Subgroups — C₂₃×#7544 #19155170

Order 5888.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×M₁₂₈	C₂	1	2	yes	maximal
2	C₄×C₂₃×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×C₃₂	C₄	1	1	yes	characteristic
4	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
5	C₄×C₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂₃×C₆₄	—	2	2	no	—
7	C₂×C₁₆×C₂₃	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₂₃	C₈	1	1	yes	characteristic
10	C₂₃×C₃₂	—	2	1	no	—
11	C₂₃×C₃₂	D₄	1	1	yes	characteristic
12	C₂₃×C₃₂	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₂₃	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₂₃	C₁₆	1	1	yes	characteristic
16	C₁₆×C₂₃	—	2	1	no	—
17	C₁₆×C₂₃	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₂₃	M₁₆	1	1	yes	characteristic
19	#7544	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂×C₄×C₂₃	—	2	1	no	—
21	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
22	C₈×C₂₃	C₃₂	1	2	yes	—
23	C₈×C₂₃	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₂₃	M₃₂	1	1	yes	characteristic
25	M₁₂₈	C₂×C₂₃	1	2	yes	—
26	C₄×C₃₂	C₂×C₂₃	1	1	yes	characteristic
27	C₂²×C₂₃	M₆₄	1	1	yes	characteristic
28	C₄×C₂₃	—	2	2	no	—
29	C₄×C₂₃	C₄⋊C₁₆	1	1	yes	characteristic
30	C₄×C₂₃	C₂×C₃₂	1	1	yes	characteristic
31	C₂×C₃₂	C₄×C₂₃	1	1	yes	characteristic
32	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
33	C₄×C₁₆	C₄×C₂₃	1	1	yes	characteristic
34	C₆₄	—	2	2	no	—
35	C₂×C₂₃	—	2	1	no	—
36	C₂×C₂₃	C₄⋊C₃₂	1	1	yes	characteristic
37	C₂×C₁₆	C₈×C₂₃	1	1	yes	characteristic
38	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
39	C₄×C₈	C₈×C₂₃	1	1	yes	characteristic
40	C₃₂	—	2	1	no	—
41	C₃₂	D₄×C₂₃	1	1	yes	characteristic
42	C₃₂	Q₈×C₂₃	1	1	yes	characteristic
43	C₂₃	#7544	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₈	C₁₆×C₂₃	1	1	yes	characteristic
45	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
46	C₄²	C₁₆×C₂₃	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	C₄⋊C₄×C₂₃	1	1	yes	characteristic
49	C₁₆	M₁₆×C₂₃	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₂×C₁₆×C₂₃	1	1	yes	characteristic
52	C₈	C₂₃×C₃₂	1	2	yes	—
53	C₈	C₂₃×C₄⋊C₈	1	1	yes	characteristic
54	C₈	C₂₃×M₃₂	1	1	yes	characteristic
55	C₂²	C₂₃×M₆₄	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	C₂₃×C₄⋊C₁₆	1	1	yes	characteristic
58	C₄	C₂×C₂₃×C₃₂	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	C₂₃×C₄⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂₃×C₃₂	D₄
commutator	C₄	C₂×C₂₃×C₃₂
Frattini, cosocle	C₂×C₃₂	C₂²×C₂₃
socle	C₂×C₂₃	C₄⋊C₃₂
prime radical	C₂²×C₂₃	M₆₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂₃×C₃₂ ◁ C₂×C₂₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
19. #7544	C₂₃	direct (N × H)
43. C₂₃	#7544	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×M₁₂₈	2
C₄×C₂₃×C₃₂	2
#7544	23

Subgroups — C23×#7544 #19155170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7544 #19155170