Subgroups — C₂⁵×D₂₁₂ #28297970

Order 13568.

111198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂⁴×D₂₁₂	C₂	1	124	yes	maximal, direct product factor
3	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×D₅₃	—	2	64	no	—
5	C₂⁵×D₅₃	C₂²	1	124	yes	complemented
6	C₂³×D₂₁₂	C₂²	1	2480	yes	direct product factor
7	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₅₃	C₂²	1	62	yes	complemented
9	C₂⁴×D₅₃	—	2	1984	no	—
10	C₂⁴×D₅₃	C₂³	1	1240	yes	complemented
11	C₂²×D₂₁₂	C₂³	1	9920	yes	direct product factor
12	C₂⁵×C₅₃	D₄	1	32	yes	complemented
13	C₂⁵×C₅₃	C₂³	1	31	yes	—
14	C₂³×C₄×C₅₃	C₂³	1	620	yes	complemented
15	C₂⁴×C₅₃	C₂×D₄	1	496	yes	complemented
16	C₂⁴×C₅₃	C₂⁴	1	155	yes	—
17	C₂³×D₅₃	—	2	9920	no	—
18	C₂³×D₅₃	C₂⁴	1	2480	yes	complemented
19	C₂²×C₄×C₅₃	C₂⁴	1	1240	yes	complemented
20	C₂×D₂₁₂	C₂⁴	1	7936	yes	direct product factor
21	C₂³×C₅₃	C₂²×D₄	1	1240	yes	complemented
22	C₂³×C₅₃	C₂⁵	1	155	yes	—
23	C₂²×D₅₃	—	2	9920	no	—
24	C₂²×D₅₃	C₂⁵	1	992	yes	complemented
25	C₂×C₄×C₅₃	C₂⁵	1	496	yes	complemented
26	D₂₁₂	C₂⁵	1	1024	yes	direct product factor
27	C₂⁵×D₄	—	53	1	no	maximal, Sylow
28	C₂²×C₅₃	C₂³×D₄	1	620	yes	complemented
29	C₂²×C₅₃	C₂⁶	1	31	yes	—
30	C₂×D₅₃	—	2	1984	no	—
31	C₂×D₅₃	C₂⁶	1	64	yes	complemented
32	C₄×C₅₃	C₂⁶	1	32	yes	complemented
33	C₂⁷	—	53	2	no	—
34	C₂⁴×D₄	—	53	124	no	—
35	C₂⁵×C₄	D₅₃	1	1	yes	characteristic, complemented
36	C₂×C₅₃	C₂⁴×D₄	1	62	yes	complemented
37	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
38	D₅₃	—	2	64	no	—
39	C₂⁶	—	53	124	no	—
40	C₂⁶	—	106	64	no	—
41	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
42	C₂³×D₄	—	53	2480	no	—
43	C₂⁴×C₄	C₂×D₅₃	1	62	yes	complemented
44	C₅₃	C₂⁵×D₄	1	1	yes	characteristic, complemented, Sylow
45	C₂⁵	—	53	1240	no	—
46	C₂⁵	—	106	1984	no	—
47	C₂⁵	D₂₁₂	1	32	yes	direct product factor
48	C₂⁵	C₂²×D₅₃	1	31	yes	—
49	C₂²×D₄	—	53	9920	no	—
50	C₂³×C₄	C₂²×D₅₃	1	620	yes	complemented
51	C₂⁴	—	53	2480	no	—
52	C₂⁴	—	106	9920	no	—
53	C₂⁴	C₂×D₂₁₂	1	496	yes	direct product factor
54	C₂⁴	C₂³×D₅₃	1	155	yes	—
55	C₂×D₄	—	53	7936	no	—
56	C₂²×C₄	C₂³×D₅₃	1	1240	yes	complemented
57	C₂³	—	53	992	no	—
58	C₂³	—	106	9920	no	—
59	C₂³	C₂²×D₂₁₂	1	1240	yes	direct product factor
60	C₂³	C₂⁴×D₅₃	1	155	yes	—
61	D₄	—	53	1024	no	—
62	C₂×C₄	C₂⁴×D₅₃	1	496	yes	complemented
63	C₂²	—	53	64	no	—
64	C₂²	—	106	1984	no	—
65	C₂²	C₂³×D₂₁₂	1	620	yes	direct product factor
66	C₂²	C₂⁵×D₅₃	1	31	yes	—
67	C₄	C₂⁵×D₅₃	1	32	yes	complemented
68	C₂	—	106	64	no	—
69	C₂	C₂⁴×D₂₁₂	1	62	yes	direct product factor
70	C₂	C₂⁶×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁶	C₂×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂⁵×C₄×C₅₃	C₂
socle	C₂⁶×C₅₃	C₂²
hypercenter	C₂⁵×C₄	D₅₃
hypocenter	C₅₃	C₂⁵×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁶ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₅₃	C₂	semidirect
2. C₂⁴×D₂₁₂	C₂	direct (N × H)
3. C₂⁵×C₄×C₅₃	C₂	semidirect
5. C₂⁵×D₅₃	C₂²	semidirect
6. C₂³×D₂₁₂	C₂²	direct (N × H)
8. C₂⁴×C₄×C₅₃	C₂²	semidirect
10. C₂⁴×D₅₃	C₂³	semidirect
11. C₂²×D₂₁₂	C₂³	direct (N × H)
12. C₂⁵×C₅₃	D₄	semidirect
14. C₂³×C₄×C₅₃	C₂³	semidirect
15. C₂⁴×C₅₃	C₂×D₄	semidirect
18. C₂³×D₅₃	C₂⁴	semidirect
19. C₂²×C₄×C₅₃	C₂⁴	semidirect
20. C₂×D₂₁₂	C₂⁴	direct (N × H)
21. C₂³×C₅₃	C₂²×D₄	semidirect
24. C₂²×D₅₃	C₂⁵	semidirect
25. C₂×C₄×C₅₃	C₂⁵	semidirect
26. D₂₁₂	C₂⁵	direct (N × H)
28. C₂²×C₅₃	C₂³×D₄	semidirect
31. C₂×D₅₃	C₂⁶	semidirect
32. C₄×C₅₃	C₂⁶	semidirect
35. C₂⁵×C₄	D₅₃	semidirect
36. C₂×C₅₃	C₂⁴×D₄	semidirect
43. C₂⁴×C₄	C₂×D₅₃	semidirect
44. C₅₃	C₂⁵×D₄	semidirect
47. C₂⁵	D₂₁₂	direct (N × H)
50. C₂³×C₄	C₂²×D₅₃	semidirect
53. C₂⁴	C₂×D₂₁₂	direct (N × H)
56. C₂²×C₄	C₂³×D₅₃	semidirect
59. C₂³	C₂²×D₂₁₂	direct (N × H)
62. C₂×C₄	C₂⁴×D₅₃	semidirect
65. C₂²	C₂³×D₂₁₂	direct (N × H)
67. C₄	C₂⁵×D₅₃	semidirect
69. C₂	C₂⁴×D₂₁₂	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×D₅₃	2
C₂⁴×D₂₁₂	2
C₂⁵×C₄×C₅₃	2
C₂⁵×D₄	53

Subgroups — C25×D212 #28297970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×D₂₁₂ #28297970