Subgroups — C₄³×D₂₅₇ #45666641

Order 32896.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×D₂₅₇	C₂	1	7	yes	maximal
2	C₄²×Dic₂₅₇	C₂	1	7	yes	maximal, complemented
3	C₄³×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₂₅₇	C₂²	1	7	yes	—
5	C₂×C₄×Dic₂₅₇	C₂²	1	21	yes	—
6	C₄²×D₂₅₇	C₄	1	56	yes	direct product factor
7	C₂×C₄²×C₂₅₇	C₂²	1	7	yes	—
8	C₂³×D₂₅₇	C₂³	1	1	yes	characteristic
9	C₂²×Dic₂₅₇	C₂³	1	7	yes	—
10	C₂×C₄×D₂₅₇	C₂×C₄	1	84	yes	—
11	C₄×Dic₂₅₇	C₂×C₄	1	84	yes	complemented
12	C₂²×C₄×C₂₅₇	C₂³	1	7	yes	—
13	C₄²×C₂₅₇	C₂×C₄	1	28	yes	complemented
14	C₂²×D₂₅₇	C₂²×C₄	1	14	yes	—
15	C₂×Dic₂₅₇	C₂²×C₄	1	42	yes	—
16	C₄×D₂₅₇	C₄²	1	112	yes	direct product factor
17	C₂³×C₂₅₇	C₂⁴	1	1	yes	characteristic
18	C₂×C₄×C₂₅₇	C₂²×C₄	1	42	yes	—
19	C₂²×C₂₅₇	C₂³×C₄	1	7	yes	—
20	C₂×D₂₅₇	C₂×C₄²	1	28	yes	—
21	C₄×C₂₅₇	C₂×C₄²	1	28	yes	complemented
22	Dic₂₅₇	C₂×C₄²	1	28	yes	complemented
23	C₂×C₂₅₇	C₂²×C₄²	1	7	yes	—
24	D₂₅₇	C₄³	1	8	yes	direct product factor
25	C₂₅₇	C₂×C₄³	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₄³	—	257	1	no	maximal, Sylow
27	C₂²×C₄²	—	257	7	no	—
28	C₄³	—	257	7	no	—
29	C₄³	D₂₅₇	1	1	yes	characteristic, direct product factor
30	C₂³×C₄	—	257	7	no	—
31	C₂×C₄²	—	257	21	no	—
32	C₂×C₄²	—	257	56	no	—
33	C₂×C₄²	C₂×D₂₅₇	1	7	yes	—
34	C₂⁴	—	257	1	no	—
35	C₂²×C₄	—	257	7	no	—
36	C₂²×C₄	—	257	84	no	—
37	C₂²×C₄	C₂²×D₂₅₇	1	7	yes	—
38	C₄²	—	257	84	no	—
39	C₄²	C₄×D₂₅₇	1	28	yes	direct product factor
40	C₂³	—	257	14	no	—
41	C₂³	C₂³×D₂₅₇	1	1	yes	characteristic
42	C₂×C₄	—	257	42	no	—
43	C₂×C₄	—	257	112	no	—
44	C₂×C₄	C₂×C₄×D₂₅₇	1	42	yes	—
45	C₂²	—	257	28	no	—
46	C₂²	C₂²×C₄×D₂₅₇	1	7	yes	—
47	C₄	—	257	28	no	—
48	C₄	C₄²×D₂₅₇	1	28	yes	direct product factor
49	C₂	—	257	8	no	—
50	C₂	C₂×C₄²×D₂₅₇	1	7	yes	—

	subgroup	G/H
center, hypercenter	C₄³	D₂₅₇
commutator, hypocenter	C₂₅₇	C₂×C₄³
Frattini	C₂³	C₂³×D₂₅₇
Fitting	C₄³×C₂₅₇	C₂
socle, cosocle	C₂³×C₂₅₇	C₂⁴
prime radical	C₂³×D₂₅₇	C₂³
LCS	G ▷ C₂₅₇
UCS	C₁ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. C₄²×Dic₂₅₇	C₂	semidirect
3. C₄³×C₂₅₇	C₂	semidirect
6. C₄²×D₂₅₇	C₄	direct (N × H)
11. C₄×Dic₂₅₇	C₂×C₄	semidirect
13. C₄²×C₂₅₇	C₂×C₄	semidirect
16. C₄×D₂₅₇	C₄²	direct (N × H)
21. C₄×C₂₅₇	C₂×C₄²	semidirect
22. Dic₂₅₇	C₂×C₄²	semidirect
24. D₂₅₇	C₄³	direct (N × H)
25. C₂₅₇	C₂×C₄³	semidirect
29. C₄³	D₂₅₇	direct (N × H)
39. C₄²	C₄×D₂₅₇	direct (N × H)
48. C₄	C₄²×D₂₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄²×D₂₅₇	2
C₄²×Dic₂₅₇	2
C₄³×C₂₅₇	2
C₂×C₄³	257

Subgroups — C43×D257 #45666641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄³×D₂₅₇ #45666641