Subgroups — C₂₃×#13731 #19161357

Order 5888.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2252	C₂	1	2	yes	maximal
2	C₂₃×#1423	C₂	1	4	yes	maximal
3	C₄×C₂₃×C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#500	—	2	2	no	—
5	C₂×C₂₃×C₃₂	C₂²	1	2	yes	—
6	C₂₃×#366	—	2	2	no	—
7	C₂₃×#366	C₂²	1	4	yes	—
8	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×C₄⋊Q₈	—	4	2	no	—
10	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
12	C₂₃×#107	—	2	4	no	—
13	C₂₃×#107	—	4	2	no	—
14	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
15	C₂₃×C₃₂	Q₈	1	4	yes	complemented
16	C₂×Q₈×C₂₃	—	8	2	no	—
17	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
18	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
19	C₄⋊C₄×C₂₃	—	4	4	no	—
20	C₄⋊C₄×C₂₃	—	8	2	no	—
21	C₄²×C₂₃	D₈	1	1	yes	characteristic
22	C₁₆×C₂₃	Q₁₆	1	2	yes	—
23	C₁₆×C₂₃	C₂×Q₈	1	2	yes	—
24	#13731	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	Q₈×C₂₃	—	16	4	no	—
26	C₂×C₄×C₂₃	—	8	4	no	—
27	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
28	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
29	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
30	C₈×C₂₃	Q₃₂	1	2	yes	—
31	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
32	C₈×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
33	#2252	C₂×C₂₃	1	2	yes	—
34	#1423	C₂×C₂₃	1	4	yes	—
35	C₄×C₃₂	C₂×C₂₃	1	1	yes	characteristic
36	C₂²×C₂₃	C₂×D₁₆	1	1	yes	characteristic
37	C₄×C₂₃	—	16	4	no	—
38	C₄×C₂₃	D₃₂	1	2	yes	—
39	C₄×C₂₃	Q₆₄	1	2	yes	—
40	C₄×C₂₃	#500	1	1	yes	characteristic
41	C₄×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
42	#500	—	2	2	no	—
43	C₂×C₃₂	C₂²×C₂₃	1	2	yes	—
44	#366	—	2	2	no	—
45	#366	C₂²×C₂₃	1	4	yes	—
46	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
47	C₂×C₂₃	#2252	1	1	yes	characteristic
48	C₂×C₂₃	C₂×D₃₂	1	1	yes	characteristic
49	C₂×C₂₃	C₂×Q₆₄	1	1	yes	characteristic
50	C₄⋊Q₈	—	4	2	no	—
51	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
52	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
53	#107	—	2	4	no	—
54	#107	—	4	2	no	—
55	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
56	C₃₂	Q₈×C₂₃	1	4	yes	complemented
57	C₂₃	#13731	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×Q₈	—	8	2	no	—
59	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
60	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
61	C₄⋊C₄	—	4	4	no	—
62	C₄⋊C₄	—	8	2	no	—
63	C₄²	D₈×C₂₃	1	1	yes	characteristic
64	C₁₆	Q₁₆×C₂₃	1	2	yes	—
65	C₁₆	C₂×Q₈×C₂₃	1	2	yes	—
66	Q₈	—	16	4	no	—
67	C₂×C₄	—	8	4	no	—
68	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
69	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
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	—	16	4	no	—
76	C₄	C₂₃×D₃₂	1	2	yes	—
77	C₄	C₂₃×Q₆₄	1	2	yes	—
78	C₄	C₂₃×#500	1	1	yes	characteristic
79	C₄	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
80	C₂	C₂₃×#2252	1	1	yes	characteristic
81	C₂	C₂×C₂₃×D₃₂	1	1	yes	characteristic
82	C₂	C₂×C₂₃×Q₆₄	1	1	yes	characteristic

	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
15. C₂₃×C₃₂	Q₈	semidirect
24. #13731	C₂₃	direct (N × H)
56. C₃₂	Q₈×C₂₃	semidirect
57. C₂₃	#13731	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×#2252	2
C₂₃×#1423	2
C₄×C₂₃×C₃₂	2
#13731	23

Subgroups — C23×#13731 #19161357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13731 #19161357