Subgroups — #30657139

Order 16448.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551584	C₂	1	2	yes	maximal
2	#22551526	C₂	1	2	yes	maximal
3	#22551524	C₂	1	2	yes	maximal
4	C₄×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₂₅₇	C₂²	1	4	yes	—
6	#16861597	—	2	4	no	—
7	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
9	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₅₇	—	2	4	no	—
11	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
12	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
13	C₈×C₂₅₇	Q₈	1	2	yes	—
14	C₈×C₂₅₇	Q₈	1	2	yes	—
15	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
16	C₄×C₂₅₇	—	2	2	no	—
17	C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
18	C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
19	Dic₂₅₇	—	4	4	no	—
20	C₂×C₂₅₇	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₂₅₇	#135	1	2	yes	—
22	C₂₅₇	#499	1	1	yes	characteristic, complemented, Sylow
23	#499	—	257	1	no	maximal, Sylow
24	#125	—	257	2	no	—
25	#107	—	257	2	no	—
26	#106	—	257	2	no	—
27	C₄×C₈	D₂₅₇	1	1	yes	characteristic
28	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
29	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
30	C₄⋊C₄	—	257	4	no	—
31	C₄⋊C₄	—	514	4	no	—
32	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
33	C₂×C₄	—	514	4	no	—
34	C₂×C₄	D₁₀₂₈	1	2	yes	—
35	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
36	C₈	Dic₅₁₄	1	2	yes	—
37	C₈	Dic₅₁₄	1	2	yes	—
38	C₂²	C₂×D₁₀₂₈	1	1	yes	characteristic
39	C₄	—	2	2	no	—
40	C₄	—	1028	4	no	—
41	C₄	C₂×Dic₅₁₄	1	1	yes	characteristic
42	C₄	C₂×Dic₅₁₄	1	1	yes	characteristic
43	C₂	#22551592	1	1	yes	characteristic
44	C₂	#22551614	1	2	yes	—

	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	#499
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
22. C₂₅₇	#499	semidirect

Maximal subgroups

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

subgroup	index
#22551584	2
#22551526	2
#22551524	2
C₄×C₈×C₂₅₇	2
#499	257