Subgroups — C₂₃×#12516 #19160142

Order 5888.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2247	C₂	1	4	yes	maximal
2	C₂₃×#1713	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×Q₃₂	—	2	8	no	—
5	C₂₃×#500	—	2	4	no	—
6	C₂₃×#494	C₂²	1	4	yes	—
7	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
8	C₈²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×Q₁₆×C₂₃	—	2	8	no	—
10	C₂₃×C₄⋊Q₈	—	2	4	no	—
11	C₂₃×Q₃₂	—	4	16	no	—
12	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
13	C₂₃×#107	—	4	8	no	—
14	C₄×C₈×C₂₃	D₄	1	2	yes	—
15	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×Q₈×C₂₃	—	4	4	no	—
17	Q₁₆×C₂₃	—	4	16	no	—
18	C₂×C₈×C₂₃	D₈	1	4	yes	—
19	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
20	C₄⋊C₄×C₂₃	—	4	8	no	—
21	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
22	C₁₆×C₂₃	D₈	1	8	yes	—
23	#12516	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	Q₈×C₂₃	—	8	8	no	—
25	C₂×C₄×C₂₃	—	8	4	no	—
26	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
28	C₈×C₂₃	Q₃₂	1	8	yes	complemented
29	C₈×C₂₃	C₂×D₈	1	4	yes	—
30	#2247	C₂×C₂₃	1	4	yes	—
31	#1713	C₂×C₂₃	1	2	yes	—
32	C₈×C₁₆	C₂×C₂₃	1	1	yes	characteristic
33	C₂²×C₂₃	#493	1	1	yes	characteristic
34	C₄×C₂₃	—	16	4	no	—
35	C₄×C₂₃	#493	1	2	yes	—
36	C₄×C₂₃	C₂×Q₃₂	1	4	yes	—
37	C₂×Q₃₂	—	2	8	no	—
38	#500	—	2	4	no	—
39	#494	C₂²×C₂₃	1	4	yes	—
40	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
41	C₈²	C₂²×C₂₃	1	1	yes	characteristic
42	C₂×C₂₃	#1712	1	1	yes	characteristic
43	C₂×C₂₃	#2247	1	2	yes	—
44	C₂×Q₁₆	—	2	8	no	—
45	C₄⋊Q₈	—	2	4	no	—
46	Q₃₂	—	4	16	no	—
47	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
48	#107	—	4	8	no	—
49	C₄×C₈	D₄×C₂₃	1	2	yes	—
50	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
51	C₂₃	#12516	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×Q₈	—	4	4	no	—
53	Q₁₆	—	4	16	no	—
54	C₂×C₈	D₈×C₂₃	1	4	yes	—
55	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
56	C₄⋊C₄	—	4	8	no	—
57	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
58	C₁₆	D₈×C₂₃	1	8	yes	—
59	Q₈	—	8	8	no	—
60	C₂×C₄	—	8	4	no	—
61	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
62	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
63	C₈	C₂₃×Q₃₂	1	8	yes	complemented
64	C₈	C₂×D₈×C₂₃	1	4	yes	—
65	C₂²	C₂₃×#493	1	1	yes	characteristic
66	C₄	—	16	4	no	—
67	C₄	C₂₃×#493	1	2	yes	—
68	C₄	C₂×C₂₃×Q₃₂	1	4	yes	—
69	C₂	C₂₃×#1712	1	1	yes	characteristic
70	C₂	C₂₃×#2247	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₂₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ C₈²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
23. #12516	C₂₃	direct (N × H)
28. C₈×C₂₃	Q₃₂	semidirect
51. C₂₃	#12516	direct (N × H)
63. C₈	C₂₃×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2247	2
C₂₃×#1713	2
C₈×C₁₆×C₂₃	2
#12516	23

Subgroups — C23×#12516 #19160142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12516 #19160142