Subgroups — C₂⁴×C₄×C₄₁₉ #41000072

Order 26816.

1362 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×C₄₁₉	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₄₁₉	C₄	1	16	yes	direct product factor
4	C₂⁴×C₄₁₉	C₂²	1	15	yes	—
5	C₂²×C₄×C₄₁₉	C₂²	1	140	yes	direct product factor
6	C₂³×C₄₁₉	C₂×C₄	1	120	yes	direct product factor
7	C₂³×C₄₁₉	C₂³	1	35	yes	—
8	C₂×C₄×C₄₁₉	C₂³	1	120	yes	direct product factor
9	C₂²×C₄₁₉	C₂²×C₄	1	140	yes	direct product factor
10	C₂²×C₄₁₉	C₂⁴	1	15	yes	—
11	C₄×C₄₁₉	C₂⁴	1	16	yes	direct product factor
12	C₂×C₄₁₉	C₂³×C₄	1	30	yes	direct product factor
13	C₂×C₄₁₉	C₂⁵	1	1	yes	characteristic
14	C₄₁₉	C₂⁴×C₄	1	1	yes	characteristic, direct product factor, Sylow
15	C₂⁴×C₄	C₄₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂⁵	C₂×C₄₁₉	1	1	yes	characteristic
17	C₂³×C₄	C₂×C₄₁₉	1	30	yes	direct product factor
18	C₂⁴	C₄×C₄₁₉	1	16	yes	direct product factor
19	C₂⁴	C₂²×C₄₁₉	1	15	yes	—
20	C₂²×C₄	C₂²×C₄₁₉	1	140	yes	direct product factor
21	C₂³	C₂×C₄×C₄₁₉	1	120	yes	direct product factor
22	C₂³	C₂³×C₄₁₉	1	35	yes	—
23	C₂×C₄	C₂³×C₄₁₉	1	120	yes	direct product factor
24	C₂²	C₂²×C₄×C₄₁₉	1	140	yes	direct product factor
25	C₂²	C₂⁴×C₄₁₉	1	15	yes	—
26	C₄	C₂⁴×C₄₁₉	1	16	yes	direct product factor
27	C₂	C₂³×C₄×C₄₁₉	1	30	yes	direct product factor
28	C₂	C₂⁵×C₄₁₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂⁵×C₄₁₉
socle, prime radical	C₂⁵×C₄₁₉	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄×C₄₁₉	C₂	direct (N × H)
3. C₂⁴×C₄₁₉	C₄	direct (N × H)
5. C₂²×C₄×C₄₁₉	C₂²	direct (N × H)
6. C₂³×C₄₁₉	C₂×C₄	direct (N × H)
8. C₂×C₄×C₄₁₉	C₂³	direct (N × H)
9. C₂²×C₄₁₉	C₂²×C₄	direct (N × H)
11. C₄×C₄₁₉	C₂⁴	direct (N × H)
12. C₂×C₄₁₉	C₂³×C₄	direct (N × H)
14. C₄₁₉	C₂⁴×C₄	direct (N × H)
15. C₂⁴×C₄	C₄₁₉	direct (N × H)
17. C₂³×C₄	C₂×C₄₁₉	direct (N × H)
18. C₂⁴	C₄×C₄₁₉	direct (N × H)
20. C₂²×C₄	C₂²×C₄₁₉	direct (N × H)
21. C₂³	C₂×C₄×C₄₁₉	direct (N × H)
23. C₂×C₄	C₂³×C₄₁₉	direct (N × H)
24. C₂²	C₂²×C₄×C₄₁₉	direct (N × H)
26. C₄	C₂⁴×C₄₁₉	direct (N × H)
27. C₂	C₂³×C₄×C₄₁₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄₁₉	2
C₂³×C₄×C₄₁₉	2
C₂⁴×C₄	419

Subgroups — C24×C4×C419 #41000072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄×C₄₁₉ #41000072