Subgroups — C₂⁴×D₅₅₆ #31821197

Order 17792.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₁₃₉	C₂	1	2	yes	maximal, complemented
2	C₂³×D₅₅₆	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₁₃₉	—	2	32	no	—
5	C₂⁴×D₁₃₉	C₂²	1	60	yes	complemented
6	C₂²×D₅₅₆	C₂²	1	560	yes	direct product factor
7	C₂⁵×C₁₃₉	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₃₉	C₂²	1	30	yes	complemented
9	C₂³×D₁₃₉	—	2	480	no	—
10	C₂³×D₁₃₉	C₂³	1	280	yes	complemented
11	C₂×D₅₅₆	C₂³	1	960	yes	direct product factor
12	C₂⁴×C₁₃₉	D₄	1	16	yes	complemented
13	C₂⁴×C₁₃₉	C₂³	1	15	yes	—
14	C₂²×C₄×C₁₃₉	C₂³	1	140	yes	complemented
15	C₂²×D₁₃₉	—	2	1120	no	—
16	C₂²×D₁₃₉	C₂⁴	1	240	yes	complemented
17	D₅₅₆	C₂⁴	1	256	yes	direct product factor
18	C₂³×C₁₃₉	C₂×D₄	1	120	yes	complemented
19	C₂³×C₁₃₉	C₂⁴	1	35	yes	—
20	C₂×C₄×C₁₃₉	C₂⁴	1	120	yes	complemented
21	C₂²×C₁₃₉	C₂²×D₄	1	140	yes	complemented
22	C₂²×C₁₃₉	C₂⁵	1	15	yes	—
23	C₂×D₁₃₉	—	2	480	no	—
24	C₂×D₁₃₉	C₂⁵	1	32	yes	complemented
25	C₄×C₁₃₉	C₂⁵	1	16	yes	complemented
26	C₂×C₁₃₉	C₂³×D₄	1	30	yes	complemented
27	C₂×C₁₃₉	C₂⁶	1	1	yes	characteristic
28	D₁₃₉	—	2	32	no	—
29	C₁₃₉	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
30	C₂⁴×D₄	—	139	1	no	maximal, Sylow
31	C₂⁶	—	139	2	no	—
32	C₂³×D₄	—	139	60	no	—
33	C₂⁴×C₄	D₁₃₉	1	1	yes	characteristic, complemented
34	C₂⁵	—	278	32	no	—
35	C₂⁵	—	139	60	no	—
36	C₂⁵	C₂×D₁₃₉	1	1	yes	characteristic
37	C₂²×D₄	—	139	560	no	—
38	C₂³×C₄	C₂×D₁₃₉	1	30	yes	complemented
39	C₂⁴	—	278	480	no	—
40	C₂⁴	—	139	280	no	—
41	C₂⁴	D₅₅₆	1	16	yes	direct product factor
42	C₂⁴	C₂²×D₁₃₉	1	15	yes	—
43	C₂×D₄	—	139	960	no	—
44	C₂²×C₄	C₂²×D₁₃₉	1	140	yes	complemented
45	C₂³	—	278	1120	no	—
46	C₂³	—	139	240	no	—
47	C₂³	C₂×D₅₅₆	1	120	yes	direct product factor
48	C₂³	C₂³×D₁₃₉	1	35	yes	—
49	D₄	—	139	256	no	—
50	C₂×C₄	C₂³×D₁₃₉	1	120	yes	complemented
51	C₂²	—	278	480	no	—
52	C₂²	—	139	32	no	—
53	C₂²	C₂²×D₅₅₆	1	140	yes	direct product factor
54	C₂²	C₂⁴×D₁₃₉	1	15	yes	—
55	C₄	C₂⁴×D₁₃₉	1	16	yes	complemented
56	C₂	—	278	32	no	—
57	C₂	C₂³×D₅₅₆	1	30	yes	direct product factor
58	C₂	C₂⁵×D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₁₃₉
commutator, cosocle	C₂×C₁₃₉	C₂⁶
Frattini	C₂	C₂⁵×D₁₃₉
Fitting	C₂⁴×C₄×C₁₃₉	C₂
socle	C₂⁵×C₁₃₉	C₂²
hypercenter	C₂⁴×C₄	D₁₃₉
hypocenter	C₁₃₉	C₂⁴×D₄
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₁₃₉	C₂	semidirect
2. C₂³×D₅₅₆	C₂	direct (N × H)
3. C₂⁴×C₄×C₁₃₉	C₂	semidirect
5. C₂⁴×D₁₃₉	C₂²	semidirect
6. C₂²×D₅₅₆	C₂²	direct (N × H)
8. C₂³×C₄×C₁₃₉	C₂²	semidirect
10. C₂³×D₁₃₉	C₂³	semidirect
11. C₂×D₅₅₆	C₂³	direct (N × H)
12. C₂⁴×C₁₃₉	D₄	semidirect
14. C₂²×C₄×C₁₃₉	C₂³	semidirect
16. C₂²×D₁₃₉	C₂⁴	semidirect
17. D₅₅₆	C₂⁴	direct (N × H)
18. C₂³×C₁₃₉	C₂×D₄	semidirect
20. C₂×C₄×C₁₃₉	C₂⁴	semidirect
21. C₂²×C₁₃₉	C₂²×D₄	semidirect
24. C₂×D₁₃₉	C₂⁵	semidirect
25. C₄×C₁₃₉	C₂⁵	semidirect
26. C₂×C₁₃₉	C₂³×D₄	semidirect
29. C₁₃₉	C₂⁴×D₄	semidirect
33. C₂⁴×C₄	D₁₃₉	semidirect
38. C₂³×C₄	C₂×D₁₃₉	semidirect
41. C₂⁴	D₅₅₆	direct (N × H)
44. C₂²×C₄	C₂²×D₁₃₉	semidirect
47. C₂³	C₂×D₅₅₆	direct (N × H)
50. C₂×C₄	C₂³×D₁₃₉	semidirect
53. C₂²	C₂²×D₅₅₆	direct (N × H)
55. C₄	C₂⁴×D₁₃₉	semidirect
57. C₂	C₂³×D₅₅₆	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×D₁₃₉	2
C₂³×D₅₅₆	2
C₂⁴×C₄×C₁₃₉	2
C₂⁴×D₄	139

Subgroups — C24×D556 #31821197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₅₅₆ #31821197