Subgroups — C₂⁴×C₄×D₁₀₉ #28377821

Order 13952.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄×D₁₀₉	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₁₀₉	C₄	1	32	yes	direct product factor
6	C₂⁴×D₁₀₉	C₂²	1	30	yes	—
7	C₂³×Dic₁₀₉	C₂²	1	30	yes	complemented
8	C₂²×C₄×D₁₀₉	C₂²	1	560	yes	direct product factor
9	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₀₉	C₂²	1	30	yes	complemented
11	C₂³×D₁₀₉	C₂×C₄	1	480	yes	direct product factor
12	C₂³×D₁₀₉	C₂³	1	140	yes	—
13	C₂²×Dic₁₀₉	C₂³	1	140	yes	complemented
14	C₂×C₄×D₁₀₉	C₂³	1	960	yes	direct product factor
15	C₂⁴×C₁₀₉	C₂×C₄	1	16	yes	complemented
16	C₂⁴×C₁₀₉	C₂³	1	15	yes	—
17	C₂²×C₄×C₁₀₉	C₂³	1	140	yes	complemented
18	C₂³×C₁₀₉	C₂²×C₄	1	120	yes	complemented
19	C₂³×C₁₀₉	C₂⁴	1	35	yes	—
20	C₂²×D₁₀₉	C₂²×C₄	1	1120	yes	direct product factor
21	C₂²×D₁₀₉	C₂⁴	1	120	yes	—
22	C₂×C₄×C₁₀₉	C₂⁴	1	120	yes	complemented
23	C₂×Dic₁₀₉	C₂⁴	1	120	yes	complemented
24	C₄×D₁₀₉	C₂⁴	1	256	yes	direct product factor
25	C₂²×C₁₀₉	C₂³×C₄	1	140	yes	complemented
26	C₂²×C₁₀₉	C₂⁵	1	15	yes	—
27	C₂×D₁₀₉	C₂³×C₄	1	480	yes	direct product factor
28	C₂×D₁₀₉	C₂⁵	1	16	yes	—
29	C₄×C₁₀₉	C₂⁵	1	16	yes	complemented
30	Dic₁₀₉	C₂⁵	1	16	yes	complemented
31	C₂×C₁₀₉	C₂⁴×C₄	1	30	yes	complemented
32	C₂×C₁₀₉	C₂⁶	1	1	yes	characteristic
33	D₁₀₉	C₂⁴×C₄	1	32	yes	direct product factor
34	C₂⁵×C₄	—	109	1	no	maximal, Sylow
35	C₁₀₉	C₂⁵×C₄	1	1	yes	characteristic, complemented, Sylow
36	C₂⁶	—	109	1	no	—
37	C₂⁴×C₄	—	109	1	no	—
38	C₂⁴×C₄	—	109	60	no	—
39	C₂⁴×C₄	D₁₀₉	1	1	yes	characteristic, direct product factor
40	C₂⁵	—	109	32	no	—
41	C₂⁵	—	109	30	no	—
42	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic
43	C₂³×C₄	—	109	30	no	—
44	C₂³×C₄	—	109	560	no	—
45	C₂³×C₄	C₂×D₁₀₉	1	30	yes	direct product factor
46	C₂⁴	—	109	480	no	—
47	C₂⁴	—	109	140	no	—
48	C₂⁴	C₄×D₁₀₉	1	16	yes	direct product factor
49	C₂⁴	C₂²×D₁₀₉	1	15	yes	—
50	C₂²×C₄	—	109	140	no	—
51	C₂²×C₄	—	109	960	no	—
52	C₂²×C₄	C₂²×D₁₀₉	1	140	yes	direct product factor
53	C₂³	—	109	1120	no	—
54	C₂³	—	109	120	no	—
55	C₂³	C₂×C₄×D₁₀₉	1	120	yes	direct product factor
56	C₂³	C₂³×D₁₀₉	1	35	yes	—
57	C₂×C₄	—	109	120	no	—
58	C₂×C₄	—	109	256	no	—
59	C₂×C₄	C₂³×D₁₀₉	1	120	yes	direct product factor
60	C₂²	—	109	480	no	—
61	C₂²	—	109	16	no	—
62	C₂²	C₂²×C₄×D₁₀₉	1	140	yes	direct product factor
63	C₂²	C₂⁴×D₁₀₉	1	15	yes	—
64	C₄	—	109	16	no	—
65	C₄	C₂⁴×D₁₀₉	1	16	yes	direct product factor
66	C₂	—	109	32	no	—
67	C₂	C₂³×C₄×D₁₀₉	1	30	yes	direct product factor
68	C₂	C₂⁵×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	D₁₀₉
commutator, hypocenter	C₁₀₉	C₂⁵×C₄
Frattini	C₂	C₂⁵×D₁₀₉
Fitting	C₂⁴×C₄×C₁₀₉	C₂
socle	C₂⁵×C₁₀₉	C₂²
prime radical	C₂⁵×D₁₀₉	C₂
cosocle	C₂×C₁₀₉	C₂⁶
LCS	G ▷ C₁₀₉
UCS	C₁ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Dic₁₀₉	C₂	semidirect
3. C₂³×C₄×D₁₀₉	C₂	direct (N × H)
4. C₂⁴×C₄×C₁₀₉	C₂	semidirect
5. C₂⁴×D₁₀₉	C₄	direct (N × H)
7. C₂³×Dic₁₀₉	C₂²	semidirect
8. C₂²×C₄×D₁₀₉	C₂²	direct (N × H)
10. C₂³×C₄×C₁₀₉	C₂²	semidirect
11. C₂³×D₁₀₉	C₂×C₄	direct (N × H)
13. C₂²×Dic₁₀₉	C₂³	semidirect
14. C₂×C₄×D₁₀₉	C₂³	direct (N × H)
15. C₂⁴×C₁₀₉	C₂×C₄	semidirect
17. C₂²×C₄×C₁₀₉	C₂³	semidirect
18. C₂³×C₁₀₉	C₂²×C₄	semidirect
20. C₂²×D₁₀₉	C₂²×C₄	direct (N × H)
22. C₂×C₄×C₁₀₉	C₂⁴	semidirect
23. C₂×Dic₁₀₉	C₂⁴	semidirect
24. C₄×D₁₀₉	C₂⁴	direct (N × H)
25. C₂²×C₁₀₉	C₂³×C₄	semidirect
27. C₂×D₁₀₉	C₂³×C₄	direct (N × H)
29. C₄×C₁₀₉	C₂⁵	semidirect
30. Dic₁₀₉	C₂⁵	semidirect
31. C₂×C₁₀₉	C₂⁴×C₄	semidirect
33. D₁₀₉	C₂⁴×C₄	direct (N × H)
35. C₁₀₉	C₂⁵×C₄	semidirect
39. C₂⁴×C₄	D₁₀₉	direct (N × H)
45. C₂³×C₄	C₂×D₁₀₉	direct (N × H)
48. C₂⁴	C₄×D₁₀₉	direct (N × H)
52. C₂²×C₄	C₂²×D₁₀₉	direct (N × H)
55. C₂³	C₂×C₄×D₁₀₉	direct (N × H)
59. C₂×C₄	C₂³×D₁₀₉	direct (N × H)
62. C₂²	C₂²×C₄×D₁₀₉	direct (N × H)
65. C₄	C₂⁴×D₁₀₉	direct (N × H)
67. C₂	C₂³×C₄×D₁₀₉	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×D₁₀₉	2
C₂⁴×Dic₁₀₉	2
C₂³×C₄×D₁₀₉	2
C₂⁴×C₄×C₁₀₉	2
C₂⁵×C₄	109

Subgroups — C24×C4×D109 #28377821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄×D₁₀₉ #28377821