Subgroups — D₂₈₄₉₆ #72292123

Order 56992.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₄₂₄₈	C₂	1	2	yes	maximal, complemented
2	D₇₁₂₄	—	2	2	no	—
3	C₁₃×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₁₃×C₁₃₇	C₂²	1	1	yes	characteristic
5	C₂×D₁₇₈₁	—	4	2	no	—
6	C₄×C₁₃×C₁₃₇	D₄	1	1	yes	characteristic
7	D₂₁₉₂	—	13	1	no	maximal, Hall
8	C₂×C₁₃×C₁₃₇	D₈	1	1	yes	characteristic
9	D₁₇₈₁	—	8	2	no	—
10	D₁₀₉₆	—	13	2	no	—
11	C₁₆×C₁₃₇	D₁₃	1	1	yes	characteristic, complemented
12	C₁₃×C₁₃₇	D₁₆	1	1	yes	characteristic, complemented, Hall
13	D₅₄₈	—	26	2	no	—
14	C₈×C₁₃₇	C₂×D₁₃	1	1	yes	characteristic
15	C₂×D₁₃₇	—	52	2	no	—
16	C₄×C₁₃₇	D₅₂	1	1	yes	characteristic
17	D₂₀₈	—	137	1	no	maximal, Hall
18	C₂×C₁₃₇	D₁₀₄	1	1	yes	characteristic
19	D₁₃₇	—	104	2	no	—
20	D₁₀₄	—	137	2	no	—
21	C₁₃×C₁₆	D₁₃₇	1	1	yes	characteristic, complemented
22	C₁₃₇	D₂₀₈	1	1	yes	characteristic, complemented, Sylow
23	D₅₂	—	274	2	no	—
24	C₈×C₁₃	C₂×D₁₃₇	1	1	yes	characteristic
25	C₂×D₁₃	—	548	2	no	—
26	C₄×C₁₃	D₅₄₈	1	1	yes	characteristic
27	D₁₆	—	1781	1	no	Sylow
28	C₂×C₁₃	D₁₀₉₆	1	1	yes	characteristic
29	D₁₃	—	1096	2	no	—
30	D₈	—	1781	2	no	—
31	C₁₆	D₁₇₈₁	1	1	yes	characteristic, complemented
32	C₁₃	D₂₁₉₂	1	1	yes	characteristic, complemented, Sylow
33	D₄	—	3562	2	no	—
34	C₈	C₂×D₁₇₈₁	1	1	yes	characteristic
35	C₂²	—	7124	2	no	—
36	C₄	D₇₁₂₄	1	1	yes	characteristic
37	C₂	—	14248	2	no	—
38	C₂	D₁₄₂₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₄₂₄₈
commutator, cosocle	C₈×C₁₃×C₁₃₇	C₂²
Frattini	C₈	C₂×D₁₇₈₁
Fitting	C₁₃×C₁₆×C₁₃₇	C₂
socle	C₂×C₁₃×C₁₃₇	D₈
hypercenter	C₁₆	D₁₇₈₁
hypocenter	C₁₃×C₁₃₇	D₁₆
LCS	G ▷ C₈×C₁₃×C₁₃₇ ▷ C₄×C₁₃×C₁₃₇ ▷ C₂×C₁₃×C₁₃₇ ▷ C₁₃×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
1. D₁₄₂₄₈	C₂	semidirect
3. C₁₃×C₁₆×C₁₃₇	C₂	semidirect
11. C₁₆×C₁₃₇	D₁₃	semidirect
12. C₁₃×C₁₃₇	D₁₆	semidirect
21. C₁₃×C₁₆	D₁₃₇	semidirect
22. C₁₃₇	D₂₀₈	semidirect
31. C₁₆	D₁₇₈₁	semidirect
32. C₁₃	D₂₁₉₂	semidirect

Maximal subgroups

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

subgroup	index
D₁₄₂₄₈	2
C₁₃×C₁₆×C₁₃₇	2
D₂₁₉₂	13
D₂₀₈	137

Subgroups — D28496 #72292123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₈₄₉₆ #72292123