Subgroups — #30657834

Order 16448.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861630	C₂	1	2	yes	maximal, complemented
2	C₂×#16861626	C₂	1	1	yes	maximal, characteristic
3	#16861630	—	2	2	no	—
4	#16861626	—	2	2	no	—
5	C₂×#14523346	C₄	1	1	yes	characteristic
6	C₂×#14523346	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₂₅₇	C₄	1	1	yes	characteristic
8	#14523346	—	2	1	no	—
9	#14523346	D₄	1	2	yes	—
10	C₂²×D₂₅₇	C₈	1	1	yes	characteristic
11	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
12	C₄×D₂₅₇	—	2	1	no	—
13	C₄×D₂₅₇	—	2	1	no	—
14	C₂×C₄×C₂₅₇	C₈	1	1	yes	characteristic
15	C₂²×C₂₅₇	C₂×C₈	1	1	yes	characteristic
16	C₂×D₂₅₇	—	2	1	no	—
17	C₂×D₂₅₇	—	2	1	no	—
18	C₂×D₂₅₇	C₁₆	1	2	yes	complemented
19	C₄×C₂₅₇	—	2	1	no	—
20	Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
21	Dic₂₅₇	M₁₆	1	1	yes	characteristic
22	C₂×C₂₅₇	C₂²⋊C₈	1	1	yes	characteristic
23	C₂×C₂₅₇	C₂×C₁₆	1	1	yes	characteristic
24	C₂×C₂₅₇	M₃₂	1	1	yes	characteristic
25	D₂₅₇	—	2	2	no	—
26	C₂₅₇	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂²⋊C₁₆	—	257	1	no	maximal, Sylow
28	C₂²×C₈	—	257	1	no	—
29	C₂×C₁₆	—	257	2	no	—
30	C₂²×C₄	—	257	1	no	—
31	C₂×C₈	—	257	1	no	—
32	C₂×C₈	—	257	1	no	—
33	C₂×C₈	—	514	2	no	—
34	C₁₆	—	514	2	no	—
35	C₂³	—	257	1	no	—
36	C₂×C₄	—	257	1	no	—
37	C₂×C₄	—	514	1	no	—
38	C₂×C₄	—	514	1	no	—
39	C₂×C₄	#14523348	1	1	yes	characteristic
40	C₈	—	257	2	no	—
41	C₈	—	514	1	no	—
42	C₂²	—	257	2	no	—
43	C₂²	—	514	1	no	—
44	C₂²	—	514	1	no	—
45	C₂²	C₂×#14523348	1	1	yes	characteristic
46	C₄	—	257	1	no	—
47	C₄	—	257	1	no	—
48	C₄	—	2	1	no	—
49	C₂	—	514	2	no	—
50	C₂	C₂²⋊#14523348	1	1	yes	characteristic
51	C₂	#22551687	1	1	yes	characteristic
52	C₂	#22551689	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×#14523348
commutator	C₂×C₂₅₇	C₂×C₁₆
Fitting	C₂×C₄×C₂₅₇	C₈
socle	C₂²×C₂₅₇	C₂×C₈
prime radical	C₂²×D₂₅₇	C₈
cosocle	C₂×#14523346	C₂²
hypercenter	C₂×C₄	#14523348
hypocenter	C₂₅₇	C₂²⋊C₁₆
LCS	G ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16861630	C₂	semidirect
18. C₂×D₂₅₇	C₁₆	semidirect
26. C₂₅₇	C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861630	2
C₂×#16861626	2
C₂²⋊C₁₆	257