Subgroups — Dic₇₅₉₆ #72271982

Order 30384.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃₇₉₈	C₂	1	2	yes	maximal
2	Dic₂₅₃₂	—	3	1	no	maximal
3	Dic₁₂₆₆	—	3	2	no	—
4	C₈×C₉×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₃×C₈×C₂₁₁	D₃	1	1	yes	characteristic
6	Dic₁₈₉₉	—	2	2	no	—
7	C₄×C₉×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₂₁₁	D₄	1	1	yes	characteristic
9	Dic₈₄₄	—	9	1	no	Hall
10	Dic₆₃₃	—	6	2	no	—
11	C₃×C₄×C₂₁₁	C₂×D₃	1	1	yes	characteristic
12	C₉×C₂₁₁	Q₁₆	1	1	yes	characteristic, complemented, Hall
13	Dic₄₂₂	—	9	2	no	—
14	C₈×C₂₁₁	D₉	1	1	yes	characteristic
15	C₂×C₃×C₂₁₁	D₁₂	1	1	yes	characteristic
16	C₄×C₂₁₁	C₂×D₉	1	1	yes	characteristic
17	Dic₂₁₁	—	18	2	no	—
18	C₃×C₂₁₁	Dic₁₂	1	1	yes	characteristic
19	C₂×C₂₁₁	D₃₆	1	1	yes	characteristic
20	C₂₁₁	Dic₃₆	1	1	yes	characteristic, complemented, Sylow
21	Dic₃₆	—	211	1	no	maximal, Hall
22	Dic₁₈	—	211	2	no	—
23	C₈×C₉	D₂₁₁	1	1	yes	characteristic
24	Dic₁₂	—	633	1	no	—
25	C₄×C₉	C₂×D₂₁₁	1	1	yes	characteristic
26	Dic₉	—	422	2	no	—
27	Dic₆	—	633	2	no	—
28	C₃×C₈	D₆₃₃	1	1	yes	characteristic
29	C₂×C₉	D₈₄₄	1	1	yes	characteristic
30	Q₁₆	—	1899	1	no	Sylow
31	C₃×C₄	C₂×D₆₃₃	1	1	yes	characteristic
32	Dic₃	—	1266	2	no	—
33	C₉	Dic₈₄₄	1	1	yes	characteristic, complemented, Sylow
34	Q₈	—	1899	2	no	—
35	C₈	D₁₈₉₉	1	1	yes	characteristic
36	C₂×C₃	D₂₅₃₂	1	1	yes	characteristic
37	C₄	—	3798	2	no	—
38	C₄	C₂×D₁₈₉₉	1	1	yes	characteristic
39	C₃	Dic₂₅₃₂	1	1	yes	characteristic
40	C₂	D₇₅₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₇₅₉₆
commutator, cosocle	C₄×C₉×C₂₁₁	C₂²
Frattini	C₃×C₄	C₂×D₆₃₃
Fitting	C₈×C₉×C₂₁₁	C₂
socle, prime radical	C₂×C₃×C₂₁₁	D₁₂
hypercenter	C₈	D₁₈₉₉
hypocenter	C₉×C₂₁₁	Q₁₆
LCS	G ▷ C₄×C₉×C₂₁₁ ▷ C₂×C₉×C₂₁₁ ▷ C₉×C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
12. C₉×C₂₁₁	Q₁₆	semidirect
20. C₂₁₁	Dic₃₆	semidirect
33. C₉	Dic₈₄₄	semidirect

Maximal subgroups

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

subgroup	index
Dic₃₇₉₈	2
Dic₂₅₃₂	3
C₈×C₉×C₂₁₁	2
Dic₃₆	211

Subgroups — Dic7596 #72271982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇₅₉₆ #72271982