Subgroups — #30657142

Order 16448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551592	C₂	1	2	yes	maximal
2	#22551526	C₂	1	4	yes	maximal
3	C₄×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₁₄	—	2	2	no	—
5	C₄⋊Dic₂₅₇	—	2	2	no	—
6	C₄⋊Dic₂₅₇	C₂²	1	4	yes	—
7	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
8	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
9	Dic₅₁₄	—	4	4	no	—
10	C₂×Dic₂₅₇	—	2	4	no	—
11	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
12	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
13	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
14	C₈×C₂₅₇	Q₈	1	4	yes	complemented
15	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
16	C₄×C₂₅₇	D₈	1	2	yes	—
17	C₄×C₂₅₇	Q₁₆	1	2	yes	—
18	C₄×C₂₅₇	C₂×Q₈	1	2	yes	—
19	Dic₂₅₇	—	4	4	no	—
20	C₂×C₂₅₇	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₂₅₇	C₂×D₈	1	1	yes	characteristic
22	C₂×C₂₅₇	C₂×Q₁₆	1	1	yes	characteristic
23	C₂₅₇	#500	1	1	yes	characteristic, complemented, Sylow
24	#500	—	257	1	no	maximal, Sylow
25	C₄⋊Q₈	—	257	2	no	—
26	#107	—	257	4	no	—
27	C₄×C₈	D₂₅₇	1	1	yes	characteristic
28	C₂×Q₈	—	514	2	no	—
29	C₂×C₈	C₂×D₂₅₇	1	2	yes	—
30	C₄⋊C₄	—	257	4	no	—
31	C₄⋊C₄	—	514	2	no	—
32	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
33	Q₈	—	1028	4	no	—
34	C₂×C₄	—	514	4	no	—
35	C₂×C₄	D₁₀₂₈	1	1	yes	characteristic
36	C₂×C₄	D₁₀₂₈	1	1	yes	characteristic
37	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
38	C₈	Dic₅₁₄	1	4	yes	complemented
39	C₂²	C₂×D₁₀₂₈	1	1	yes	characteristic
40	C₄	—	1028	4	no	—
41	C₄	D₂₀₅₆	1	2	yes	—
42	C₄	Dic₁₀₂₈	1	2	yes	—
43	C₄	C₂×Dic₅₁₄	1	2	yes	—
44	C₂	#22551592	1	1	yes	characteristic
45	C₂	C₂×D₂₀₅₆	1	1	yes	characteristic
46	C₂	C₂×Dic₁₀₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₂₈
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	C₄×C₈×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	C₂×D₄
hypercenter	C₄×C₈	D₂₅₇
hypocenter	C₂₅₇	#500
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
14. C₈×C₂₅₇	Q₈	semidirect
23. C₂₅₇	#500	semidirect
38. C₈	Dic₅₁₄	semidirect

Maximal subgroups

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

subgroup	index
#22551592	2
#22551526	2
C₄×C₈×C₂₅₇	2
#500	257