Subgroups — Dic₁₄₀₄ #72211942

Order 5616.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₇₀₂	C₂	1	2	yes	maximal
2	C₈×C₁₃×C₂₇	C₂	1	1	yes	maximal, characteristic
3	Dic₄₆₈	—	3	1	no	maximal
4	C₄×C₁₃×C₂₇	C₂²	1	1	yes	characteristic
5	Dic₃₅₁	—	2	2	no	—
6	Dic₂₃₄	—	3	2	no	—
7	C₈×C₉×C₁₃	D₃	1	1	yes	characteristic
8	C₂×C₁₃×C₂₇	D₄	1	1	yes	characteristic
9	Dic₁₅₆	—	9	1	no	—
10	C₄×C₉×C₁₃	C₂×D₃	1	1	yes	characteristic
11	Dic₁₁₇	—	6	2	no	—
12	Dic₁₀₈	—	13	1	no	maximal, Hall
13	C₁₃×C₂₇	Q₁₆	1	1	yes	characteristic, complemented, Hall
14	Dic₇₈	—	9	2	no	—
15	C₃×C₈×C₁₃	D₉	1	1	yes	characteristic
16	C₂×C₉×C₁₃	D₁₂	1	1	yes	characteristic
17	Dic₅₄	—	13	2	no	—
18	C₈×C₂₇	D₁₃	1	1	yes	characteristic
19	Dic₅₂	—	27	1	no	Hall
20	C₃×C₄×C₁₃	C₂×D₉	1	1	yes	characteristic
21	Dic₃₉	—	18	2	no	—
22	Dic₃₆	—	39	1	no	—
23	C₉×C₁₃	Dic₁₂	1	1	yes	characteristic
24	C₄×C₂₇	C₂×D₁₃	1	1	yes	characteristic
25	Dic₂₇	—	26	2	no	—
26	Dic₂₆	—	27	2	no	—
27	C₈×C₁₃	D₂₇	1	1	yes	characteristic
28	C₂×C₃×C₁₃	D₃₆	1	1	yes	characteristic
29	Dic₁₈	—	39	2	no	—
30	C₈×C₉	D₃₉	1	1	yes	characteristic
31	C₂×C₂₇	D₅₂	1	1	yes	characteristic
32	C₄×C₁₃	C₂×D₂₇	1	1	yes	characteristic
33	Dic₁₃	—	54	2	no	—
34	Dic₁₂	—	117	1	no	—
35	C₃×C₁₃	Dic₃₆	1	1	yes	characteristic
36	C₄×C₉	C₂×D₃₉	1	1	yes	characteristic
37	Dic₉	—	78	2	no	—
38	C₂₇	Dic₅₂	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₁₃	D₁₀₈	1	1	yes	characteristic
40	Dic₆	—	117	2	no	—
41	C₃×C₈	D₁₁₇	1	1	yes	characteristic
42	C₂×C₉	D₁₅₆	1	1	yes	characteristic
43	Q₁₆	—	351	1	no	Sylow
44	C₁₃	Dic₁₀₈	1	1	yes	characteristic, complemented, Sylow
45	C₃×C₄	C₂×D₁₁₇	1	1	yes	characteristic
46	Dic₃	—	234	2	no	—
47	C₉	Dic₁₅₆	1	1	yes	characteristic
48	Q₈	—	351	2	no	—
49	C₈	D₃₅₁	1	1	yes	characteristic
50	C₂×C₃	D₄₆₈	1	1	yes	characteristic
51	C₄	—	702	2	no	—
52	C₄	C₂×D₃₅₁	1	1	yes	characteristic
53	C₃	Dic₄₆₈	1	1	yes	characteristic
54	C₂	D₁₄₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₄₀₄
commutator, cosocle	C₄×C₁₃×C₂₇	C₂²
Frattini	C₄×C₉	C₂×D₃₉
Fitting	C₈×C₁₃×C₂₇	C₂
socle, prime radical	C₂×C₃×C₁₃	D₃₆
hypercenter	C₈	D₃₅₁
hypocenter	C₁₃×C₂₇	Q₁₆
LCS	G ▷ C₄×C₁₃×C₂₇ ▷ C₂×C₁₃×C₂₇ ▷ C₁₃×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
13. C₁₃×C₂₇	Q₁₆	semidirect
38. C₂₇	Dic₅₂	semidirect
44. C₁₃	Dic₁₀₈	semidirect

Maximal subgroups

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

subgroup	index
Dic₇₀₂	2
C₈×C₁₃×C₂₇	2
Dic₄₆₈	3
Dic₁₀₈	13

Subgroups — Dic1404 #72211942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₄₀₄ #72211942