Subgroups — C₂₃×Q₂₅₆ #19155179

Order 5888.

42 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×Q₁₂₈	C₂	1	2	yes	maximal
2	C₂₃×C₁₂₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×Q₆₄	—	2	2	no	—
4	C₂₃×C₆₄	C₂²	1	1	yes	characteristic
5	C₂₃×Q₃₂	—	4	2	no	—
6	C₂₃×C₃₂	D₄	1	1	yes	characteristic
7	Q₁₆×C₂₃	—	8	2	no	—
8	C₁₆×C₂₃	D₈	1	1	yes	characteristic
9	Q₂₅₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	Q₈×C₂₃	—	16	2	no	—
11	C₈×C₂₃	D₁₆	1	1	yes	characteristic
12	Q₁₂₈	C₂×C₂₃	1	2	yes	—
13	C₁₂₈	C₂×C₂₃	1	1	yes	characteristic
14	C₄×C₂₃	—	32	2	no	—
15	C₄×C₂₃	D₃₂	1	1	yes	characteristic
16	Q₆₄	—	2	2	no	—
17	C₆₄	C₂²×C₂₃	1	1	yes	characteristic
18	C₂×C₂₃	D₆₄	1	1	yes	characteristic
19	Q₃₂	—	4	2	no	—
20	C₃₂	D₄×C₂₃	1	1	yes	characteristic
21	C₂₃	Q₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
22	Q₁₆	—	8	2	no	—
23	C₁₆	D₈×C₂₃	1	1	yes	characteristic
24	Q₈	—	16	2	no	—
25	C₈	C₂₃×D₁₆	1	1	yes	characteristic
26	C₄	—	32	2	no	—
27	C₄	C₂₃×D₃₂	1	1	yes	characteristic
28	C₂	C₂₃×D₆₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₂₃	D₆₄
commutator, Frattini, cosocle	C₆₄	C₂²×C₂₃
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₄×C₂₃ ◁ C₈×C₂₃ ◁ C₁₆×C₂₃ ◁ C₂₃×C₃₂ ◁ C₂₃×C₆₄ ◁ G

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

N	H ≅ G/N	type
9. Q₂₅₆	C₂₃	direct (N × H)
21. C₂₃	Q₂₅₆	direct (N × H)

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