Subgroups — C₂⁴×D₁₂₄ #16860504

Order 3968.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₃₁	C₂	1	2	yes	maximal, complemented
2	C₂³×D₁₂₄	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
5	C₂⁴×D₃₁	—	2	32	no	—
6	C₂⁴×D₃₁	C₂²	1	60	yes	complemented
7	C₂³×C₄×C₃₁	C₂²	1	30	yes	complemented
8	C₂²×D₁₂₄	C₂²	1	560	yes	direct product factor
9	C₂⁴×C₃₁	D₄	1	16	yes	complemented
10	C₂⁴×C₃₁	C₂³	1	15	yes	—
11	C₂³×D₃₁	—	2	480	no	—
12	C₂³×D₃₁	C₂³	1	280	yes	complemented
13	C₂²×C₄×C₃₁	C₂³	1	140	yes	complemented
14	C₂×D₁₂₄	C₂³	1	960	yes	direct product factor
15	C₂³×C₃₁	C₂×D₄	1	120	yes	complemented
16	C₂³×C₃₁	C₂⁴	1	35	yes	—
17	C₂²×D₃₁	—	2	1120	no	—
18	C₂²×D₃₁	C₂⁴	1	240	yes	complemented
19	C₂×C₄×C₃₁	C₂⁴	1	120	yes	complemented
20	D₁₂₄	C₂⁴	1	256	yes	direct product factor
21	C₂⁴×D₄	—	31	1	no	maximal, Sylow
22	C₂²×C₃₁	C₂²×D₄	1	140	yes	complemented
23	C₂²×C₃₁	C₂⁵	1	15	yes	—
24	C₂×D₃₁	—	2	480	no	—
25	C₂×D₃₁	C₂⁵	1	32	yes	complemented
26	C₄×C₃₁	C₂⁵	1	16	yes	complemented
27	C₂⁶	—	31	2	no	—
28	C₂³×D₄	—	31	60	no	—
29	C₂⁴×C₄	D₃₁	1	1	yes	characteristic, complemented
30	C₂×C₃₁	C₂³×D₄	1	30	yes	complemented
31	C₂×C₃₁	C₂⁶	1	1	yes	characteristic
32	D₃₁	—	2	32	no	—
33	C₂⁵	—	31	60	no	—
34	C₂⁵	—	62	32	no	—
35	C₂⁵	C₂×D₃₁	1	1	yes	characteristic
36	C₂²×D₄	—	31	560	no	—
37	C₂³×C₄	C₂×D₃₁	1	30	yes	complemented
38	C₃₁	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
39	C₂⁴	—	31	280	no	—
40	C₂⁴	—	62	480	no	—
41	C₂⁴	D₁₂₄	1	16	yes	direct product factor
42	C₂⁴	C₂²×D₃₁	1	15	yes	—
43	C₂×D₄	—	31	960	no	—
44	C₂²×C₄	C₂²×D₃₁	1	140	yes	complemented
45	C₂³	—	31	240	no	—
46	C₂³	—	62	1120	no	—
47	C₂³	C₂×D₁₂₄	1	120	yes	direct product factor
48	C₂³	C₂³×D₃₁	1	35	yes	—
49	D₄	—	31	256	no	—
50	C₂×C₄	C₂³×D₃₁	1	120	yes	complemented
51	C₂²	—	31	32	no	—
52	C₂²	—	62	480	no	—
53	C₂²	C₂²×D₁₂₄	1	140	yes	direct product factor
54	C₂²	C₂⁴×D₃₁	1	15	yes	—
55	C₄	C₂⁴×D₃₁	1	16	yes	complemented
56	C₂	—	62	32	no	—
57	C₂	C₂³×D₁₂₄	1	30	yes	direct product factor
58	C₂	C₂⁵×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₃₁
commutator, cosocle	C₂×C₃₁	C₂⁶
Frattini	C₂	C₂⁵×D₃₁
Fitting	C₂⁴×C₄×C₃₁	C₂
socle	C₂⁵×C₃₁	C₂²
hypercenter	C₂⁴×C₄	D₃₁
hypocenter	C₃₁	C₂⁴×D₄
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₃₁	C₂	semidirect
2. C₂³×D₁₂₄	C₂	direct (N × H)
3. C₂⁴×C₄×C₃₁	C₂	semidirect
6. C₂⁴×D₃₁	C₂²	semidirect
7. C₂³×C₄×C₃₁	C₂²	semidirect
8. C₂²×D₁₂₄	C₂²	direct (N × H)
9. C₂⁴×C₃₁	D₄	semidirect
12. C₂³×D₃₁	C₂³	semidirect
13. C₂²×C₄×C₃₁	C₂³	semidirect
14. C₂×D₁₂₄	C₂³	direct (N × H)
15. C₂³×C₃₁	C₂×D₄	semidirect
18. C₂²×D₃₁	C₂⁴	semidirect
19. C₂×C₄×C₃₁	C₂⁴	semidirect
20. D₁₂₄	C₂⁴	direct (N × H)
22. C₂²×C₃₁	C₂²×D₄	semidirect
25. C₂×D₃₁	C₂⁵	semidirect
26. C₄×C₃₁	C₂⁵	semidirect
29. C₂⁴×C₄	D₃₁	semidirect
30. C₂×C₃₁	C₂³×D₄	semidirect
37. C₂³×C₄	C₂×D₃₁	semidirect
38. C₃₁	C₂⁴×D₄	semidirect
41. C₂⁴	D₁₂₄	direct (N × H)
44. C₂²×C₄	C₂²×D₃₁	semidirect
47. C₂³	C₂×D₁₂₄	direct (N × H)
50. C₂×C₄	C₂³×D₃₁	semidirect
53. C₂²	C₂²×D₁₂₄	direct (N × H)
55. C₄	C₂⁴×D₃₁	semidirect
57. C₂	C₂³×D₁₂₄	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×D₃₁	2
C₂³×D₁₂₄	2
C₂⁴×C₄×C₃₁	2
C₂⁴×D₄	31

Subgroups — C24×D124 #16860504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₁₂₄ #16860504