Subgroups — C₂⁴×M₁₆ #63072

Order 256.

6655 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₈	C₂	1	2	yes	maximal, complemented
4	C₂⁶	C₄	1	1	yes	characteristic
5	C₂⁴×C₄	C₄	1	1	yes	characteristic
6	C₂⁴×C₄	C₄	1	30	yes	—
7	C₂⁴×C₄	C₂²	1	30	yes	—
8	C₂⁴×C₄	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆	C₂²	1	560	yes	direct product factor
10	C₂³×C₈	C₂²	1	60	yes	complemented
11	C₂⁵	—	2	16	no	—
12	C₂⁵	C₂×C₄	1	30	yes	—
13	C₂⁵	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄	C₂×C₄	1	420	yes	—
15	C₂³×C₄	C₂×C₄	1	30	yes	—
16	C₂³×C₄	C₂×C₄	1	15	yes	—
17	C₂³×C₄	C₂³	1	15	yes	—
18	C₂³×C₄	C₂³	1	140	yes	—
19	C₂×M₁₆	C₂³	1	960	yes	direct product factor
20	C₂²×C₈	C₂³	1	280	yes	complemented
21	C₂⁴	—	2	240	no	—
22	C₂⁴	M₁₆	1	16	yes	direct product factor
23	C₂⁴	C₂²×C₄	1	140	yes	—
24	C₂⁴	C₂²×C₄	1	15	yes	—
25	C₂²×C₄	C₂²×C₄	1	840	yes	—
26	C₂²×C₄	C₂²×C₄	1	140	yes	—
27	C₂²×C₄	C₂²×C₄	1	105	yes	—
28	C₂²×C₄	C₂⁴	1	35	yes	—
29	C₂²×C₄	C₂⁴	1	120	yes	—
30	M₁₆	C₂⁴	1	256	yes	direct product factor
31	C₂×C₈	C₂⁴	1	240	yes	complemented
32	C₂³	—	2	560	no	—
33	C₂³	C₂×M₁₆	1	120	yes	direct product factor
34	C₂³	C₂³×C₄	1	120	yes	—
35	C₂³	C₂³×C₄	1	35	yes	—
36	C₂×C₄	C₂³×C₄	1	240	yes	—
37	C₂×C₄	C₂³×C₄	1	105	yes	—
38	C₂×C₄	C₂³×C₄	1	120	yes	—
39	C₂×C₄	C₂⁵	1	15	yes	—
40	C₂×C₄	C₂⁵	1	16	yes	—
41	C₈	C₂⁵	1	32	yes	complemented
42	C₂²	—	2	240	no	—
43	C₂²	C₂²×M₁₆	1	140	yes	direct product factor
44	C₂²	C₂⁴×C₄	1	16	yes	—
45	C₂²	C₂⁴×C₄	1	15	yes	—
46	C₄	C₂⁴×C₄	1	15	yes	—
47	C₄	C₂⁴×C₄	1	16	yes	—
48	C₄	C₂⁶	1	1	yes	characteristic
49	C₂	—	2	16	no	—
50	C₂	C₂³×M₁₆	1	30	yes	direct product factor
51	C₂	C₂⁵×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂²
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₄	C₂⁶
socle	C₂⁵	C₂×C₄
prime radical	C₂⁶	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂³×M₁₆	C₂	direct (N × H)
3. C₂⁴×C₈	C₂	semidirect
9. C₂²×M₁₆	C₂²	direct (N × H)
10. C₂³×C₈	C₂²	semidirect
19. C₂×M₁₆	C₂³	direct (N × H)
20. C₂²×C₈	C₂³	semidirect
22. C₂⁴	M₁₆	direct (N × H)
30. M₁₆	C₂⁴	direct (N × H)
31. C₂×C₈	C₂⁴	semidirect
33. C₂³	C₂×M₁₆	direct (N × H)
41. C₈	C₂⁵	semidirect
43. C₂²	C₂²×M₁₆	direct (N × H)
50. C₂	C₂³×M₁₆	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×C₄	2
C₂³×M₁₆	2
C₂⁴×C₈	2

Subgroups — C24×M16 #63072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×M₁₆ #63072