Subgroups — Dic₆₁₄₄ #72224350

Order 24576.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃₀₇₂	C₂	1	2	yes	maximal
2	Dic₁₅₃₆	—	2	2	no	—
3	Dic₇₆₈	—	4	2	no	—
4	Dic₃₈₄	—	8	2	no	—
5	Q₈₁₉₂	—	3	1	no	maximal, Sylow
6	Q₄₀₉₆	—	3	2	no	—
7	Q₂₀₄₈	—	6	2	no	—
8	Q₁₀₂₄	—	12	2	no	—
9	C₃×C₄₀₉₆	C₂	1	1	yes	maximal, characteristic
10	C₃×C₂₀₄₈	C₂²	1	1	yes	characteristic
11	C₄₀₉₆	D₃	1	1	yes	characteristic
12	C₃×C₁₀₂₄	D₄	1	1	yes	characteristic
13	C₂₀₄₈	C₂×D₃	1	1	yes	characteristic
14	C₃×C₅₁₂	D₈	1	1	yes	characteristic
15	C₁₀₂₄	D₁₂	1	1	yes	characteristic
16	Dic₁₉₂	—	16	2	no	—
17	C₃×C₂₅₆	D₁₆	1	1	yes	characteristic
18	Q₅₁₂	—	24	2	no	—
19	C₅₁₂	D₂₄	1	1	yes	characteristic
20	Dic₉₆	—	32	2	no	—
21	C₃×C₁₂₈	D₃₂	1	1	yes	characteristic
22	Q₂₅₆	—	48	2	no	—
23	C₂₅₆	D₄₈	1	1	yes	characteristic
24	Dic₄₈	—	64	2	no	—
25	C₃×C₆₄	D₆₄	1	1	yes	characteristic
26	Q₁₂₈	—	96	2	no	—
27	C₁₂₈	D₉₆	1	1	yes	characteristic
28	Dic₂₄	—	128	2	no	—
29	C₃×C₃₂	D₁₂₈	1	1	yes	characteristic
30	Q₆₄	—	192	2	no	—
31	C₆₄	D₁₉₂	1	1	yes	characteristic
32	Dic₁₂	—	256	2	no	—
33	C₃×C₁₆	D₂₅₆	1	1	yes	characteristic
34	Q₃₂	—	384	2	no	—
35	C₃₂	D₃₈₄	1	1	yes	characteristic
36	Dic₆	—	512	2	no	—
37	C₃×C₈	D₅₁₂	1	1	yes	characteristic
38	Q₁₆	—	768	2	no	—
39	C₁₆	D₇₆₈	1	1	yes	characteristic
40	C₃×C₄	D₁₀₂₄	1	1	yes	characteristic
41	Dic₃	—	1024	2	no	—
42	Q₈	—	1536	2	no	—
43	C₈	D₁₅₃₆	1	1	yes	characteristic
44	C₂×C₃	D₂₀₄₈	1	1	yes	characteristic
45	C₄	—	3072	2	no	—
46	C₄	D₃₀₇₂	1	1	yes	characteristic
47	C₃	Q₈₁₉₂	1	1	yes	characteristic, complemented, Sylow
48	C₂	D₆₁₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₆₁₄₄
commutator, cosocle	C₃×C₂₀₄₈	C₂²
Frattini	C₂₀₄₈	C₂×D₃
Fitting	C₃×C₄₀₉₆	C₂
socle, prime radical	C₂×C₃	D₂₀₄₈
hypercenter	C₄₀₉₆	D₃
hypocenter	C₃	Q₈₁₉₂
LCS	G ▷ C₃×C₂₀₄₈ ▷ C₃×C₁₀₂₄ ▷ C₃×C₅₁₂ ▷ C₃×C₂₅₆ ▷ C₃×C₁₂₈ ▷ C₃×C₆₄ ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄ ◁ C₁₂₈ ◁ C₂₅₆ ◁ C₅₁₂ ◁ C₁₀₂₄ ◁ C₂₀₄₈ ◁ C₄₀₉₆

Complements

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

N	H ≅ G/N	type
47. C₃	Q₈₁₉₂	semidirect

Maximal subgroups

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

subgroup	index
Dic₃₀₇₂	2
Q₈₁₉₂	3
C₃×C₄₀₉₆	2

Subgroups — Dic6144 #72224350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₆₁₄₄ #72224350