Subgroups — C₂³×C₄×Dic₁₀₉ #28376187

Order 13952.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₀₉	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₁₀₉	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₀₉	C₄	1	32	yes	direct product factor
5	C₂³×Dic₁₀₉	C₂²	1	28	yes	—
6	C₂×C₄×Dic₁₀₉	C₂²	1	112	yes	direct product factor
7	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₀₉	C₄	1	16	yes	complemented
9	C₂³×C₄×C₁₀₉	C₂²	1	14	yes	—
10	C₂²×Dic₁₀₉	C₂×C₄	1	224	yes	direct product factor
11	C₂²×Dic₁₀₉	C₂³	1	56	yes	—
12	C₄×Dic₁₀₉	C₂³	1	64	yes	direct product factor
13	C₂⁴×C₁₀₉	C₂×C₄	1	16	yes	—
14	C₂⁴×C₁₀₉	C₂×C₄	1	8	yes	—
15	C₂⁴×C₁₀₉	C₂³	1	7	yes	—
16	C₂²×C₄×C₁₀₉	C₂×C₄	1	112	yes	complemented
17	C₂²×C₄×C₁₀₉	C₂³	1	28	yes	—
18	C₂³×C₁₀₉	C₄²	1	64	yes	complemented
19	C₂³×C₁₀₉	C₂²×C₄	1	56	yes	—
20	C₂³×C₁₀₉	C₂²×C₄	1	28	yes	—
21	C₂³×C₁₀₉	C₂⁴	1	7	yes	—
22	C₂×C₄×C₁₀₉	C₂²×C₄	1	112	yes	complemented
23	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
24	C₂×Dic₁₀₉	C₂²×C₄	1	224	yes	direct product factor
25	C₂×Dic₁₀₉	C₂⁴	1	16	yes	—
26	C₂²×C₁₀₉	C₂×C₄²	1	112	yes	complemented
27	C₂²×C₁₀₉	C₂³×C₄	1	28	yes	—
28	C₂²×C₁₀₉	C₂³×C₄	1	14	yes	—
29	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
30	C₄×C₁₀₉	C₂³×C₄	1	16	yes	complemented
31	Dic₁₀₉	C₂³×C₄	1	32	yes	direct product factor
32	C₂×C₁₀₉	C₂²×C₄²	1	28	yes	complemented
33	C₂×C₁₀₉	C₂⁴×C₄	1	2	yes	—
34	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
35	C₂³×C₄²	—	109	1	no	maximal, Sylow
36	C₁₀₉	C₂³×C₄²	1	1	yes	characteristic, complemented, Sylow
37	C₂⁴×C₄	—	109	2	no	—
38	C₂⁴×C₄	D₁₀₉	1	1	yes	characteristic
39	C₂²×C₄²	—	109	28	no	—
40	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic
41	C₂³×C₄	—	109	32	no	—
42	C₂³×C₄	—	109	28	no	—
43	C₂³×C₄	Dic₁₀₉	1	16	yes	direct product factor
44	C₂³×C₄	C₂×D₁₀₉	1	14	yes	—
45	C₂×C₄²	—	109	112	no	—
46	C₂⁴	C₄×D₁₀₉	1	16	yes	—
47	C₂⁴	C₂×Dic₁₀₉	1	8	yes	—
48	C₂⁴	C₂²×D₁₀₉	1	7	yes	—
49	C₂²×C₄	—	109	224	no	—
50	C₂²×C₄	—	109	56	no	—
51	C₂²×C₄	C₂×Dic₁₀₉	1	112	yes	direct product factor
52	C₂²×C₄	C₂²×D₁₀₉	1	28	yes	—
53	C₄²	—	109	64	no	—
54	C₂³	C₄×Dic₁₀₉	1	64	yes	direct product factor
55	C₂³	C₂×C₄×D₁₀₉	1	56	yes	—
56	C₂³	C₂²×Dic₁₀₉	1	28	yes	—
57	C₂³	C₂³×D₁₀₉	1	7	yes	—
58	C₂×C₄	—	109	224	no	—
59	C₂×C₄	—	109	16	no	—
60	C₂×C₄	C₂²×Dic₁₀₉	1	112	yes	direct product factor
61	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
62	C₂²	C₂×C₄×Dic₁₀₉	1	112	yes	direct product factor
63	C₂²	C₂²×C₄×D₁₀₉	1	28	yes	—
64	C₂²	C₂³×Dic₁₀₉	1	14	yes	—
65	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
66	C₄	—	109	32	no	—
67	C₄	C₂³×Dic₁₀₉	1	16	yes	direct product factor
68	C₂	C₂²×C₄×Dic₁₀₉	1	28	yes	direct product factor
69	C₂	C₂³×C₄×D₁₀₉	1	2	yes	—
70	C₂	C₂⁴×Dic₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	D₁₀₉
commutator, hypocenter	C₁₀₉	C₂³×C₄²
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂⁴×C₄×C₁₀₉	C₂
socle, prime radical	C₂⁵×C₁₀₉	C₂²
cosocle	C₂²×C₁₀₉	C₂⁵
LCS	G ▷ C₁₀₉
UCS	C₁ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Dic₁₀₉	C₂	direct (N × H)
4. C₂³×Dic₁₀₉	C₄	direct (N × H)
6. C₂×C₄×Dic₁₀₉	C₂²	direct (N × H)
8. C₂³×C₄×C₁₀₉	C₄	semidirect
10. C₂²×Dic₁₀₉	C₂×C₄	direct (N × H)
12. C₄×Dic₁₀₉	C₂³	direct (N × H)
16. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
18. C₂³×C₁₀₉	C₄²	semidirect
22. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
24. C₂×Dic₁₀₉	C₂²×C₄	direct (N × H)
26. C₂²×C₁₀₉	C₂×C₄²	semidirect
30. C₄×C₁₀₉	C₂³×C₄	semidirect
31. Dic₁₀₉	C₂³×C₄	direct (N × H)
32. C₂×C₁₀₉	C₂²×C₄²	semidirect
36. C₁₀₉	C₂³×C₄²	semidirect
43. C₂³×C₄	Dic₁₀₉	direct (N × H)
51. C₂²×C₄	C₂×Dic₁₀₉	direct (N × H)
54. C₂³	C₄×Dic₁₀₉	direct (N × H)
60. C₂×C₄	C₂²×Dic₁₀₉	direct (N × H)
62. C₂²	C₂×C₄×Dic₁₀₉	direct (N × H)
67. C₄	C₂³×Dic₁₀₉	direct (N × H)
68. C₂	C₂²×C₄×Dic₁₀₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×Dic₁₀₉	2
C₂²×C₄×Dic₁₀₉	2
C₂⁴×C₄×C₁₀₉	2
C₂³×C₄²	109

Subgroups — C23×C4×Dic109 #28376187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×Dic₁₀₉ #28376187