Subgroups — C₂⁴×C₂₅₁⋊C₈ #44565064

Order 32128.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂₅₁⋊C₈	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₅₁⋊C₈	C₂²	1	140	yes	direct product factor
4	C₂⁵×C₂₅₁	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₂₅₁	C₄	1	15	yes	—
6	C₂³×C₄×C₂₅₁	C₂²	1	15	yes	—
7	C₂×C₂₅₁⋊C₈	C₂³	1	120	yes	direct product factor
8	C₂⁴×C₂₅₁	C₈	1	16	yes	complemented
9	C₂⁴×C₂₅₁	C₂×C₄	1	15	yes	—
10	C₂²×C₄×C₂₅₁	C₂×C₄	1	105	yes	—
11	C₂²×C₄×C₂₅₁	C₂³	1	35	yes	—
12	C₂₅₁⋊C₈	C₂⁴	1	16	yes	direct product factor
13	C₂³×C₂₅₁	C₂×C₈	1	120	yes	complemented
14	C₂³×C₂₅₁	C₂²×C₄	1	35	yes	—
15	C₂×C₄×C₂₅₁	C₂²×C₄	1	105	yes	—
16	C₂×C₄×C₂₅₁	C₂⁴	1	15	yes	—
17	C₂²×C₂₅₁	C₂²×C₈	1	140	yes	complemented
18	C₂²×C₂₅₁	C₂³×C₄	1	15	yes	—
19	C₄×C₂₅₁	C₂³×C₄	1	15	yes	—
20	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
21	C₂×C₂₅₁	C₂³×C₈	1	30	yes	complemented
22	C₂×C₂₅₁	C₂⁴×C₄	1	1	yes	characteristic
23	C₂₅₁	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
24	C₂⁴×C₈	—	251	1	no	maximal, Sylow
25	C₂⁴×C₄	D₂₅₁	1	1	yes	characteristic
26	C₂³×C₈	—	251	30	no	—
27	C₂⁵	Dic₂₅₁	1	1	yes	characteristic
28	C₂³×C₄	Dic₂₅₁	1	15	yes	—
29	C₂³×C₄	C₂×D₂₅₁	1	15	yes	—
30	C₂²×C₈	—	251	140	no	—
31	C₂⁴	C₂₅₁⋊C₈	1	16	yes	direct product factor
32	C₂⁴	C₂×Dic₂₅₁	1	15	yes	—
33	C₂²×C₄	C₂×Dic₂₅₁	1	105	yes	—
34	C₂²×C₄	C₂²×D₂₅₁	1	35	yes	—
35	C₂×C₈	—	251	120	no	—
36	C₂³	C₂×C₂₅₁⋊C₈	1	120	yes	direct product factor
37	C₂³	C₂²×Dic₂₅₁	1	35	yes	—
38	C₂×C₄	C₂²×Dic₂₅₁	1	105	yes	—
39	C₂×C₄	C₂³×D₂₅₁	1	15	yes	—
40	C₈	—	251	16	no	—
41	C₂²	C₂²×C₂₅₁⋊C₈	1	140	yes	direct product factor
42	C₂²	C₂³×Dic₂₅₁	1	15	yes	—
43	C₄	C₂³×Dic₂₅₁	1	15	yes	—
44	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
45	C₂	C₂³×C₂₅₁⋊C₈	1	30	yes	direct product factor
46	C₂	C₂⁴×Dic₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	D₂₅₁
commutator, hypocenter	C₂₅₁	C₂⁴×C₈
Frattini	C₄	C₂⁴×D₂₅₁
Fitting	C₂⁴×C₄×C₂₅₁	C₂
socle, prime radical	C₂⁵×C₂₅₁	C₄
cosocle	C₄×C₂₅₁	C₂⁵
LCS	G ▷ C₂₅₁
UCS	C₁ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₂₅₁⋊C₈	C₂	direct (N × H)
3. C₂²×C₂₅₁⋊C₈	C₂²	direct (N × H)
7. C₂×C₂₅₁⋊C₈	C₂³	direct (N × H)
8. C₂⁴×C₂₅₁	C₈	semidirect
12. C₂₅₁⋊C₈	C₂⁴	direct (N × H)
13. C₂³×C₂₅₁	C₂×C₈	semidirect
17. C₂²×C₂₅₁	C₂²×C₈	semidirect
21. C₂×C₂₅₁	C₂³×C₈	semidirect
23. C₂₅₁	C₂⁴×C₈	semidirect
31. C₂⁴	C₂₅₁⋊C₈	direct (N × H)
36. C₂³	C₂×C₂₅₁⋊C₈	direct (N × H)
41. C₂²	C₂²×C₂₅₁⋊C₈	direct (N × H)
45. C₂	C₂³×C₂₅₁⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂₅₁⋊C₈	2
C₂⁴×C₄×C₂₅₁	2
C₂⁴×C₈	251

Subgroups — C24×C251⋊C8 #44565064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₂₅₁⋊C₈ #44565064