Subgroups — C₂₃×#2264 #15669883

Order 2944.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×Q₆₄	C₂	1	2	yes	maximal, complemented
4	C₂₃×SD₆₄	C₂	1	2	yes	maximal, complemented
5	C₂₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#135	—	2	1	no	—
7	C₂×Q₁₆×C₂₃	—	2	1	no	—
8	C₂₃×Q₃₂	—	2	1	no	—
9	C₂₃×Q₃₂	C₂²	1	2	yes	—
10	C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
11	C₂₃×SD₃₂	—	2	1	no	—
12	C₂₃×D₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₃₂	C₂²	1	2	yes	—
15	#41×C₂₃	—	4	1	no	—
16	C₂×Q₈×C₂₃	—	4	1	no	—
17	Q₁₆×C₂₃	—	2	1	no	—
18	Q₁₆×C₂₃	—	2	2	no	—
19	Q₁₆×C₂₃	—	4	1	no	—
20	SD₁₆×C₂₃	—	4	1	no	—
21	D₈×C₂₃	—	2	1	no	—
22	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₁₆×C₂₃	D₄	1	1	yes	characteristic
24	C₁₆×C₂₃	C₂³	1	1	yes	characteristic
25	Q₈×C₂₃	—	4	1	no	—
26	Q₈×C₂₃	—	4	2	no	—
27	Q₈×C₂₃	—	8	1	no	—
28	D₄×C₂₃	—	4	1	no	—
29	D₄×C₂₃	—	8	1	no	—
30	C₂×C₄×C₂₃	—	8	1	no	—
31	C₂×C₄×C₂₃	—	8	1	no	—
32	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
33	C₈×C₂₃	D₈	1	1	yes	characteristic
34	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
35	#2264	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₂₃	—	8	1	no	—
37	C₂²×C₂₃	D₁₆	1	1	yes	characteristic
38	C₄×C₂₃	—	8	1	no	—
39	C₄×C₂₃	—	8	2	no	—
40	C₄×C₂₃	D₁₆	1	1	yes	characteristic
41	C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
42	#508	C₂×C₂₃	1	1	yes	characteristic
43	C₂×Q₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
44	Q₆₄	C₂×C₂₃	1	2	yes	complemented
45	SD₆₄	C₂×C₂₃	1	2	yes	complemented
46	M₆₄	C₂×C₂₃	1	1	yes	characteristic, complemented
47	C₂×C₂₃	—	2	1	no	—
48	C₂×C₂₃	—	16	1	no	—
49	C₂×C₂₃	C₂×D₁₆	1	1	yes	characteristic
50	#135	—	2	1	no	—
51	C₂×Q₁₆	—	2	1	no	—
52	Q₃₂	—	2	1	no	—
53	Q₃₂	C₂²×C₂₃	1	2	yes	—
54	Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
55	SD₃₂	—	2	1	no	—
56	D₁₆	C₂²×C₂₃	1	1	yes	characteristic
57	C₂×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
58	C₃₂	C₂²×C₂₃	1	2	yes	—
59	C₂₃	#2264	1	1	yes	characteristic, direct product factor, Sylow
60	#41	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	Q₁₆	—	2	1	no	—
63	Q₁₆	—	2	2	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	1	no	—
67	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
68	C₁₆	D₄×C₂₃	1	1	yes	characteristic
69	C₁₆	C₂³×C₂₃	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	8	1	no	—
73	D₄	—	4	1	no	—
74	D₄	—	8	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	8	1	no	—
77	C₂×C₄	D₈×C₂₃	1	1	yes	characteristic
78	C₈	D₈×C₂₃	1	1	yes	characteristic
79	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
80	C₂²	—	8	1	no	—
81	C₂²	C₂₃×D₁₆	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	8	2	no	—
84	C₄	C₂₃×D₁₆	1	1	yes	characteristic
85	C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	16	1	no	—
88	C₂	C₂×C₂₃×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₂₃
prime radical	C₂₃×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂×C₄×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×Q₃₂	C₂	semidirect
3. C₂₃×Q₆₄	C₂	semidirect
4. C₂₃×SD₆₄	C₂	semidirect
5. C₂₃×M₆₄	C₂	semidirect
35. #2264	C₂₃	direct (N × H)
43. C₂×Q₃₂	C₂×C₂₃	semidirect
44. Q₆₄	C₂×C₂₃	semidirect
45. SD₆₄	C₂×C₂₃	semidirect
46. M₆₄	C₂×C₂₃	semidirect
59. C₂₃	#2264	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×#508	2
C₂×C₂₃×Q₃₂	2
C₂₃×Q₆₄	2
C₂₃×SD₆₄	2
C₂₃×M₆₄	2
#2264	23

Subgroups — C23×#2264 #15669883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2264 #15669883