Subgroups — C₂⁴×D₄×C₂₂₃ #43252603

Order 28544.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	C₂³×D₄×C₂₂₃	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×C₂₂₃	—	2	32	no	—
5	C₂⁵×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂⁵×C₂₂₃	C₂²	1	60	yes	complemented
7	C₂²×D₄×C₂₂₃	C₂²	1	560	yes	direct product factor
8	C₂³×C₄×C₂₂₃	C₂²	1	30	yes	complemented
9	C₂⁴×C₂₂₃	—	2	480	no	—
10	C₂⁴×C₂₂₃	D₄	1	16	yes	direct product factor
11	C₂⁴×C₂₂₃	C₂³	1	15	yes	—
12	C₂⁴×C₂₂₃	C₂³	1	280	yes	complemented
13	C₂×D₄×C₂₂₃	C₂³	1	960	yes	direct product factor
14	C₂²×C₄×C₂₂₃	C₂³	1	140	yes	complemented
15	C₂³×C₂₂₃	—	2	1120	no	—
16	C₂³×C₂₂₃	C₂×D₄	1	120	yes	direct product factor
17	C₂³×C₂₂₃	C₂⁴	1	35	yes	—
18	C₂³×C₂₂₃	C₂⁴	1	240	yes	complemented
19	D₄×C₂₂₃	C₂⁴	1	256	yes	direct product factor
20	C₂×C₄×C₂₂₃	C₂⁴	1	120	yes	complemented
21	C₂²×C₂₂₃	—	2	480	no	—
22	C₂²×C₂₂₃	C₂²×D₄	1	140	yes	direct product factor
23	C₂²×C₂₂₃	C₂⁵	1	15	yes	—
24	C₂²×C₂₂₃	C₂⁵	1	32	yes	complemented
25	C₄×C₂₂₃	C₂⁵	1	16	yes	complemented
26	C₂×C₂₂₃	—	2	32	no	—
27	C₂×C₂₂₃	C₂³×D₄	1	30	yes	direct product factor
28	C₂×C₂₂₃	C₂⁶	1	1	yes	characteristic
29	C₂₂₃	C₂⁴×D₄	1	1	yes	characteristic, direct product factor, Sylow
30	C₂⁴×D₄	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂⁶	C₂×C₂₂₃	1	2	yes	complemented
32	C₂³×D₄	C₂×C₂₂₃	1	60	yes	direct product factor
33	C₂⁴×C₄	C₂×C₂₂₃	1	1	yes	characteristic, complemented
34	C₂⁵	—	2	32	no	—
35	C₂⁵	C₂²×C₂₂₃	1	1	yes	characteristic
36	C₂⁵	C₂²×C₂₂₃	1	60	yes	complemented
37	C₂²×D₄	C₂²×C₂₂₃	1	560	yes	direct product factor
38	C₂³×C₄	C₂²×C₂₂₃	1	30	yes	complemented
39	C₂⁴	—	2	480	no	—
40	C₂⁴	D₄×C₂₂₃	1	16	yes	direct product factor
41	C₂⁴	C₂³×C₂₂₃	1	15	yes	—
42	C₂⁴	C₂³×C₂₂₃	1	280	yes	complemented
43	C₂×D₄	C₂³×C₂₂₃	1	960	yes	direct product factor
44	C₂²×C₄	C₂³×C₂₂₃	1	140	yes	complemented
45	C₂³	—	2	1120	no	—
46	C₂³	C₂×D₄×C₂₂₃	1	120	yes	direct product factor
47	C₂³	C₂⁴×C₂₂₃	1	35	yes	—
48	C₂³	C₂⁴×C₂₂₃	1	240	yes	complemented
49	D₄	C₂⁴×C₂₂₃	1	256	yes	direct product factor
50	C₂×C₄	C₂⁴×C₂₂₃	1	120	yes	complemented
51	C₂²	—	2	480	no	—
52	C₂²	C₂²×D₄×C₂₂₃	1	140	yes	direct product factor
53	C₂²	C₂⁵×C₂₂₃	1	15	yes	—
54	C₂²	C₂⁵×C₂₂₃	1	32	yes	complemented
55	C₄	C₂⁵×C₂₂₃	1	16	yes	complemented
56	C₂	—	2	32	no	—
57	C₂	C₂³×D₄×C₂₂₃	1	30	yes	direct product factor
58	C₂	C₂⁶×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₂₂₃	C₂²
commutator, Frattini, cosocle	C₂	C₂⁶×C₂₂₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁶×C₂₂₃	C₂	semidirect
2. C₂³×D₄×C₂₂₃	C₂	direct (N × H)
3. C₂⁴×C₄×C₂₂₃	C₂	semidirect
6. C₂⁵×C₂₂₃	C₂²	semidirect
7. C₂²×D₄×C₂₂₃	C₂²	direct (N × H)
8. C₂³×C₄×C₂₂₃	C₂²	semidirect
10. C₂⁴×C₂₂₃	D₄	direct (N × H)
12. C₂⁴×C₂₂₃	C₂³	semidirect
13. C₂×D₄×C₂₂₃	C₂³	direct (N × H)
14. C₂²×C₄×C₂₂₃	C₂³	semidirect
16. C₂³×C₂₂₃	C₂×D₄	direct (N × H)
18. C₂³×C₂₂₃	C₂⁴	semidirect
19. D₄×C₂₂₃	C₂⁴	direct (N × H)
20. C₂×C₄×C₂₂₃	C₂⁴	semidirect
22. C₂²×C₂₂₃	C₂²×D₄	direct (N × H)
24. C₂²×C₂₂₃	C₂⁵	semidirect
25. C₄×C₂₂₃	C₂⁵	semidirect
27. C₂×C₂₂₃	C₂³×D₄	direct (N × H)
29. C₂₂₃	C₂⁴×D₄	direct (N × H)
30. C₂⁴×D₄	C₂₂₃	direct (N × H)
31. C₂⁶	C₂×C₂₂₃	semidirect
32. C₂³×D₄	C₂×C₂₂₃	direct (N × H)
33. C₂⁴×C₄	C₂×C₂₂₃	semidirect
36. C₂⁵	C₂²×C₂₂₃	semidirect
37. C₂²×D₄	C₂²×C₂₂₃	direct (N × H)
38. C₂³×C₄	C₂²×C₂₂₃	semidirect
40. C₂⁴	D₄×C₂₂₃	direct (N × H)
42. C₂⁴	C₂³×C₂₂₃	semidirect
43. C₂×D₄	C₂³×C₂₂₃	direct (N × H)
44. C₂²×C₄	C₂³×C₂₂₃	semidirect
46. C₂³	C₂×D₄×C₂₂₃	direct (N × H)
48. C₂³	C₂⁴×C₂₂₃	semidirect
49. D₄	C₂⁴×C₂₂₃	direct (N × H)
50. C₂×C₄	C₂⁴×C₂₂₃	semidirect
52. C₂²	C₂²×D₄×C₂₂₃	direct (N × H)
54. C₂²	C₂⁵×C₂₂₃	semidirect
55. C₄	C₂⁵×C₂₂₃	semidirect
57. C₂	C₂³×D₄×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×C₂₂₃	2
C₂³×D₄×C₂₂₃	2
C₂⁴×C₄×C₂₂₃	2
C₂⁴×D₄	223

Subgroups — C24×D4×C223 #43252603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₄×C₂₂₃ #43252603