₂₅₁	#1368	1	1	yes	characteristic, complemented, Sylow
33	#1368	—	251	1	no	maximal, Sylow
34	#346	—	251	1	no	—
35	#334	—	251	1	no	—
36	#334	D₂₅₁	1	1	yes	characteristic
37	C₂×C₁₆	—	251	1	no	—
38	C₂×C₁₆	—	251	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	502	1	no	—
41	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	1004	1	no	—
44	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
45	C₂×C₈	Dic₅₀₂	1	1	yes	characteristic
46	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
47	C₁₆	—	502	1	no	—
48	C₁₆	—	251	2	no	—
49	C₂×C₄	Dic₂₅₁⋊C₄	1	1	yes	characteristic
50	C₂×C₄	#16860797	1	1	yes	characteristic
51	C₂×C₄	#16860798	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	8	1	no	—
54	C₈	—	2008	1	no	—
55	C₈	C₈×D₂₅₁	1	1	yes	characteristic
56	C₈	#16860793	1	1	yes	characteristic
57	C₂²	#22548646	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	Dic₂₅₁⋊C₈	1	1	yes	characteristic
61	C₄	#22548650	1	1	yes	characteristic
62	C₂	#30631437	1	1	yes	characteristic
63	C₂	#30631509	1	1	yes	characteristic
64	C₂	#30631511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22548646
commutator	C₈×C₂₅₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₅₁
Fitting	#334×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#106
cosocle	C₄×C₈×C₂₅₁	C₂²
hypercenter	#334	D₂₅₁
hypocenter	C₂₅₁	#1368
LCS	G ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
11. C₂₅₁⋊C₁₆	C₈	semidirect
32. C₂₅₁	#1368	semidirect

Maximal subgroups

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

subgroup	index
#30631463	2
#30631438	2
#334×C₂₅₁	2
#1368	251