Subgroups — C₂₃×#13732 #19161358

Order 5888.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2253	C₂	1	2	yes	maximal
2	C₂₃×#1424	C₂	1	2	yes	maximal
3	C₂₃×#1423	C₂	1	2	yes	maximal
4	C₄×C₂₃×C₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#499	—	2	2	no	—
6	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
8	C₂₃×#367	—	2	2	no	—
9	C₂₃×#366	C₂²	1	4	yes	—
10	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#125	—	4	2	no	—
12	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
14	C₂₃×#107	—	2	4	no	—
15	C₂₃×#106	—	4	2	no	—
16	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂₃×C₃₂	Q₈	1	2	yes	—
18	C₂₃×C₃₂	Q₈	1	2	yes	—
19	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
20	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₂₃	—	4	4	no	—
22	C₄⋊C₄×C₂₃	—	8	4	no	—
23	C₄²×C₂₃	D₈	1	1	yes	characteristic
24	C₁₆×C₂₃	Q₁₆	1	2	yes	—
25	C₁₆×C₂₃	C₂×Q₈	1	1	yes	characteristic
26	C₁₆×C₂₃	C₂×Q₈	1	1	yes	characteristic
27	#13732	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₄×C₂₃	—	8	4	no	—
29	C₂×C₄×C₂₃	D₁₆	1	2	yes	—
30	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
31	C₈×C₂₃	Q₃₂	1	2	yes	—
32	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
33	C₈×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
34	#2253	C₂×C₂₃	1	2	yes	—
35	#1424	C₂×C₂₃	1	2	yes	—
36	#1423	C₂×C₂₃	1	2	yes	—
37	C₄×C₃₂	C₂×C₂₃	1	1	yes	characteristic
38	C₂²×C₂₃	C₂×D₁₆	1	1	yes	characteristic
39	C₄×C₂₃	—	2	2	no	—
40	C₄×C₂₃	—	16	4	no	—
41	C₄×C₂₃	#500	1	1	yes	characteristic
42	C₄×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
43	#499	—	2	2	no	—
44	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
45	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
46	#367	—	2	2	no	—
47	#366	C₂²×C₂₃	1	4	yes	—
48	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
49	C₂×C₂₃	#2252	1	1	yes	characteristic
50	C₂×C₂₃	#2262	1	2	yes	—
51	#125	—	4	2	no	—
52	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
53	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
54	#107	—	2	4	no	—
55	#106	—	4	2	no	—
56	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
57	C₃₂	Q₈×C₂₃	1	2	yes	—
58	C₃₂	Q₈×C₂₃	1	2	yes	—
59	C₂₃	#13732	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
61	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
62	C₄⋊C₄	—	4	4	no	—
63	C₄⋊C₄	—	8	4	no	—
64	C₄²	D₈×C₂₃	1	1	yes	characteristic
65	C₁₆	Q₁₆×C₂₃	1	2	yes	—
66	C₁₆	C₂×Q₈×C₂₃	1	1	yes	characteristic
67	C₁₆	C₂×Q₈×C₂₃	1	1	yes	characteristic
68	C₂×C₄	—	8	4	no	—
69	C₂×C₄	C₂₃×D₁₆	1	2	yes	—
70	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
71	C₈	C₂₃×Q₃₂	1	2	yes	—
72	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
73	C₈	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
74	C₂²	C₂×C₂₃×D₁₆	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	16	4	no	—
77	C₄	C₂₃×#500	1	1	yes	characteristic
78	C₄	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
79	C₂	C₂₃×#2252	1	1	yes	characteristic
80	C₂	C₂₃×#2262	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ C₄×C₈×C₂₃ ◁ C₄×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
27. #13732	C₂₃	direct (N × H)
59. C₂₃	#13732	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×#2253	2
C₂₃×#1424	2
C₂₃×#1423	2
C₄×C₂₃×C₃₂	2
#13732	23

Subgroups — C23×#13732 #19161358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13732 #19161358