Subgroups — Dic₇₁₆₈ #72236519

Order 28672.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃₅₈₄	C₂	1	2	yes	maximal
2	Dic₁₇₉₂	—	2	2	no	—
3	Dic₈₉₆	—	4	2	no	—
4	Q₄₀₉₆	—	7	1	no	maximal, Sylow
5	Q₂₀₄₈	—	7	2	no	—
6	Q₁₀₂₄	—	14	2	no	—
7	C₇×C₂₀₄₈	C₂	1	1	yes	maximal, characteristic
8	C₇×C₁₀₂₄	C₂²	1	1	yes	characteristic
9	C₇×C₅₁₂	D₄	1	1	yes	characteristic
10	C₂₀₄₈	D₇	1	1	yes	characteristic
11	C₁₀₂₄	C₂×D₇	1	1	yes	characteristic
12	Dic₄₄₈	—	8	2	no	—
13	C₇×C₂₅₆	D₈	1	1	yes	characteristic
14	Dic₂₂₄	—	16	2	no	—
15	C₇×C₁₂₈	D₁₆	1	1	yes	characteristic
16	Q₅₁₂	—	28	2	no	—
17	C₅₁₂	D₂₈	1	1	yes	characteristic
18	Dic₁₁₂	—	32	2	no	—
19	C₇×C₆₄	D₃₂	1	1	yes	characteristic
20	Q₂₅₆	—	56	2	no	—
21	C₂₅₆	D₅₆	1	1	yes	characteristic
22	Dic₅₆	—	64	2	no	—
23	C₇×C₃₂	D₆₄	1	1	yes	characteristic
24	Q₁₂₈	—	112	2	no	—
25	C₁₂₈	D₁₁₂	1	1	yes	characteristic
26	Dic₂₈	—	128	2	no	—
27	C₇×C₁₆	D₁₂₈	1	1	yes	characteristic
28	Q₆₄	—	224	2	no	—
29	C₆₄	D₂₂₄	1	1	yes	characteristic
30	Dic₁₄	—	256	2	no	—
31	C₇×C₈	D₂₅₆	1	1	yes	characteristic
32	Q₃₂	—	448	2	no	—
33	C₃₂	D₄₄₈	1	1	yes	characteristic
34	C₄×C₇	D₅₁₂	1	1	yes	characteristic
35	Dic₇	—	512	2	no	—
36	Q₁₆	—	896	2	no	—
37	C₁₆	D₈₉₆	1	1	yes	characteristic
38	C₂×C₇	D₁₀₂₄	1	1	yes	characteristic
39	Q₈	—	1792	2	no	—
40	C₈	D₁₇₉₂	1	1	yes	characteristic
41	C₇	Q₄₀₉₆	1	1	yes	characteristic, complemented, Sylow
42	C₄	—	3584	2	no	—
43	C₄	D₃₅₈₄	1	1	yes	characteristic
44	C₂	D₇₁₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₇₁₆₈
commutator, cosocle	C₇×C₁₀₂₄	C₂²
Frattini	C₁₀₂₄	C₂×D₇
Fitting	C₇×C₂₀₄₈	C₂
socle, prime radical	C₂×C₇	D₁₀₂₄
hypercenter	C₂₀₄₈	D₇
hypocenter	C₇	Q₄₀₉₆
LCS	G ▷ C₇×C₁₀₂₄ ▷ C₇×C₅₁₂ ▷ C₇×C₂₅₆ ▷ C₇×C₁₂₈ ▷ C₇×C₆₄ ▷ C₇×C₃₂ ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄ ◁ C₁₂₈ ◁ C₂₅₆ ◁ C₅₁₂ ◁ C₁₀₂₄ ◁ C₂₀₄₈

Complements

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

N	H ≅ G/N	type
41. C₇	Q₄₀₉₆	semidirect

Maximal subgroups

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

subgroup	index
Dic₃₅₈₄	2
Q₄₀₉₆	7
C₇×C₂₀₄₈	2

Subgroups — Dic7168 #72236519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇₁₆₈ #72236519