Subgroups — C₂₃×#13737 #19161363

Order 5888.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2258	C₂	1	1	yes	maximal, characteristic
2	C₂₃×M₁₂₈	C₂	1	3	yes	maximal, complemented
3	C₂×C₂₃×C₆₄	C₂	1	3	yes	maximal, complemented
4	C₂₃×#504	C₄	1	1	yes	characteristic
5	C₂₃×M₆₄	C₄	1	3	yes	—
6	C₂×C₂₃×C₃₂	C₂²	1	3	yes	—
7	C₂₃×C₆₄	C₂²	1	3	yes	—
8	C₂₃×C₆₄	C₂²	1	1	yes	characteristic
9	C₂₃×#131	C₈	1	1	yes	characteristic
10	C₂₃×M₃₂	C₈	1	3	yes	—
11	C₂×C₁₆×C₂₃	C₂×C₄	1	3	yes	—
12	C₂₃×C₃₂	C₂×C₄	1	3	yes	—
13	C₂₃×C₃₂	C₂³	1	1	yes	characteristic
14	#41×C₂₃	C₁₆	1	1	yes	characteristic
15	M₁₆×C₂₃	C₁₆	1	3	yes	—
16	C₂×C₈×C₂₃	C₂×C₈	1	3	yes	—
17	C₁₆×C₂₃	C₂×C₈	1	3	yes	—
18	C₁₆×C₂₃	C₂²×C₄	1	1	yes	characteristic
19	#13737	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	Q₈×C₂₃	C₃₂	1	1	yes	characteristic
21	D₄×C₂₃	C₃₂	1	3	yes	—
22	C₂×C₄×C₂₃	C₂×C₁₆	1	3	yes	—
23	C₈×C₂₃	C₂×C₁₆	1	3	yes	—
24	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
25	#2258	C₂×C₂₃	1	1	yes	characteristic
26	M₁₂₈	C₂×C₂₃	1	3	yes	complemented
27	C₂×C₆₄	C₂×C₂₃	1	3	yes	complemented
28	C₂²×C₂₃	C₂×C₃₂	1	3	yes	—
29	C₄×C₂₃	C₂×C₃₂	1	3	yes	—
30	C₄×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
31	#504	C₄×C₂₃	1	1	yes	characteristic
32	M₆₄	C₄×C₂₃	1	3	yes	—
33	C₂×C₃₂	C₂²×C₂₃	1	3	yes	—
34	C₆₄	C₂²×C₂₃	1	3	yes	—
35	C₆₄	C₂²×C₂₃	1	1	yes	characteristic
36	C₂×C₂₃	—	2	3	no	—
37	C₂×C₂₃	C₂²×C₃₂	1	1	yes	characteristic
38	#131	C₈×C₂₃	1	1	yes	characteristic
39	M₃₂	C₈×C₂₃	1	3	yes	—
40	C₂×C₁₆	C₂×C₄×C₂₃	1	3	yes	—
41	C₃₂	C₂×C₄×C₂₃	1	3	yes	—
42	C₃₂	C₂³×C₂₃	1	1	yes	characteristic
43	C₂₃	#13737	1	1	yes	characteristic, direct product factor, Sylow
44	#41	C₁₆×C₂₃	1	1	yes	characteristic
45	M₁₆	C₁₆×C₂₃	1	3	yes	—
46	C₂×C₈	C₂×C₈×C₂₃	1	3	yes	—
47	C₁₆	C₂×C₈×C₂₃	1	3	yes	—
48	C₁₆	C₂²×C₄×C₂₃	1	1	yes	characteristic
49	Q₈	C₂₃×C₃₂	1	1	yes	characteristic
50	D₄	C₂₃×C₃₂	1	3	yes	—
51	C₂×C₄	C₂×C₁₆×C₂₃	1	3	yes	—
52	C₈	C₂×C₁₆×C₂₃	1	3	yes	—
53	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
54	C₂²	C₂×C₂₃×C₃₂	1	3	yes	—
55	C₄	C₂×C₂₃×C₃₂	1	3	yes	—
56	C₄	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₂₃×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂₃×C₆₄	C₂²
commutator	C₂	C₂²×C₂₃×C₃₂
Frattini, cosocle	C₃₂	C₂³×C₂₃
socle	C₂×C₂₃	C₂²×C₃₂
prime radical	#41×C₂₃	C₁₆
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×M₁₂₈	C₂	semidirect
3. C₂×C₂₃×C₆₄	C₂	semidirect
19. #13737	C₂₃	direct (N × H)
26. M₁₂₈	C₂×C₂₃	semidirect
27. C₂×C₆₄	C₂×C₂₃	semidirect
43. C₂₃	#13737	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×#2258	2
C₂₃×M₁₂₈	2
C₂×C₂₃×C₆₄	2
#13737	23

Subgroups — C23×#13737 #19161363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13737 #19161363