Subgroups — #30657632

Order 16448.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861621	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#16861621	—	2	2	no	—
4	C₂×C₄×D₂₅₇	C₄	1	1	yes	characteristic
5	C₂×#14523339	C₂²	1	1	yes	characteristic
6	C₈×D₂₅₇	—	2	2	no	—
7	C₂×C₈×C₂₅₇	C₄	1	1	yes	characteristic
8	C₂²×D₂₅₇	C₈	1	1	yes	characteristic
9	C₂×Dic₂₅₇	C₈	1	1	yes	characteristic
10	C₄×D₂₅₇	—	2	1	no	—
11	C₄×D₂₅₇	—	2	1	no	—
12	#14523339	D₄	1	2	yes	—
13	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₈×C₂₅₇	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₂₅₇	M₁₆	1	1	yes	characteristic
21	Dic₂₅₇	—	2	1	no	—
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	—	514	2	no	—
33	C₂×C₈	#11759967	1	1	yes	characteristic
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	C₂×#11759967	1	1	yes	characteristic
40	C₈	—	257	2	no	—
41	C₈	—	2	1	no	—
42	C₂²	—	257	2	no	—
43	C₂²	—	514	1	no	—
44	C₂²	—	514	1	no	—
45	C₂²	#16861626	1	1	yes	characteristic
46	C₄	—	514	1	no	—
47	C₄	C₂²⋊#11759967	1	1	yes	characteristic
48	C₄	#16861628	1	1	yes	characteristic
49	C₂	—	514	2	no	—
50	C₂	#22551649	1	1	yes	characteristic
51	C₂	#22551665	1	1	yes	characteristic
52	C₂	#22551667	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#11759967
commutator	C₂×C₂₅₇	C₂×C₁₆
Fitting	C₂×C₈×C₂₅₇	C₄
socle	C₂²×C₂₅₇	C₂×C₈
prime radical	C₂²×D₂₅₇	C₈
cosocle	C₂×#14523339	C₂²
hypercenter	C₂×C₈	#11759967
hypocenter	C₂₅₇	C₂²⋊C₁₆
LCS	G ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16861621	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×#16861621	2
C₂×C₈×D₂₅₇	2
C₂²⋊C₁₆	257