Subgroups — C₂₃×#8818 #19156444

Order 5888.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1845	C₂	1	6	yes	maximal
2	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#107	—	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×#107	—	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	D₈	1	12	yes	—
17	C₂×C₈×C₂₃	Q₁₆	1	12	yes	—
18	C₄⋊C₄×C₂₃	—	4	24	no	—
19	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
20	C₄²×C₂₃	C₂×Q₈	1	3	yes	—
21	#8818	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₂₃	#127	1	1	yes	characteristic
23	C₂×C₄×C₂₃	—	8	12	no	—
24	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
25	C₂×C₄×C₂₃	C₄⋊Q₈	1	3	yes	—
26	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
27	C₂×C₄×C₂₃	C₂×Q₁₆	1	6	yes	—
28	C₈×C₂₃	#107	1	24	yes	complemented
29	#1845	C₂×C₂₃	1	6	yes	—
30	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
31	C₂²×C₂₃	#384	1	1	yes	characteristic
32	C₂²×C₂₃	#493	1	3	yes	—
33	C₂²×C₂₃	#494	1	3	yes	—
34	C₄×C₂₃	—	16	4	no	—
35	C₄×C₂₃	C₂×#107	1	6	yes	—
36	C₄×C₂₃	#500	1	6	yes	—
37	C₂×#107	—	2	12	no	—
38	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
39	#384	C₂²×C₂₃	1	4	yes	—
40	C₈²	C₄×C₂₃	1	4	yes	complemented
41	C₂×C₂₃	#1712	1	1	yes	characteristic
42	C₂×C₂₃	#1713	1	3	yes	—
43	C₂×C₂₃	#1845	1	3	yes	—
44	C₂²×C₈	D₄×C₂₃	1	6	yes	—
45	C₂×C₄⋊C₄	—	2	12	no	—
46	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
47	#107	—	4	24	no	—
48	C₄×C₈	C₂×C₄×C₂₃	1	6	yes	—
49	C₄×C₈	Q₈×C₂₃	1	6	yes	—
50	C₂₃	#8818	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²×C₄	—	4	4	no	—
52	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
53	C₂×C₈	C₄⋊C₄×C₂₃	1	12	yes	—
54	C₂×C₈	D₈×C₂₃	1	12	yes	—
55	C₂×C₈	Q₁₆×C₂₃	1	12	yes	—
56	C₄⋊C₄	—	4	24	no	—
57	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
58	C₄²	C₂×Q₈×C₂₃	1	3	yes	—
59	C₂³	C₂₃×#127	1	1	yes	characteristic
60	C₂×C₄	—	8	12	no	—
61	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
62	C₂×C₄	C₂₃×C₄⋊Q₈	1	3	yes	—
63	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
64	C₂×C₄	C₂×Q₁₆×C₂₃	1	6	yes	—
65	C₈	C₂₃×#107	1	24	yes	complemented
66	C₂²	C₂₃×#384	1	1	yes	characteristic
67	C₂²	C₂₃×#493	1	3	yes	—
68	C₂²	C₂₃×#494	1	3	yes	—
69	C₄	—	16	4	no	—
70	C₄	C₂×C₂₃×#107	1	6	yes	—
71	C₄	C₂₃×#500	1	6	yes	—
72	C₂	C₂₃×#1712	1	1	yes	characteristic
73	C₂	C₂₃×#1713	1	3	yes	—
74	C₂	C₂₃×#1845	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
21. #8818	C₂₃	direct (N × H)
28. C₈×C₂₃	#107	semidirect
40. C₈²	C₄×C₂₃	semidirect
50. C₂₃	#8818	direct (N × H)
65. C₈	C₂₃×#107	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#1845	2
C₂×C₈²×C₂₃	2
#8818	23

Subgroups — C23×#8818 #19156444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8818 #19156444