Subgroups — C₂³×C₄⋊Dic₁₀₉ #28376193

Order 13952.

2488 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	—	2	16	no	—
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	—	2	112	no	—
11	C₂²×Dic₁₀₉	C₂³	1	56	yes	—
12	C₄⋊Dic₁₀₉	C₂³	1	64	yes	direct product factor
13	C₂⁴×C₁₀₉	C₂×C₄	1	8	yes	—
14	C₂⁴×C₁₀₉	D₄	1	8	yes	—
15	C₂⁴×C₁₀₉	Q₈	1	8	yes	—
16	C₂⁴×C₁₀₉	C₂³	1	7	yes	—
17	C₂²×C₄×C₁₀₉	C₂×C₄	1	112	yes	complemented
18	C₂²×C₄×C₁₀₉	C₂³	1	28	yes	—
19	C₂³×C₁₀₉	C₄⋊C₄	1	64	yes	complemented
20	C₂³×C₁₀₉	C₂²×C₄	1	28	yes	—
21	C₂³×C₁₀₉	C₂×D₄	1	28	yes	—
22	C₂³×C₁₀₉	C₂×Q₈	1	28	yes	—
23	C₂³×C₁₀₉	C₂⁴	1	7	yes	—
24	C₂×C₄×C₁₀₉	C₂²×C₄	1	112	yes	complemented
25	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
26	C₂×Dic₁₀₉	—	2	112	no	—
27	C₂×Dic₁₀₉	C₂⁴	1	16	yes	—
28	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	112	yes	complemented
29	C₂²×C₁₀₉	C₂³×C₄	1	14	yes	—
30	C₂²×C₁₀₉	C₂²×D₄	1	14	yes	—
31	C₂²×C₁₀₉	C₂²×Q₈	1	14	yes	—
32	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
33	C₄×C₁₀₉	C₂³×C₄	1	16	yes	complemented
34	Dic₁₀₉	—	2	16	no	—
35	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	28	yes	complemented
36	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
37	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
38	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
39	C₂³×C₄⋊C₄	—	109	1	no	maximal, Sylow
40	C₁₀₉	C₂³×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
41	C₂⁴×C₄	—	109	2	no	—
42	C₂⁴×C₄	D₁₀₉	1	1	yes	characteristic
43	C₂²×C₄⋊C₄	—	109	28	no	—
44	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic
45	C₂³×C₄	—	109	28	no	—
46	C₂³×C₄	—	218	16	no	—
47	C₂³×C₄	Dic₁₀₉	1	16	yes	complemented
48	C₂³×C₄	C₂×D₁₀₉	1	14	yes	—
49	C₂×C₄⋊C₄	—	109	112	no	—
50	C₂⁴	Dic₂₁₈	1	8	yes	—
51	C₂⁴	D₄₃₆	1	8	yes	—
52	C₂⁴	C₂×Dic₁₀₉	1	8	yes	—
53	C₂⁴	C₂²×D₁₀₉	1	7	yes	—
54	C₂²×C₄	—	109	56	no	—
55	C₂²×C₄	—	218	112	no	—
56	C₂²×C₄	C₂×Dic₁₀₉	1	112	yes	complemented
57	C₂²×C₄	C₂²×D₁₀₉	1	28	yes	—
58	C₄⋊C₄	—	109	64	no	—
59	C₂³	C₄⋊Dic₁₀₉	1	64	yes	direct product factor
60	C₂³	C₂²×Dic₁₀₉	1	28	yes	—
61	C₂³	C₂×D₄₃₆	1	28	yes	—
62	C₂³	C₂×Dic₂₁₈	1	28	yes	—
63	C₂³	C₂³×D₁₀₉	1	7	yes	—
64	C₂×C₄	—	109	16	no	—
65	C₂×C₄	—	218	112	no	—
66	C₂×C₄	C₂²×Dic₁₀₉	1	112	yes	complemented
67	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
68	C₂²	C₂×C₄⋊Dic₁₀₉	1	112	yes	direct product factor
69	C₂²	C₂³×Dic₁₀₉	1	14	yes	—
70	C₂²	C₂²×D₄₃₆	1	14	yes	—
71	C₂²	C₂²×Dic₂₁₈	1	14	yes	—
72	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
73	C₄	—	218	16	no	—
74	C₄	C₂³×Dic₁₀₉	1	16	yes	complemented
75	C₂	C₂²×C₄⋊Dic₁₀₉	1	28	yes	direct product factor
76	C₂	C₂⁴×Dic₁₀₉	1	1	yes	characteristic
77	C₂	C₂³×D₄₃₆	1	1	yes	characteristic
78	C₂	C₂³×Dic₂₁₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂⁴×C₄×C₁₀₉	C₂
socle, prime radical	C₂⁵×C₁₀₉	C₂²
cosocle	C₂²×C₁₀₉	C₂⁵
hypercenter	C₂⁴×C₄	D₁₀₉
hypocenter	C₁₀₉	C₂³×C₄⋊C₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁵ ◁ 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)
6. C₂×C₄⋊Dic₁₀₉	C₂²	direct (N × H)
8. C₂³×C₄×C₁₀₉	C₄	semidirect
12. C₄⋊Dic₁₀₉	C₂³	direct (N × H)
17. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
19. C₂³×C₁₀₉	C₄⋊C₄	semidirect
24. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
28. C₂²×C₁₀₉	C₂×C₄⋊C₄	semidirect
33. C₄×C₁₀₉	C₂³×C₄	semidirect
35. C₂×C₁₀₉	C₂²×C₄⋊C₄	semidirect
40. C₁₀₉	C₂³×C₄⋊C₄	semidirect
47. C₂³×C₄	Dic₁₀₉	semidirect
56. C₂²×C₄	C₂×Dic₁₀₉	semidirect
59. C₂³	C₄⋊Dic₁₀₉	direct (N × H)
66. C₂×C₄	C₂²×Dic₁₀₉	semidirect
68. C₂²	C₂×C₄⋊Dic₁₀₉	direct (N × H)
74. C₄	C₂³×Dic₁₀₉	semidirect
75. 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×C₄⋊C₄	109

Subgroups — C23×C4⋊Dic109 #28376193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊Dic₁₀₉ #28376193