Subgroups — Dic₇₆₀₂ #72271997

Order 30408.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₂₅₃₄	—	3	1	no	maximal, Hall
2	Dic₁₀₈₆	—	7	1	no	maximal, Hall
3	Dic₃₈₀₁	C₂	1	2	yes	maximal
4	C₃×C₄×C₇×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×C₇×C₁₈₁	C₂²	1	1	yes	characteristic
6	Dic₁₂₆₇	—	3	2	no	—
7	C₄×C₇×C₁₈₁	D₃	1	1	yes	characteristic
8	C₃×C₇×C₁₈₁	Q₈	1	1	yes	characteristic, complemented, Hall
9	C₂×C₇×C₁₈₁	C₂×D₃	1	1	yes	characteristic
10	Dic₅₄₃	—	7	2	no	—
11	C₃×C₄×C₁₈₁	D₇	1	1	yes	characteristic
12	Dic₃₆₂	—	21	1	no	Hall
13	C₇×C₁₈₁	Dic₆	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₁₈₁	C₂×D₇	1	1	yes	characteristic
15	C₄×C₁₈₁	D₂₁	1	1	yes	characteristic
16	Dic₁₈₁	—	21	2	no	—
17	C₃×C₁₈₁	Dic₁₄	1	1	yes	characteristic, complemented, Hall
18	C₂×C₁₈₁	C₂×D₂₁	1	1	yes	characteristic
19	C₁₈₁	Dic₄₂	1	1	yes	characteristic, complemented, Sylow
20	Dic₄₂	—	181	1	no	maximal, Hall
21	C₃×C₄×C₇	D₁₈₁	1	1	yes	characteristic
22	Dic₂₁	—	181	2	no	—
23	Dic₁₄	—	543	1	no	Hall
24	C₂×C₃×C₇	C₂×D₁₈₁	1	1	yes	characteristic
25	C₄×C₇	D₅₄₃	1	1	yes	characteristic
26	Dic₇	—	543	2	no	—
27	Dic₆	—	1267	1	no	Hall
28	C₃×C₇	Dic₃₆₂	1	1	yes	characteristic, complemented, Hall
29	C₂×C₇	C₂×D₅₄₃	1	1	yes	characteristic
30	C₃×C₄	D₁₂₆₇	1	1	yes	characteristic
31	Dic₃	—	1267	2	no	—
32	Q₈	—	3801	1	no	Sylow
33	C₇	Dic₁₀₈₆	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃	C₂×D₁₂₆₇	1	1	yes	characteristic
35	C₄	—	3801	2	no	—
36	C₄	D₃₈₀₁	1	1	yes	characteristic
37	C₃	Dic₂₅₃₄	1	1	yes	characteristic, complemented, Sylow
38	C₂	C₂×D₃₈₀₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₈₀₁
commutator, socle, prime radical, cosocle	C₂×C₃×C₇×C₁₈₁	C₂²
Fitting	C₃×C₄×C₇×C₁₈₁	C₂
hypercenter	C₄	D₃₈₀₁
hypocenter	C₃×C₇×C₁₈₁	Q₈
LCS	G ▷ C₂×C₃×C₇×C₁₈₁ ▷ C₃×C₇×C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
8. C₃×C₇×C₁₈₁	Q₈	semidirect
13. C₇×C₁₈₁	Dic₆	semidirect
17. C₃×C₁₈₁	Dic₁₄	semidirect
19. C₁₈₁	Dic₄₂	semidirect
28. C₃×C₇	Dic₃₆₂	semidirect
33. C₇	Dic₁₀₈₆	semidirect
37. C₃	Dic₂₅₃₄	semidirect

Maximal subgroups

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

subgroup	index
Dic₂₅₃₄	3
Dic₁₀₈₆	7
Dic₃₈₀₁	2
C₃×C₄×C₇×C₁₈₁	2
Dic₄₂	181

Subgroups — Dic7602 #72271997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇₆₀₂ #72271997