Subgroups — C₂³×M₁₆×C₂₂₃ #43252585

Order 28544.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₂₂₃	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	C₂⁵×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₂₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₂₂₃	C₄	1	14	yes	—
7	C₂³×C₄×C₂₂₃	C₂²	1	14	yes	—
8	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₂₃	C₂²	1	112	yes	direct product factor
10	C₂²×C₈×C₂₂₃	C₂²	1	28	yes	complemented
11	C₂⁴×C₂₂₃	—	2	8	no	—
12	C₂⁴×C₂₂₃	C₂×C₄	1	14	yes	—
13	C₂⁴×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₂₃	C₂×C₄	1	84	yes	—
15	C₂²×C₄×C₂₂₃	C₂×C₄	1	14	yes	—
16	C₂²×C₄×C₂₂₃	C₂×C₄	1	7	yes	—
17	C₂²×C₄×C₂₂₃	C₂³	1	7	yes	—
18	C₂²×C₄×C₂₂₃	C₂³	1	28	yes	—
19	M₁₆×C₂₂₃	C₂³	1	64	yes	direct product factor
20	C₂×C₈×C₂₂₃	C₂³	1	56	yes	complemented
21	C₂³×C₂₂₃	—	2	56	no	—
22	C₂³×C₂₂₃	M₁₆	1	8	yes	direct product factor
23	C₂³×C₂₂₃	C₂²×C₄	1	28	yes	—
24	C₂³×C₂₂₃	C₂²×C₄	1	7	yes	—
25	C₂×C₄×C₂₂₃	C₂²×C₄	1	56	yes	—
26	C₂×C₄×C₂₂₃	C₂²×C₄	1	28	yes	—
27	C₂×C₄×C₂₂₃	C₂²×C₄	1	21	yes	—
28	C₂×C₄×C₂₂₃	C₂⁴	1	7	yes	—
29	C₂×C₄×C₂₂₃	C₂⁴	1	8	yes	—
30	C₈×C₂₂₃	C₂⁴	1	16	yes	complemented
31	C₂²×C₂₂₃	—	2	56	no	—
32	C₂²×C₂₂₃	C₂×M₁₆	1	28	yes	direct product factor
33	C₂²×C₂₂₃	C₂³×C₄	1	8	yes	—
34	C₂²×C₂₂₃	C₂³×C₄	1	7	yes	—
35	C₄×C₂₂₃	C₂³×C₄	1	7	yes	—
36	C₄×C₂₂₃	C₂³×C₄	1	8	yes	—
37	C₄×C₂₂₃	C₂⁵	1	1	yes	characteristic
38	C₂×C₂₂₃	—	2	8	no	—
39	C₂×C₂₂₃	C₂²×M₁₆	1	14	yes	direct product factor
40	C₂×C₂₂₃	C₂⁴×C₄	1	1	yes	characteristic
41	C₂₂₃	C₂³×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
42	C₂³×M₁₆	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂⁴×C₄	C₂×C₂₂₃	1	1	yes	characteristic
44	C₂²×M₁₆	C₂×C₂₂₃	1	28	yes	direct product factor
45	C₂³×C₈	C₂×C₂₂₃	1	2	yes	complemented
46	C₂⁵	C₄×C₂₂₃	1	1	yes	characteristic
47	C₂³×C₄	C₄×C₂₂₃	1	14	yes	—
48	C₂³×C₄	C₄×C₂₂₃	1	1	yes	characteristic
49	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
50	C₂³×C₄	C₂²×C₂₂₃	1	14	yes	—
51	C₂×M₁₆	C₂²×C₂₂₃	1	112	yes	direct product factor
52	C₂²×C₈	C₂²×C₂₂₃	1	28	yes	complemented
53	C₂⁴	—	2	8	no	—
54	C₂⁴	C₂×C₄×C₂₂₃	1	14	yes	—
55	C₂⁴	C₂×C₄×C₂₂₃	1	1	yes	characteristic
56	C₂²×C₄	C₂×C₄×C₂₂₃	1	84	yes	—
57	C₂²×C₄	C₂×C₄×C₂₂₃	1	14	yes	—
58	C₂²×C₄	C₂×C₄×C₂₂₃	1	7	yes	—
59	C₂²×C₄	C₂³×C₂₂₃	1	7	yes	—
60	C₂²×C₄	C₂³×C₂₂₃	1	28	yes	—
61	M₁₆	C₂³×C₂₂₃	1	64	yes	direct product factor
62	C₂×C₈	C₂³×C₂₂₃	1	56	yes	complemented
63	C₂³	—	2	56	no	—
64	C₂³	M₁₆×C₂₂₃	1	8	yes	direct product factor
65	C₂³	C₂²×C₄×C₂₂₃	1	28	yes	—
66	C₂³	C₂²×C₄×C₂₂₃	1	7	yes	—
67	C₂×C₄	C₂²×C₄×C₂₂₃	1	56	yes	—
68	C₂×C₄	C₂²×C₄×C₂₂₃	1	21	yes	—
69	C₂×C₄	C₂²×C₄×C₂₂₃	1	28	yes	—
70	C₂×C₄	C₂⁴×C₂₂₃	1	7	yes	—
71	C₂×C₄	C₂⁴×C₂₂₃	1	8	yes	—
72	C₈	C₂⁴×C₂₂₃	1	16	yes	complemented
73	C₂²	—	2	56	no	—
74	C₂²	C₂×M₁₆×C₂₂₃	1	28	yes	direct product factor
75	C₂²	C₂³×C₄×C₂₂₃	1	8	yes	—
76	C₂²	C₂³×C₄×C₂₂₃	1	7	yes	—
77	C₄	C₂³×C₄×C₂₂₃	1	7	yes	—
78	C₄	C₂³×C₄×C₂₂₃	1	8	yes	—
79	C₄	C₂⁵×C₂₂₃	1	1	yes	characteristic
80	C₂	—	2	8	no	—
81	C₂	C₂²×M₁₆×C₂₂₃	1	14	yes	direct product factor
82	C₂	C₂⁴×C₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₂₂₃	C₂²
commutator	C₂	C₂⁴×C₄×C₂₂₃
Frattini, cosocle	C₄	C₂⁵×C₂₂₃
socle	C₂⁴×C₂₂₃	C₂×C₄
prime radical	C₂⁵×C₂₂₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₁₆×C₂₂₃	C₂	direct (N × H)
3. C₂³×C₈×C₂₂₃	C₂	semidirect
9. C₂×M₁₆×C₂₂₃	C₂²	direct (N × H)
10. C₂²×C₈×C₂₂₃	C₂²	semidirect
19. M₁₆×C₂₂₃	C₂³	direct (N × H)
20. C₂×C₈×C₂₂₃	C₂³	semidirect
22. C₂³×C₂₂₃	M₁₆	direct (N × H)
30. C₈×C₂₂₃	C₂⁴	semidirect
32. C₂²×C₂₂₃	C₂×M₁₆	direct (N × H)
39. C₂×C₂₂₃	C₂²×M₁₆	direct (N × H)
41. C₂₂₃	C₂³×M₁₆	direct (N × H)
42. C₂³×M₁₆	C₂₂₃	direct (N × H)
44. C₂²×M₁₆	C₂×C₂₂₃	direct (N × H)
45. C₂³×C₈	C₂×C₂₂₃	semidirect
51. C₂×M₁₆	C₂²×C₂₂₃	direct (N × H)
52. C₂²×C₈	C₂²×C₂₂₃	semidirect
61. M₁₆	C₂³×C₂₂₃	direct (N × H)
62. C₂×C₈	C₂³×C₂₂₃	semidirect
64. C₂³	M₁₆×C₂₂₃	direct (N × H)
72. C₈	C₂⁴×C₂₂₃	semidirect
74. C₂²	C₂×M₁₆×C₂₂₃	direct (N × H)
81. C₂	C₂²×M₁₆×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₂₂₃	2
C₂²×M₁₆×C₂₂₃	2
C₂³×C₈×C₂₂₃	2
C₂³×M₁₆	223

Subgroups — C23×M16×C223 #43252585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×M₁₆×C₂₂₃ #43252585