Subgroups — C₂₃×M₂₅₆ #19155176

Order 5888.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂₃×C₁₂₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×C₃₂	C₄	1	1	yes	characteristic
4	C₂₃×C₆₄	C₄	1	1	yes	characteristic
5	C₂₃×C₆₄	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₂₃	C₈	1	1	yes	characteristic
7	C₂₃×C₃₂	C₈	1	1	yes	characteristic
8	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₃	C₁₆	1	1	yes	characteristic
10	C₁₆×C₂₃	C₁₆	1	1	yes	characteristic
11	C₁₆×C₂₃	C₂×C₈	1	1	yes	characteristic
12	M₂₅₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₄×C₂₃	C₃₂	1	1	yes	characteristic
14	C₈×C₂₃	C₃₂	1	1	yes	characteristic
15	C₈×C₂₃	C₂×C₁₆	1	1	yes	characteristic
16	C₂×C₆₄	C₂×C₂₃	1	1	yes	characteristic
17	C₁₂₈	C₂×C₂₃	1	2	yes	complemented
18	C₂²×C₂₃	C₆₄	1	1	yes	characteristic
19	C₄×C₂₃	C₆₄	1	1	yes	characteristic
20	C₄×C₂₃	C₂×C₃₂	1	1	yes	characteristic
21	C₂×C₃₂	C₄×C₂₃	1	1	yes	characteristic
22	C₆₄	C₄×C₂₃	1	1	yes	characteristic
23	C₆₄	C₂²×C₂₃	1	1	yes	characteristic
24	C₂×C₂₃	—	2	1	no	—
25	C₂×C₂₃	C₂×C₆₄	1	1	yes	characteristic
26	C₂×C₁₆	C₈×C₂₃	1	1	yes	characteristic
27	C₃₂	C₈×C₂₃	1	1	yes	characteristic
28	C₃₂	C₂×C₄×C₂₃	1	1	yes	characteristic
29	C₂₃	M₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₈	C₁₆×C₂₃	1	1	yes	characteristic
31	C₁₆	C₁₆×C₂₃	1	1	yes	characteristic
32	C₁₆	C₂×C₈×C₂₃	1	1	yes	characteristic
33	C₂×C₄	C₂₃×C₃₂	1	1	yes	characteristic
34	C₈	C₂₃×C₃₂	1	1	yes	characteristic
35	C₈	C₂×C₁₆×C₂₃	1	1	yes	characteristic
36	C₂²	C₂₃×C₆₄	1	1	yes	characteristic
37	C₄	C₂₃×C₆₄	1	1	yes	characteristic
38	C₄	C₂×C₂₃×C₃₂	1	1	yes	characteristic
39	C₂	—	2	1	no	—
40	C₂	C₂×C₂₃×C₆₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×C₁₂₈	C₂	semidirect
12. M₂₅₆	C₂₃	direct (N × H)
17. C₁₂₈	C₂×C₂₃	semidirect
29. C₂₃	M₂₅₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₆₄	2
C₂₃×C₁₂₈	2
M₂₅₆	23

Subgroups — C23×M256 #19155176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×M₂₅₆ #19155176