Subgroups — C₂₃×#455 #13092004

Order 1472.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#137	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×#100	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₂₃	—	2	2	no	—
7	#41×C₂₃	C₂²	1	2	yes	—
8	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
9	C₂×D₄×C₂₃	—	2	2	no	—
10	C₂×D₄×C₂₃	—	2	2	no	—
11	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	Q₁₆×C₂₃	—	2	2	no	—
13	SD₁₆×C₂₃	—	2	2	no	—
14	M₁₆×C₂₃	C₂²	1	2	yes	complemented
15	C₂²⋊C₄×C₂₃	—	2	2	no	—
16	C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₂₃	—	2	2	no	—
18	C₂³×C₂₃	D₄	1	2	yes	—
19	Q₈×C₂₃	—	2	2	no	—
20	Q₈×C₂₃	D₄	1	2	yes	—
21	D₄×C₂₃	—	2	4	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	2	no	—
26	C₂×C₄×C₂₃	—	2	2	no	—
27	C₂×C₄×C₂₃	—	2	1	no	—
28	C₂×C₄×C₂₃	—	2	1	no	—
29	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₈×C₂₃	—	2	2	no	—
31	C₂²×C₂₃	—	2	2	no	—
32	C₂²×C₂₃	—	2	2	no	—
33	C₂²×C₂₃	—	4	2	no	—
34	C₂²×C₂₃	—	8	2	no	—
35	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₃	—	2	2	no	—
37	C₄×C₂₃	—	4	1	no	—
38	C₄×C₂₃	—	4	1	no	—
39	C₄×C₂₃	C₂×D₄	1	2	yes	—
40	#455	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×C₂₃	—	2	1	no	—
42	C₂×C₂₃	—	4	2	no	—
43	C₂×C₂₃	—	4	2	no	—
44	C₂×C₂₃	#120	1	1	yes	characteristic
45	H(5,𝔽₂)	C₂×C₂₃	1	1	yes	characteristic
46	#137	C₂×C₂₃	1	2	yes	complemented
47	#124	C₂×C₂₃	1	1	yes	characteristic, complemented
48	#104	C₂×C₂₃	1	2	yes	complemented
49	#100	C₂×C₂₃	1	1	yes	characteristic, complemented
50	C₂₃	#455	1	1	yes	characteristic, direct product factor, Sylow
51	#41	—	2	2	no	—
52	#41	C₂²×C₂₃	1	2	yes	—
53	C₂×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
54	C₂×D₄	—	2	2	no	—
55	C₂×D₄	—	2	2	no	—
56	C₂×D₄	C₂²×C₂₃	1	1	yes	characteristic
57	Q₁₆	—	2	2	no	—
58	SD₁₆	—	2	2	no	—
59	M₁₆	C₂²×C₂₃	1	2	yes	complemented
60	C₂²⋊C₄	—	2	2	no	—
61	C₄²	C₂²×C₂₃	1	1	yes	characteristic, complemented
62	C₂³	—	2	2	no	—
63	C₂³	D₄×C₂₃	1	2	yes	—
64	Q₈	—	2	2	no	—
65	Q₈	D₄×C₂₃	1	2	yes	—
66	D₄	—	2	4	no	—
67	D₄	—	2	2	no	—
68	D₄	—	2	2	no	—
69	D₄	D₄×C₂₃	1	2	yes	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	C₂³×C₂₃	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	2	2	no	—
78	C₂²	—	4	2	no	—
79	C₂²	—	8	2	no	—
80	C₂²	C₂×D₄×C₂₃	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	4	1	no	—
83	C₄	—	4	1	no	—
84	C₄	C₂×D₄×C₂₃	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	4	2	no	—
87	C₂	—	4	2	no	—
88	C₂	C₂₃×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₃
prime radical	C₂₃×H(5,𝔽₂)	C₂
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
2. C₂₃×#137	C₂	semidirect
3. C₂₃×#124	C₂	semidirect
4. C₂₃×#104	C₂	semidirect
5. C₂₃×#100	C₂	semidirect
8. C₂×Q₈×C₂₃	C₂²	semidirect
14. M₁₆×C₂₃	C₂²	semidirect
16. C₄²×C₂₃	C₂²	semidirect
40. #455	C₂₃	direct (N × H)
46. #137	C₂×C₂₃	semidirect
47. #124	C₂×C₂₃	semidirect
48. #104	C₂×C₂₃	semidirect
49. #100	C₂×C₂₃	semidirect
50. C₂₃	#455	direct (N × H)
53. C₂×Q₈	C₂²×C₂₃	semidirect
59. M₁₆	C₂²×C₂₃	semidirect
61. C₄²	C₂²×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×H(5,𝔽₂)	2
C₂₃×#137	2
C₂₃×#124	2
C₂₃×#104	2
C₂₃×#100	2
#455	23

Subgroups — C23×#455 #13092004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#455 #13092004