Subgroups — C₂⁴×C₂²⋊C₄ #62639

Order 256.

21839 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×C₄	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊C₄	C₂	1	60	yes	maximal, direct product factor
4	C₂⁶	—	2	32	no	—
5	C₂⁶	C₄	1	32	yes	complemented
6	C₂⁶	C₂²	1	30	yes	—
7	C₂⁶	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄	—	2	32	no	—
9	C₂⁴×C₄	C₂²	1	60	yes	complemented
10	C₂²×C₂²⋊C₄	C₂²	1	560	yes	direct product factor
11	C₂⁵	—	2	512	no	—
12	C₂⁵	—	2	480	no	—
13	C₂⁵	C₂×C₄	1	16	yes	—
14	C₂⁵	C₂×C₄	1	480	yes	complemented
15	C₂⁵	D₄	1	32	yes	—
16	C₂⁵	C₂³	1	140	yes	—
17	C₂⁵	C₂³	1	15	yes	—
18	C₂³×C₄	—	2	480	no	—
19	C₂³×C₄	C₂³	1	280	yes	complemented
20	C₂×C₂²⋊C₄	C₂³	1	960	yes	direct product factor
21	C₂⁴	—	2	3840	no	—
22	C₂⁴	—	2	1120	no	—
23	C₂⁴	C₂²⋊C₄	1	256	yes	direct product factor
24	C₂⁴	C₂²×C₄	1	120	yes	—
25	C₂⁴	C₂²×C₄	1	1120	yes	complemented
26	C₂⁴	C₂×D₄	1	240	yes	—
27	C₂⁴	C₂⁴	1	120	yes	—
28	C₂⁴	C₂⁴	1	35	yes	—
29	C₂²×C₄	—	2	1120	no	—
30	C₂²×C₄	C₂⁴	1	240	yes	complemented
31	C₂²⋊C₄	C₂⁴	1	256	yes	direct product factor
32	C₂³	—	2	4480	no	—
33	C₂³	—	2	480	no	—
34	C₂³	C₂×C₂²⋊C₄	1	960	yes	direct product factor
35	C₂³	C₂³×C₄	1	140	yes	—
36	C₂³	C₂³×C₄	1	480	yes	complemented
37	C₂³	C₂²×D₄	1	280	yes	—
38	C₂³	C₂⁵	1	16	yes	—
39	C₂³	C₂⁵	1	15	yes	—
40	C₂×C₄	—	2	480	no	—
41	C₂×C₄	C₂⁵	1	32	yes	complemented
42	C₂²	—	2	32	no	—
43	C₂²	—	2	960	no	—
44	C₂²	C₂²×C₂²⋊C₄	1	560	yes	direct product factor
45	C₂²	C₂⁴×C₄	1	30	yes	—
46	C₂²	C₂⁴×C₄	1	32	yes	complemented
47	C₂²	C₂³×D₄	1	60	yes	—
48	C₂²	C₂⁶	1	1	yes	characteristic
49	C₄	—	2	32	no	—
50	C₂	—	2	32	no	—
51	C₂	C₂³×C₂²⋊C₄	1	60	yes	direct product factor
52	C₂	C₂⁵×C₄	1	1	yes	characteristic
53	C₂	C₂⁴×D₄	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁶	C₂²
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂⁷	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂⁵×C₄	C₂	semidirect
3. C₂³×C₂²⋊C₄	C₂	direct (N × H)
5. C₂⁶	C₄	semidirect
9. C₂⁴×C₄	C₂²	semidirect
10. C₂²×C₂²⋊C₄	C₂²	direct (N × H)
14. C₂⁵	C₂×C₄	semidirect
19. C₂³×C₄	C₂³	semidirect
20. C₂×C₂²⋊C₄	C₂³	direct (N × H)
23. C₂⁴	C₂²⋊C₄	direct (N × H)
25. C₂⁴	C₂²×C₄	semidirect
30. C₂²×C₄	C₂⁴	semidirect
31. C₂²⋊C₄	C₂⁴	direct (N × H)
34. C₂³	C₂×C₂²⋊C₄	direct (N × H)
36. C₂³	C₂³×C₄	semidirect
41. C₂×C₄	C₂⁵	semidirect
44. C₂²	C₂²×C₂²⋊C₄	direct (N × H)
46. C₂²	C₂⁴×C₄	semidirect
51. C₂	C₂³×C₂²⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁷	2
C₂⁵×C₄	2
C₂³×C₂²⋊C₄	2

Subgroups — C24×C22⋊C4 #62639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₂²⋊C₄ #62639