Subgroups — C₂₃×#454 #13092003

Order 1472.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#104	C₂	1	2	yes	maximal, complemented
5	C₂₃×#101	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₂₃	—	2	2	no	—
7	#41×C₂₃	—	2	2	no	—
8	#41×C₂₃	C₂²	1	2	yes	—
9	C₂×Q₈×C₂₃	—	2	2	no	—
10	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	SD₁₆×C₂₃	—	2	2	no	—
13	D₈×C₂₃	—	2	2	no	—
14	M₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂²⋊C₄×C₂₃	—	2	2	no	—
16	C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂³×C₂₃	—	2	1	no	—
18	Q₈×C₂₃	—	2	2	no	—
19	Q₈×C₂₃	—	2	2	no	—
20	Q₈×C₂₃	D₄	1	2	yes	—
21	D₄×C₂₃	—	2	2	no	—
22	D₄×C₂₃	—	2	2	no	—
23	D₄×C₂₃	—	2	2	no	—
24	D₄×C₂₃	D₄	1	2	yes	—
25	C₂×C₄×C₂₃	—	2	4	no	—
26	C₂×C₄×C₂₃	—	2	2	no	—
27	C₂×C₄×C₂₃	—	2	1	no	—
28	C₂×C₄×C₂₃	D₄	1	2	yes	—
29	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₈×C₂₃	—	2	2	no	—
31	C₂²×C₂₃	—	2	2	no	—
32	C₂²×C₂₃	—	4	2	no	—
33	C₂²×C₂₃	—	4	1	no	—
34	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₃	—	2	2	no	—
36	C₄×C₂₃	—	2	2	no	—
37	C₄×C₂₃	—	4	1	no	—
38	C₄×C₂₃	C₂×D₄	1	2	yes	—
39	#454	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₂₃	—	2	1	no	—
41	C₂×C₂₃	—	4	2	no	—
42	C₂×C₂₃	—	8	1	no	—
43	C₂×C₂₃	#120	1	1	yes	characteristic
44	2₋¹⁺⁴	C₂×C₂₃	1	1	yes	characteristic, complemented
45	AGL(1,ℤ/8)	C₂×C₂₃	1	2	yes	complemented
46	#124	C₂×C₂₃	1	1	yes	characteristic, complemented
47	#104	C₂×C₂₃	1	2	yes	complemented
48	#101	C₂×C₂₃	1	1	yes	characteristic, complemented
49	C₂₃	#454	1	1	yes	characteristic, direct product factor, Sylow
50	#41	—	2	2	no	—
51	#41	—	2	2	no	—
52	#41	C₂²×C₂₃	1	2	yes	—
53	C₂×Q₈	—	2	2	no	—
54	C₂×Q₈	C₂²×C₂₃	1	1	yes	characteristic
55	C₂×D₄	C₂²×C₂₃	1	1	yes	characteristic
56	SD₁₆	—	2	2	no	—
57	D₈	—	2	2	no	—
58	M₁₆	C₂²×C₂₃	1	2	yes	—
59	C₂²⋊C₄	—	2	2	no	—
60	C₄²	C₂²×C₂₃	1	1	yes	characteristic
61	C₂³	—	2	1	no	—
62	Q₈	—	2	2	no	—
63	Q₈	—	2	2	no	—
64	Q₈	D₄×C₂₃	1	2	yes	—
65	D₄	—	2	2	no	—
66	D₄	—	2	2	no	—
67	D₄	—	2	2	no	—
68	D₄	D₄×C₂₃	1	2	yes	—
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	D₄×C₂₃	1	2	yes	—
73	C₂×C₄	C₂³×C₂₃	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	4	2	no	—
77	C₂²	—	4	1	no	—
78	C₂²	C₂×D₄×C₂₃	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	C₂×D₄×C₂₃	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	4	2	no	—
85	C₂	—	8	1	no	—
86	C₂	C₂₃×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×2₋¹⁺⁴	C₂	semidirect
2. C₂₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂₃×#124	C₂	semidirect
4. C₂₃×#104	C₂	semidirect
5. C₂₃×#101	C₂	semidirect
39. #454	C₂₃	direct (N × H)
44. 2₋¹⁺⁴	C₂×C₂₃	semidirect
45. AGL(1,ℤ/8)	C₂×C₂₃	semidirect
46. #124	C₂×C₂₃	semidirect
47. #104	C₂×C₂₃	semidirect
48. #101	C₂×C₂₃	semidirect
49. C₂₃	#454	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×2₋¹⁺⁴	2
C₂₃×AGL(1,ℤ/8)	2
C₂₃×#124	2
C₂₃×#104	2
C₂₃×#101	2
#454	23

Subgroups — C23×#454 #13092003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#454 #13092003