Subgroups — C₂⁴×C₂²⋊D₁₀₉ #28377824

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, complemented
2	C₂³×C₂²⋊D₁₀₉	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁶×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₁₀₉	—	2	16	no	—
6	C₂⁴×D₁₀₉	C₂²	1	30	yes	complemented
7	C₂²×C₂²⋊D₁₀₉	C₂²	1	560	yes	direct product factor
8	C₂³×Dic₁₀₉	C₂²	1	30	yes	complemented
9	C₂⁵×C₁₀₉	—	2	16	no	—
10	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂⁵×C₁₀₉	C₂²	1	30	yes	complemented
12	C₂³×D₁₀₉	—	2	240	no	—
13	C₂³×D₁₀₉	C₂³	1	140	yes	complemented
14	C₂×C₂²⋊D₁₀₉	C₂³	1	960	yes	direct product factor
15	C₂²×Dic₁₀₉	C₂³	1	140	yes	complemented
16	C₂⁴×C₁₀₉	—	2	240	no	—
17	C₂⁴×C₁₀₉	D₄	1	16	yes	complemented
18	C₂⁴×C₁₀₉	C₂³	1	15	yes	—
19	C₂⁴×C₁₀₉	C₂³	1	140	yes	complemented
20	C₂³×C₁₀₉	—	2	560	no	—
21	C₂³×C₁₀₉	C₂×D₄	1	120	yes	complemented
22	C₂³×C₁₀₉	C₂⁴	1	35	yes	—
23	C₂³×C₁₀₉	C₂⁴	1	120	yes	complemented
24	C₂²×D₁₀₉	—	2	560	no	—
25	C₂²×D₁₀₉	C₂⁴	1	120	yes	complemented
26	C₂²⋊D₁₀₉	C₂⁴	1	256	yes	direct product factor
27	C₂×Dic₁₀₉	C₂⁴	1	120	yes	complemented
28	C₂²×C₁₀₉	—	2	240	no	—
29	C₂²×C₁₀₉	C₂²×D₄	1	140	yes	complemented
30	C₂²×C₁₀₉	C₂⁵	1	15	yes	—
31	C₂²×C₁₀₉	C₂⁵	1	16	yes	complemented
32	C₂×D₁₀₉	—	2	240	no	—
33	C₂×D₁₀₉	C₂⁵	1	16	yes	complemented
34	Dic₁₀₉	C₂⁵	1	16	yes	complemented
35	C₂×C₁₀₉	—	2	16	no	—
36	C₂×C₁₀₉	C₂³×D₄	1	30	yes	complemented
37	C₂×C₁₀₉	C₂⁶	1	1	yes	characteristic
38	D₁₀₉	—	2	16	no	—
39	C₂⁴×D₄	—	109	1	no	maximal, Sylow
40	C₁₀₉	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
41	C₂⁶	—	109	1	no	—
42	C₂⁶	D₁₀₉	1	1	yes	characteristic, complemented
43	C₂³×D₄	—	109	60	no	—
44	C₂⁴×C₄	—	109	1	no	—
45	C₂⁵	—	2	16	no	—
46	C₂⁵	—	109	30	no	—
47	C₂⁵	—	218	16	no	—
48	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic
49	C₂⁵	C₂×D₁₀₉	1	30	yes	complemented
50	C₂²×D₄	—	109	560	no	—
51	C₂³×C₄	—	109	30	no	—
52	C₂⁴	—	2	240	no	—
53	C₂⁴	—	109	140	no	—
54	C₂⁴	—	218	240	no	—
55	C₂⁴	C₂²⋊D₁₀₉	1	16	yes	direct product factor
56	C₂⁴	C₂²×D₁₀₉	1	15	yes	—
57	C₂⁴	C₂²×D₁₀₉	1	140	yes	complemented
58	C₂×D₄	—	109	960	no	—
59	C₂²×C₄	—	109	140	no	—
60	C₂³	—	2	560	no	—
61	C₂³	—	109	120	no	—
62	C₂³	—	218	560	no	—
63	C₂³	C₂×C₂²⋊D₁₀₉	1	120	yes	direct product factor
64	C₂³	C₂³×D₁₀₉	1	35	yes	—
65	C₂³	C₂³×D₁₀₉	1	120	yes	complemented
66	D₄	—	109	256	no	—
67	C₂×C₄	—	109	120	no	—
68	C₂²	—	2	240	no	—
69	C₂²	—	109	16	no	—
70	C₂²	—	218	240	no	—
71	C₂²	C₂²×C₂²⋊D₁₀₉	1	140	yes	direct product factor
72	C₂²	C₂⁴×D₁₀₉	1	15	yes	—
73	C₂²	C₂⁴×D₁₀₉	1	16	yes	complemented
74	C₄	—	109	16	no	—
75	C₂	—	2	16	no	—
76	C₂	—	218	16	no	—
77	C₂	C₂³×C₂²⋊D₁₀₉	1	30	yes	direct product factor
78	C₂	C₂⁵×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₁₀₉
commutator, cosocle	C₂×C₁₀₉	C₂⁶
Frattini	C₂	C₂⁵×D₁₀₉
Fitting	C₂⁶×C₁₀₉	C₂
socle	C₂⁵×C₁₀₉	C₂²
hypercenter	C₂⁶	D₁₀₉
hypocenter	C₁₀₉	C₂⁴×D₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁵ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₁₀₉	C₂	semidirect
2. C₂³×C₂²⋊D₁₀₉	C₂	direct (N × H)
3. C₂⁴×Dic₁₀₉	C₂	semidirect
4. C₂⁶×C₁₀₉	C₂	semidirect
6. C₂⁴×D₁₀₉	C₂²	semidirect
7. C₂²×C₂²⋊D₁₀₉	C₂²	direct (N × H)
8. C₂³×Dic₁₀₉	C₂²	semidirect
11. C₂⁵×C₁₀₉	C₂²	semidirect
13. C₂³×D₁₀₉	C₂³	semidirect
14. C₂×C₂²⋊D₁₀₉	C₂³	direct (N × H)
15. C₂²×Dic₁₀₉	C₂³	semidirect
17. C₂⁴×C₁₀₉	D₄	semidirect
19. C₂⁴×C₁₀₉	C₂³	semidirect
21. C₂³×C₁₀₉	C₂×D₄	semidirect
23. C₂³×C₁₀₉	C₂⁴	semidirect
25. C₂²×D₁₀₉	C₂⁴	semidirect
26. C₂²⋊D₁₀₉	C₂⁴	direct (N × H)
27. C₂×Dic₁₀₉	C₂⁴	semidirect
29. C₂²×C₁₀₉	C₂²×D₄	semidirect
31. C₂²×C₁₀₉	C₂⁵	semidirect
33. C₂×D₁₀₉	C₂⁵	semidirect
34. Dic₁₀₉	C₂⁵	semidirect
36. C₂×C₁₀₉	C₂³×D₄	semidirect
40. C₁₀₉	C₂⁴×D₄	semidirect
42. C₂⁶	D₁₀₉	semidirect
49. C₂⁵	C₂×D₁₀₉	semidirect
55. C₂⁴	C₂²⋊D₁₀₉	direct (N × H)
57. C₂⁴	C₂²×D₁₀₉	semidirect
63. C₂³	C₂×C₂²⋊D₁₀₉	direct (N × H)
65. C₂³	C₂³×D₁₀₉	semidirect
71. C₂²	C₂²×C₂²⋊D₁₀₉	direct (N × H)
73. C₂²	C₂⁴×D₁₀₉	semidirect
77. C₂	C₂³×C₂²⋊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
C₂³×C₂²⋊D₁₀₉	2
C₂⁴×Dic₁₀₉	2
C₂⁶×C₁₀₉	2
C₂⁴×D₄	109

Subgroups — C24×C22⋊D109 #28377824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₂²⋊D₁₀₉ #28377824