Subgroups — Dic₆₃₉₆ #72236088

Order 25584.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃₁₉₈	C₂	1	2	yes	maximal
2	Dic₂₁₃₂	—	3	1	no	maximal, Hall
3	Dic₁₀₆₆	—	3	2	no	—
4	C₃×C₈×C₁₃×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₃×C₄₁	D₃	1	1	yes	characteristic
6	Dic₁₅₉₉	—	2	2	no	—
7	C₃×C₄×C₁₃×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₁₃×C₄₁	D₄	1	1	yes	characteristic
9	Dic₅₃₃	—	6	2	no	—
10	C₄×C₁₃×C₄₁	C₂×D₃	1	1	yes	characteristic
11	Dic₄₉₂	—	13	1	no	maximal, Hall
12	C₃×C₁₃×C₄₁	Q₁₆	1	1	yes	characteristic, complemented, Hall
13	C₂×C₁₃×C₄₁	D₁₂	1	1	yes	characteristic
14	Dic₂₄₆	—	13	2	no	—
15	C₃×C₈×C₄₁	D₁₃	1	1	yes	characteristic
16	Dic₁₆₄	—	39	1	no	Hall
17	Dic₁₅₆	—	41	1	no	maximal, Hall
18	C₁₃×C₄₁	Dic₁₂	1	1	yes	characteristic, complemented, Hall
19	C₃×C₄×C₄₁	C₂×D₁₃	1	1	yes	characteristic
20	Dic₁₂₃	—	26	2	no	—
21	Dic₈₂	—	39	2	no	—
22	C₈×C₄₁	D₃₉	1	1	yes	characteristic
23	Dic₇₈	—	41	2	no	—
24	C₃×C₈×C₁₃	D₄₁	1	1	yes	characteristic
25	C₂×C₃×C₄₁	D₅₂	1	1	yes	characteristic
26	Dic₅₂	—	123	1	no	Hall
27	C₄×C₄₁	C₂×D₃₉	1	1	yes	characteristic
28	Dic₄₁	—	78	2	no	—
29	C₃×C₄×C₁₃	C₂×D₄₁	1	1	yes	characteristic
30	Dic₃₉	—	82	2	no	—
31	C₃×C₄₁	Dic₅₂	1	1	yes	characteristic, complemented, Hall
32	Dic₂₆	—	123	2	no	—
33	C₈×C₁₃	D₁₂₃	1	1	yes	characteristic
34	C₂×C₄₁	D₁₅₆	1	1	yes	characteristic
35	C₂×C₃×C₁₃	D₁₆₄	1	1	yes	characteristic
36	C₄×C₁₃	C₂×D₁₂₃	1	1	yes	characteristic
37	Dic₁₃	—	246	2	no	—
38	Dic₁₂	—	533	1	no	Hall
39	C₄₁	Dic₁₅₆	1	1	yes	characteristic, complemented, Sylow
40	C₃×C₁₃	Dic₁₆₄	1	1	yes	characteristic, complemented, Hall
41	C₂×C₁₃	D₄₉₂	1	1	yes	characteristic
42	Dic₆	—	533	2	no	—
43	C₃×C₈	D₅₃₃	1	1	yes	characteristic
44	Q₁₆	—	1599	1	no	Sylow
45	C₁₃	Dic₄₉₂	1	1	yes	characteristic, complemented, Sylow
46	C₃×C₄	C₂×D₅₃₃	1	1	yes	characteristic
47	Dic₃	—	1066	2	no	—
48	Q₈	—	1599	2	no	—
49	C₈	D₁₅₉₉	1	1	yes	characteristic
50	C₂×C₃	D₂₁₃₂	1	1	yes	characteristic
51	C₄	—	3198	2	no	—
52	C₄	C₂×D₁₅₉₉	1	1	yes	characteristic
53	C₃	Dic₂₁₃₂	1	1	yes	characteristic, complemented, Sylow
54	C₂	D₆₃₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₆₃₉₆
commutator, cosocle	C₃×C₄×C₁₃×C₄₁	C₂²
Frattini	C₄	C₂×D₁₅₉₉
Fitting	C₃×C₈×C₁₃×C₄₁	C₂
socle, prime radical	C₂×C₃×C₁₃×C₄₁	D₄
hypercenter	C₈	D₁₅₉₉
hypocenter	C₃×C₁₃×C₄₁	Q₁₆
LCS	G ▷ C₃×C₄×C₁₃×C₄₁ ▷ C₂×C₃×C₁₃×C₄₁ ▷ C₃×C₁₃×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
12. C₃×C₁₃×C₄₁	Q₁₆	semidirect
18. C₁₃×C₄₁	Dic₁₂	semidirect
31. C₃×C₄₁	Dic₅₂	semidirect
39. C₄₁	Dic₁₅₆	semidirect
40. C₃×C₁₃	Dic₁₆₄	semidirect
45. C₁₃	Dic₄₉₂	semidirect
53. C₃	Dic₂₁₃₂	semidirect

Maximal subgroups

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

subgroup	index
Dic₃₁₉₈	2
Dic₂₁₃₂	3
C₃×C₈×C₁₃×C₄₁	2
Dic₄₉₂	13
Dic₁₅₆	41

Subgroups — Dic6396 #72236088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₆₃₉₆ #72236088