Subgroups — C₂⁵×D₂₈ #14267674

Order 1792.

111198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₇	C₂	1	2	yes	maximal, complemented
2	C₂⁵×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×D₂₈	C₂	1	124	yes	maximal, direct product factor
4	C₂⁶×C₇	C₂²	1	1	yes	characteristic
5	C₂⁵×D₇	—	2	64	no	—
6	C₂⁵×D₇	C₂²	1	124	yes	complemented
7	C₂⁴×C₄×C₇	C₂²	1	62	yes	complemented
8	C₂³×D₂₈	C₂²	1	2480	yes	direct product factor
9	C₂⁵×D₄	—	7	1	no	maximal, Sylow
10	C₂⁵×C₇	D₄	1	32	yes	complemented
11	C₂⁵×C₇	C₂³	1	31	yes	—
12	C₂⁴×D₇	—	2	1984	no	—
13	C₂⁴×D₇	C₂³	1	1240	yes	complemented
14	C₂³×C₄×C₇	C₂³	1	620	yes	complemented
15	C₂²×D₂₈	C₂³	1	9920	yes	direct product factor
16	C₂⁷	—	7	2	no	—
17	C₂⁴×D₄	—	7	124	no	—
18	C₂⁵×C₄	D₇	1	1	yes	characteristic, complemented
19	C₂⁴×C₇	C₂×D₄	1	496	yes	complemented
20	C₂⁴×C₇	C₂⁴	1	155	yes	—
21	C₂³×D₇	—	2	9920	no	—
22	C₂³×D₇	C₂⁴	1	2480	yes	complemented
23	C₂²×C₄×C₇	C₂⁴	1	1240	yes	complemented
24	C₂×D₂₈	C₂⁴	1	7936	yes	direct product factor
25	C₂⁶	—	7	124	no	—
26	C₂⁶	—	14	64	no	—
27	C₂⁶	C₂×D₇	1	1	yes	characteristic
28	C₂³×D₄	—	7	2480	no	—
29	C₂⁴×C₄	C₂×D₇	1	62	yes	complemented
30	C₂³×C₇	C₂²×D₄	1	1240	yes	complemented
31	C₂³×C₇	C₂⁵	1	155	yes	—
32	C₂²×D₇	—	2	9920	no	—
33	C₂²×D₇	C₂⁵	1	992	yes	complemented
34	C₂×C₄×C₇	C₂⁵	1	496	yes	complemented
35	D₂₈	C₂⁵	1	1024	yes	direct product factor
36	C₂⁵	—	7	1240	no	—
37	C₂⁵	—	14	1984	no	—
38	C₂⁵	D₂₈	1	32	yes	direct product factor
39	C₂⁵	C₂²×D₇	1	31	yes	—
40	C₂²×D₄	—	7	9920	no	—
41	C₂³×C₄	C₂²×D₇	1	620	yes	complemented
42	C₂²×C₇	C₂³×D₄	1	620	yes	complemented
43	C₂²×C₇	C₂⁶	1	31	yes	—
44	C₂×D₇	—	2	1984	no	—
45	C₂×D₇	C₂⁶	1	64	yes	complemented
46	C₄×C₇	C₂⁶	1	32	yes	complemented
47	C₂⁴	—	7	2480	no	—
48	C₂⁴	—	14	9920	no	—
49	C₂⁴	C₂×D₂₈	1	496	yes	direct product factor
50	C₂⁴	C₂³×D₇	1	155	yes	—
51	C₂×D₄	—	7	7936	no	—
52	C₂²×C₄	C₂³×D₇	1	1240	yes	complemented
53	C₂×C₇	C₂⁴×D₄	1	62	yes	complemented
54	C₂×C₇	C₂⁷	1	1	yes	characteristic
55	D₇	—	2	64	no	—
56	C₂³	—	7	992	no	—
57	C₂³	—	14	9920	no	—
58	C₂³	C₂²×D₂₈	1	1240	yes	direct product factor
59	C₂³	C₂⁴×D₇	1	155	yes	—
60	D₄	—	7	1024	no	—
61	C₂×C₄	C₂⁴×D₇	1	496	yes	complemented
62	C₇	C₂⁵×D₄	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	7	64	no	—
64	C₂²	—	14	1984	no	—
65	C₂²	C₂³×D₂₈	1	620	yes	direct product factor
66	C₂²	C₂⁵×D₇	1	31	yes	—
67	C₄	C₂⁵×D₇	1	32	yes	complemented
68	C₂	—	14	64	no	—
69	C₂	C₂⁴×D₂₈	1	62	yes	direct product factor
70	C₂	C₂⁶×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁶	C₂×D₇
commutator, cosocle	C₂×C₇	C₂⁷
Frattini	C₂	C₂⁶×D₇
Fitting	C₂⁵×C₄×C₇	C₂
socle	C₂⁶×C₇	C₂²
hypercenter	C₂⁵×C₄	D₇
hypocenter	C₇	C₂⁵×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁶ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₇	C₂	semidirect
2. C₂⁵×C₄×C₇	C₂	semidirect
3. C₂⁴×D₂₈	C₂	direct (N × H)
6. C₂⁵×D₇	C₂²	semidirect
7. C₂⁴×C₄×C₇	C₂²	semidirect
8. C₂³×D₂₈	C₂²	direct (N × H)
10. C₂⁵×C₇	D₄	semidirect
13. C₂⁴×D₇	C₂³	semidirect
14. C₂³×C₄×C₇	C₂³	semidirect
15. C₂²×D₂₈	C₂³	direct (N × H)
18. C₂⁵×C₄	D₇	semidirect
19. C₂⁴×C₇	C₂×D₄	semidirect
22. C₂³×D₇	C₂⁴	semidirect
23. C₂²×C₄×C₇	C₂⁴	semidirect
24. C₂×D₂₈	C₂⁴	direct (N × H)
29. C₂⁴×C₄	C₂×D₇	semidirect
30. C₂³×C₇	C₂²×D₄	semidirect
33. C₂²×D₇	C₂⁵	semidirect
34. C₂×C₄×C₇	C₂⁵	semidirect
35. D₂₈	C₂⁵	direct (N × H)
38. C₂⁵	D₂₈	direct (N × H)
41. C₂³×C₄	C₂²×D₇	semidirect
42. C₂²×C₇	C₂³×D₄	semidirect
45. C₂×D₇	C₂⁶	semidirect
46. C₄×C₇	C₂⁶	semidirect
49. C₂⁴	C₂×D₂₈	direct (N × H)
52. C₂²×C₄	C₂³×D₇	semidirect
53. C₂×C₇	C₂⁴×D₄	semidirect
58. C₂³	C₂²×D₂₈	direct (N × H)
61. C₂×C₄	C₂⁴×D₇	semidirect
62. C₇	C₂⁵×D₄	semidirect
65. C₂²	C₂³×D₂₈	direct (N × H)
67. C₄	C₂⁵×D₇	semidirect
69. C₂	C₂⁴×D₂₈	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×D₇	2
C₂⁵×C₄×C₇	2
C₂⁴×D₂₈	2
C₂⁵×D₄	7

Subgroups — C25×D28 #14267674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×D₂₈ #14267674