Subgroups — Dic₇₄₁₆ #72236613

Order 29664.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃₇₀₈	C₂	1	2	yes	maximal
2	Dic₂₄₇₂	—	3	1	no	maximal
3	Dic₁₈₅₄	—	2	2	no	—
4	Dic₁₂₃₆	—	3	2	no	—
5	Dic₆₁₈	—	6	2	no	—
6	C₉×C₁₆×C₁₀₃	C₂	1	1	yes	maximal, characteristic
7	C₈×C₉×C₁₀₃	C₂²	1	1	yes	characteristic
8	C₃×C₁₆×C₁₀₃	D₃	1	1	yes	characteristic
9	C₃×C₈×C₁₀₃	C₂×D₃	1	1	yes	characteristic
10	Dic₉₂₇	—	4	2	no	—
11	C₄×C₉×C₁₀₃	D₄	1	1	yes	characteristic
12	Dic₈₂₄	—	9	1	no	Hall
13	C₂×C₉×C₁₀₃	D₈	1	1	yes	characteristic
14	Dic₄₁₂	—	9	2	no	—
15	C₁₆×C₁₀₃	D₉	1	1	yes	characteristic
16	C₃×C₄×C₁₀₃	D₁₂	1	1	yes	characteristic
17	Dic₃₀₉	—	12	2	no	—
18	C₉×C₁₀₃	Q₃₂	1	1	yes	characteristic, complemented, Hall
19	Dic₂₀₆	—	18	2	no	—
20	C₈×C₁₀₃	C₂×D₉	1	1	yes	characteristic
21	C₂×C₃×C₁₀₃	D₂₄	1	1	yes	characteristic
22	C₄×C₁₀₃	D₃₆	1	1	yes	characteristic
23	Dic₁₀₃	—	36	2	no	—
24	C₃×C₁₀₃	Dic₂₄	1	1	yes	characteristic
25	Dic₇₂	—	103	1	no	maximal, Hall
26	C₂×C₁₀₃	D₇₂	1	1	yes	characteristic
27	Dic₃₆	—	103	2	no	—
28	C₉×C₁₆	D₁₀₃	1	1	yes	characteristic
29	C₁₀₃	Dic₇₂	1	1	yes	characteristic, complemented, Sylow
30	Dic₂₄	—	309	1	no	—
31	Dic₁₈	—	206	2	no	—
32	C₈×C₉	C₂×D₁₀₃	1	1	yes	characteristic
33	Dic₁₂	—	309	2	no	—
34	C₃×C₁₆	D₃₀₉	1	1	yes	characteristic
35	C₄×C₉	D₄₁₂	1	1	yes	characteristic
36	Dic₉	—	412	2	no	—
37	Q₃₂	—	927	1	no	Sylow
38	Dic₆	—	618	2	no	—
39	C₃×C₈	C₂×D₃₀₉	1	1	yes	characteristic
40	C₂×C₉	D₈₂₄	1	1	yes	characteristic
41	Q₁₆	—	927	2	no	—
42	C₁₆	D₉₂₇	1	1	yes	characteristic
43	C₃×C₄	D₁₂₃₆	1	1	yes	characteristic
44	Dic₃	—	1236	2	no	—
45	C₉	Dic₈₂₄	1	1	yes	characteristic, complemented, Sylow
46	Q₈	—	1854	2	no	—
47	C₈	C₂×D₉₂₇	1	1	yes	characteristic
48	C₂×C₃	D₂₄₇₂	1	1	yes	characteristic
49	C₄	—	3708	2	no	—
50	C₄	D₃₇₀₈	1	1	yes	characteristic
51	C₃	Dic₂₄₇₂	1	1	yes	characteristic
52	C₂	D₇₄₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₇₄₁₆
commutator, cosocle	C₈×C₉×C₁₀₃	C₂²
Frattini	C₃×C₈	C₂×D₃₀₉
Fitting	C₉×C₁₆×C₁₀₃	C₂
socle, prime radical	C₂×C₃×C₁₀₃	D₂₄
hypercenter	C₁₆	D₉₂₇
hypocenter	C₉×C₁₀₃	Q₃₂
LCS	G ▷ C₈×C₉×C₁₀₃ ▷ C₄×C₉×C₁₀₃ ▷ C₂×C₉×C₁₀₃ ▷ C₉×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
18. C₉×C₁₀₃	Q₃₂	semidirect
29. C₁₀₃	Dic₇₂	semidirect
45. C₉	Dic₈₂₄	semidirect

Maximal subgroups

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

subgroup	index
Dic₃₇₀₈	2
Dic₂₄₇₂	3
C₉×C₁₆×C₁₀₃	2
Dic₇₂	103

Subgroups — Dic7416 #72236613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇₄₁₆ #72236613