Subgroups — C₂₃×SD₂₅₆ #19155178

Order 5888.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×D₆₄	C₂	1	1	yes	maximal, characteristic
3	C₂₃×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×Q₆₄	—	2	1	no	—
5	C₂₃×D₃₂	—	2	1	no	—
6	C₂₃×C₆₄	C₂²	1	1	yes	characteristic
7	C₂₃×Q₃₂	—	4	1	no	—
8	C₂₃×D₁₆	—	4	1	no	—
9	C₂₃×C₃₂	D₄	1	1	yes	characteristic
10	Q₁₆×C₂₃	—	8	1	no	—
11	D₈×C₂₃	—	8	1	no	—
12	C₁₆×C₂₃	D₈	1	1	yes	characteristic
13	SD₂₅₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	Q₈×C₂₃	—	16	1	no	—
15	D₄×C₂₃	—	16	1	no	—
16	C₈×C₂₃	D₁₆	1	1	yes	characteristic
17	Q₁₂₈	C₂×C₂₃	1	1	yes	characteristic, complemented
18	D₆₄	C₂×C₂₃	1	1	yes	characteristic
19	C₁₂₈	C₂×C₂₃	1	1	yes	characteristic, complemented
20	C₂²×C₂₃	—	32	1	no	—
21	C₄×C₂₃	—	32	1	no	—
22	C₄×C₂₃	D₃₂	1	1	yes	characteristic
23	Q₆₄	—	2	1	no	—
24	D₃₂	—	2	1	no	—
25	C₆₄	C₂²×C₂₃	1	1	yes	characteristic
26	C₂×C₂₃	—	64	1	no	—
27	C₂×C₂₃	D₆₄	1	1	yes	characteristic
28	Q₃₂	—	4	1	no	—
29	D₁₆	—	4	1	no	—
30	C₃₂	D₄×C₂₃	1	1	yes	characteristic
31	C₂₃	SD₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
32	Q₁₆	—	8	1	no	—
33	D₈	—	8	1	no	—
34	C₁₆	D₈×C₂₃	1	1	yes	characteristic
35	Q₈	—	16	1	no	—
36	D₄	—	16	1	no	—
37	C₈	C₂₃×D₁₆	1	1	yes	characteristic
38	C₂²	—	32	1	no	—
39	C₄	—	32	1	no	—
40	C₄	C₂₃×D₃₂	1	1	yes	characteristic
41	C₂	—	64	1	no	—
42	C₂	C₂₃×D₆₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×Q₁₂₈	C₂	semidirect
3. C₂₃×C₁₂₈	C₂	semidirect
13. SD₂₅₆	C₂₃	direct (N × H)
17. Q₁₂₈	C₂×C₂₃	semidirect
19. C₁₂₈	C₂×C₂₃	semidirect
31. C₂₃	SD₂₅₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×Q₁₂₈	2
C₂₃×D₆₄	2
C₂₃×C₁₂₈	2
SD₂₅₆	23

Subgroups — C23×SD256 #19155178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×SD₂₅₆ #19155178