Subgroups — C₂⁵×D₂₆₈ #31769709

Order 17152.

111198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₆₇	C₂	1	2	yes	maximal, complemented
2	C₂⁴×D₂₆₈	C₂	1	124	yes	maximal, direct product factor
3	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×D₆₇	—	2	64	no	—
5	C₂⁵×D₆₇	C₂²	1	124	yes	complemented
6	C₂³×D₂₆₈	C₂²	1	2480	yes	direct product factor
7	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	C₂²	1	62	yes	complemented
9	C₂⁴×D₆₇	—	2	1984	no	—
10	C₂⁴×D₆₇	C₂³	1	1240	yes	complemented
11	C₂²×D₂₆₈	C₂³	1	9920	yes	direct product factor
12	C₂⁵×C₆₇	D₄	1	32	yes	complemented
13	C₂⁵×C₆₇	C₂³	1	31	yes	—
14	C₂³×C₄×C₆₇	C₂³	1	620	yes	complemented
15	C₂³×D₆₇	—	2	9920	no	—
16	C₂³×D₆₇	C₂⁴	1	2480	yes	complemented
17	C₂×D₂₆₈	C₂⁴	1	7936	yes	direct product factor
18	C₂⁴×C₆₇	C₂×D₄	1	496	yes	complemented
19	C₂⁴×C₆₇	C₂⁴	1	155	yes	—
20	C₂²×C₄×C₆₇	C₂⁴	1	1240	yes	complemented
21	C₂³×C₆₇	C₂²×D₄	1	1240	yes	complemented
22	C₂³×C₆₇	C₂⁵	1	155	yes	—
23	C₂²×D₆₇	—	2	9920	no	—
24	C₂²×D₆₇	C₂⁵	1	992	yes	complemented
25	C₂×C₄×C₆₇	C₂⁵	1	496	yes	complemented
26	D₂₆₈	C₂⁵	1	1024	yes	direct product factor
27	C₂²×C₆₇	C₂³×D₄	1	620	yes	complemented
28	C₂²×C₆₇	C₂⁶	1	31	yes	—
29	C₂×D₆₇	—	2	1984	no	—
30	C₂×D₆₇	C₂⁶	1	64	yes	complemented
31	C₄×C₆₇	C₂⁶	1	32	yes	complemented
32	C₂⁵×D₄	—	67	1	no	maximal, Sylow
33	C₂×C₆₇	C₂⁴×D₄	1	62	yes	complemented
34	C₂×C₆₇	C₂⁷	1	1	yes	characteristic
35	D₆₇	—	2	64	no	—
36	C₂⁷	—	67	2	no	—
37	C₂⁴×D₄	—	67	124	no	—
38	C₂⁵×C₄	D₆₇	1	1	yes	characteristic, complemented
39	C₆₇	C₂⁵×D₄	1	1	yes	characteristic, complemented, Sylow
40	C₂⁶	—	67	124	no	—
41	C₂⁶	—	134	64	no	—
42	C₂⁶	C₂×D₆₇	1	1	yes	characteristic
43	C₂³×D₄	—	67	2480	no	—
44	C₂⁴×C₄	C₂×D₆₇	1	62	yes	complemented
45	C₂⁵	—	67	1240	no	—
46	C₂⁵	—	134	1984	no	—
47	C₂⁵	D₂₆₈	1	32	yes	direct product factor
48	C₂⁵	C₂²×D₆₇	1	31	yes	—
49	C₂²×D₄	—	67	9920	no	—
50	C₂³×C₄	C₂²×D₆₇	1	620	yes	complemented
51	C₂⁴	—	67	2480	no	—
52	C₂⁴	—	134	9920	no	—
53	C₂⁴	C₂×D₂₆₈	1	496	yes	direct product factor
54	C₂⁴	C₂³×D₆₇	1	155	yes	—
55	C₂×D₄	—	67	7936	no	—
56	C₂²×C₄	C₂³×D₆₇	1	1240	yes	complemented
57	C₂³	—	67	992	no	—
58	C₂³	—	134	9920	no	—
59	C₂³	C₂²×D₂₆₈	1	1240	yes	direct product factor
60	C₂³	C₂⁴×D₆₇	1	155	yes	—
61	D₄	—	67	1024	no	—
62	C₂×C₄	C₂⁴×D₆₇	1	496	yes	complemented
63	C₂²	—	67	64	no	—
64	C₂²	—	134	1984	no	—
65	C₂²	C₂³×D₂₆₈	1	620	yes	direct product factor
66	C₂²	C₂⁵×D₆₇	1	31	yes	—
67	C₄	C₂⁵×D₆₇	1	32	yes	complemented
68	C₂	—	134	64	no	—
69	C₂	C₂⁴×D₂₆₈	1	62	yes	direct product factor
70	C₂	C₂⁶×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁶	C₂×D₆₇
commutator, cosocle	C₂×C₆₇	C₂⁷
Frattini	C₂	C₂⁶×D₆₇
Fitting	C₂⁵×C₄×C₆₇	C₂
socle	C₂⁶×C₆₇	C₂²
hypercenter	C₂⁵×C₄	D₆₇
hypocenter	C₆₇	C₂⁵×D₄
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁶ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₆₇	C₂	semidirect
2. C₂⁴×D₂₆₈	C₂	direct (N × H)
3. C₂⁵×C₄×C₆₇	C₂	semidirect
5. C₂⁵×D₆₇	C₂²	semidirect
6. C₂³×D₂₆₈	C₂²	direct (N × H)
8. C₂⁴×C₄×C₆₇	C₂²	semidirect
10. C₂⁴×D₆₇	C₂³	semidirect
11. C₂²×D₂₆₈	C₂³	direct (N × H)
12. C₂⁵×C₆₇	D₄	semidirect
14. C₂³×C₄×C₆₇	C₂³	semidirect
16. C₂³×D₆₇	C₂⁴	semidirect
17. C₂×D₂₆₈	C₂⁴	direct (N × H)
18. C₂⁴×C₆₇	C₂×D₄	semidirect
20. C₂²×C₄×C₆₇	C₂⁴	semidirect
21. C₂³×C₆₇	C₂²×D₄	semidirect
24. C₂²×D₆₇	C₂⁵	semidirect
25. C₂×C₄×C₆₇	C₂⁵	semidirect
26. D₂₆₈	C₂⁵	direct (N × H)
27. C₂²×C₆₇	C₂³×D₄	semidirect
30. C₂×D₆₇	C₂⁶	semidirect
31. C₄×C₆₇	C₂⁶	semidirect
33. C₂×C₆₇	C₂⁴×D₄	semidirect
38. C₂⁵×C₄	D₆₇	semidirect
39. C₆₇	C₂⁵×D₄	semidirect
44. C₂⁴×C₄	C₂×D₆₇	semidirect
47. C₂⁵	D₂₆₈	direct (N × H)
50. C₂³×C₄	C₂²×D₆₇	semidirect
53. C₂⁴	C₂×D₂₆₈	direct (N × H)
56. C₂²×C₄	C₂³×D₆₇	semidirect
59. C₂³	C₂²×D₂₆₈	direct (N × H)
62. C₂×C₄	C₂⁴×D₆₇	semidirect
65. C₂²	C₂³×D₂₆₈	direct (N × H)
67. C₄	C₂⁵×D₆₇	semidirect
69. C₂	C₂⁴×D₂₆₈	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×D₆₇	2
C₂⁴×D₂₆₈	2
C₂⁵×C₄×C₆₇	2
C₂⁵×D₄	67

Subgroups — C25×D268 #31769709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×D₂₆₈ #31769709