Subgroups — C₂₃×#13741 #19161367

Order 5888.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2262	C₂	1	2	yes	maximal, complemented
2	C₂₃×Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×SD₁₂₈	C₂	1	2	yes	maximal, complemented
4	C₂₃×D₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#508	—	2	2	no	—
7	C₂₃×Q₆₄	C₂²	1	2	yes	—
8	C₂₃×SD₆₄	—	2	2	no	—
9	C₂₃×D₃₂	C₂²	1	2	yes	—
10	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
11	C₂₃×C₆₄	C₂²	1	1	yes	characteristic
12	C₂₃×C₆₄	C₂²	1	1	yes	characteristic
13	C₂₃×#135	—	4	2	no	—
14	C₂₃×Q₃₂	—	2	2	no	—
15	C₂₃×SD₃₂	—	4	2	no	—
16	C₂₃×D₁₆	—	2	2	no	—
17	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
18	C₂₃×C₃₂	D₄	1	1	yes	characteristic
19	C₂₃×C₃₂	C₂³	1	1	yes	characteristic
20	#41×C₂₃	—	8	2	no	—
21	Q₁₆×C₂₃	—	4	2	no	—
22	SD₁₆×C₂₃	—	8	2	no	—
23	D₈×C₂₃	—	4	2	no	—
24	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
25	C₁₆×C₂₃	D₈	1	1	yes	characteristic
26	C₁₆×C₂₃	C₂×D₄	1	1	yes	characteristic
27	#13741	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	Q₈×C₂₃	—	8	2	no	—
29	D₄×C₂₃	—	8	2	no	—
30	D₄×C₂₃	—	16	2	no	—
31	C₂×C₄×C₂₃	—	16	2	no	—
32	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
33	C₈×C₂₃	D₁₆	1	1	yes	characteristic
34	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
35	#2262	C₂×C₂₃	1	2	yes	complemented
36	Q₁₂₈	C₂×C₂₃	1	1	yes	characteristic, complemented
37	SD₁₂₈	C₂×C₂₃	1	2	yes	complemented
38	D₆₄	C₂×C₂₃	1	1	yes	characteristic, complemented
39	C₂×C₆₄	C₂×C₂₃	1	1	yes	characteristic, complemented
40	C₂²×C₂₃	—	16	2	no	—
41	C₂²×C₂₃	D₃₂	1	1	yes	characteristic
42	C₄×C₂₃	—	16	2	no	—
43	C₄×C₂₃	D₃₂	1	1	yes	characteristic
44	C₄×C₂₃	C₂×D₁₆	1	1	yes	characteristic
45	#508	—	2	2	no	—
46	Q₆₄	C₂²×C₂₃	1	2	yes	—
47	SD₆₄	—	2	2	no	—
48	D₃₂	C₂²×C₂₃	1	2	yes	—
49	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
50	C₆₄	C₂²×C₂₃	1	1	yes	characteristic
51	C₆₄	C₂²×C₂₃	1	1	yes	characteristic
52	C₂×C₂₃	—	2	1	no	—
53	C₂×C₂₃	—	32	2	no	—
54	C₂×C₂₃	C₂×D₃₂	1	1	yes	characteristic
55	#135	—	4	2	no	—
56	Q₃₂	—	2	2	no	—
57	SD₃₂	—	4	2	no	—
58	D₁₆	—	2	2	no	—
59	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
60	C₃₂	D₄×C₂₃	1	1	yes	characteristic
61	C₃₂	C₂³×C₂₃	1	1	yes	characteristic
62	C₂₃	#13741	1	1	yes	characteristic, direct product factor, Sylow
63	#41	—	8	2	no	—
64	Q₁₆	—	4	2	no	—
65	SD₁₆	—	8	2	no	—
66	D₈	—	4	2	no	—
67	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
68	C₁₆	D₈×C₂₃	1	1	yes	characteristic
69	C₁₆	C₂×D₄×C₂₃	1	1	yes	characteristic
70	Q₈	—	8	2	no	—
71	D₄	—	8	2	no	—
72	D₄	—	16	2	no	—
73	C₂×C₄	—	16	2	no	—
74	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
75	C₈	C₂₃×D₁₆	1	1	yes	characteristic
76	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
77	C₂²	—	16	2	no	—
78	C₂²	C₂₃×D₃₂	1	1	yes	characteristic
79	C₄	—	16	2	no	—
80	C₄	C₂₃×D₃₂	1	1	yes	characteristic
81	C₄	C₂×C₂₃×D₁₆	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	32	2	no	—
84	C₂	C₂×C₂₃×D₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₂₃
socle	C₂×C₂₃	C₂×D₃₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×C₄×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₁₆×C₂₃ ◁ C₂×C₂₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2262	C₂	semidirect
2. C₂₃×Q₁₂₈	C₂	semidirect
3. C₂₃×SD₁₂₈	C₂	semidirect
4. C₂₃×D₆₄	C₂	semidirect
5. C₂×C₂₃×C₆₄	C₂	semidirect
27. #13741	C₂₃	direct (N × H)
35. #2262	C₂×C₂₃	semidirect
36. Q₁₂₈	C₂×C₂₃	semidirect
37. SD₁₂₈	C₂×C₂₃	semidirect
38. D₆₄	C₂×C₂₃	semidirect
39. C₂×C₆₄	C₂×C₂₃	semidirect
62. C₂₃	#13741	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×#2262	2
C₂₃×Q₁₂₈	2
C₂₃×SD₁₂₈	2
C₂₃×D₆₄	2
C₂×C₂₃×C₆₄	2
#13741	23

Subgroups — C23×#13741 #19161367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13741 #19161367