Subgroups — C₂₃×#8813 #19156439

Order 5888.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1844	C₂	1	6	yes	maximal
2	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#106	—	2	12	no	—
4	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
5	C₂₃×#384	C₂²	1	4	yes	—
6	C₈²×C₂₃	C₄	1	4	yes	complemented
7	C₂²×C₈×C₂₃	D₄	1	6	yes	—
8	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
9	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
10	C₂₃×#106	—	4	24	no	—
11	C₄×C₈×C₂₃	C₂×C₄	1	6	yes	—
12	C₄×C₈×C₂₃	Q₈	1	6	yes	—
13	C₂²×C₄×C₂₃	—	4	4	no	—
14	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
15	C₂×C₈×C₂₃	C₄⋊C₄	1	12	yes	—
16	C₂×C₈×C₂₃	SD₁₆	1	24	yes	—
17	C₄⋊C₄×C₂₃	—	4	24	no	—
18	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
19	C₄²×C₂₃	C₂×Q₈	1	3	yes	—
20	#8813	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₂₃	#127	1	1	yes	characteristic
22	C₂×C₄×C₂₃	—	8	12	no	—
23	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
24	C₂×C₄×C₂₃	C₄⋊Q₈	1	3	yes	—
25	C₂×C₄×C₂₃	C₂×SD₁₆	1	12	yes	—
26	C₈×C₂₃	#106	1	24	yes	complemented
27	#1844	C₂×C₂₃	1	6	yes	—
28	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
29	C₂²×C₂₃	#384	1	1	yes	characteristic
30	C₂²×C₂₃	#492	1	6	yes	—
31	C₄×C₂₃	—	16	4	no	—
32	C₄×C₂₃	C₂×#106	1	6	yes	—
33	C₄×C₂₃	#498	1	6	yes	—
34	C₂×#106	—	2	12	no	—
35	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
36	#384	C₂²×C₂₃	1	4	yes	—
37	C₈²	C₄×C₂₃	1	4	yes	complemented
38	C₂×C₂₃	#1705	1	4	yes	—
39	C₂×C₂₃	#1844	1	3	yes	—
40	C₂²×C₈	D₄×C₂₃	1	6	yes	—
41	C₂×C₄⋊C₄	—	2	12	no	—
42	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
43	#106	—	4	24	no	—
44	C₄×C₈	C₂×C₄×C₂₃	1	6	yes	—
45	C₄×C₈	Q₈×C₂₃	1	6	yes	—
46	C₂₃	#8813	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²×C₄	—	4	4	no	—
48	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
49	C₂×C₈	C₄⋊C₄×C₂₃	1	12	yes	—
50	C₂×C₈	SD₁₆×C₂₃	1	24	yes	—
51	C₄⋊C₄	—	4	24	no	—
52	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
53	C₄²	C₂×Q₈×C₂₃	1	3	yes	—
54	C₂³	C₂₃×#127	1	1	yes	characteristic
55	C₂×C₄	—	8	12	no	—
56	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
57	C₂×C₄	C₂₃×C₄⋊Q₈	1	3	yes	—
58	C₂×C₄	C₂×SD₁₆×C₂₃	1	12	yes	—
59	C₈	C₂₃×#106	1	24	yes	complemented
60	C₂²	C₂₃×#384	1	1	yes	characteristic
61	C₂²	C₂₃×#492	1	6	yes	—
62	C₄	—	16	4	no	—
63	C₄	C₂×C₂₃×#106	1	6	yes	—
64	C₄	C₂₃×#498	1	6	yes	—
65	C₂	C₂₃×#1705	1	4	yes	—
66	C₂	C₂₃×#1844	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₈²×C₂₃	C₄	semidirect
20. #8813	C₂₃	direct (N × H)
26. C₈×C₂₃	#106	semidirect
37. C₈²	C₄×C₂₃	semidirect
46. C₂₃	#8813	direct (N × H)
59. C₈	C₂₃×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#1844	2
C₂×C₈²×C₂₃	2
#8813	23

Subgroups — C23×#8813 #19156439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8813 #19156439