Subgroups — C₂³×Q₁₆×C₂₂₃ #43252591

Order 28544.

1464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₂₂₃	C₂	1	2	yes	maximal
2	C₂²×Q₁₆×C₂₂₃	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₂₂₃	—	2	14	no	—
5	C₂²×Q₈×C₂₂₃	C₂²	1	28	yes	—
6	C₂³×C₄×C₂₂₃	—	2	2	no	—
7	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₂×Q₁₆×C₂₂₃	C₂²	1	112	yes	direct product factor
9	C₂²×C₈×C₂₂₃	C₂²	1	14	yes	—
10	C₂⁴×C₂₂₃	D₄	1	1	yes	characteristic
11	C₂×Q₈×C₂₂₃	—	2	84	no	—
12	C₂×Q₈×C₂₂₃	C₂³	1	56	yes	—
13	C₂²×C₄×C₂₂₃	—	2	28	no	—
14	C₂²×C₄×C₂₂₃	D₄	1	7	yes	—
15	C₂²×C₄×C₂₂₃	C₂³	1	7	yes	—
16	Q₁₆×C₂₂₃	C₂³	1	64	yes	direct product factor
17	C₂×C₈×C₂₂₃	C₂³	1	28	yes	—
18	C₂³×C₂₂₃	Q₁₆	1	8	yes	direct product factor
19	C₂³×C₂₂₃	C₂×D₄	1	7	yes	—
20	Q₈×C₂₂₃	—	2	56	no	—
21	Q₈×C₂₂₃	C₂⁴	1	16	yes	—
22	C₂×C₄×C₂₂₃	—	2	56	no	—
23	C₂×C₄×C₂₂₃	C₂×D₄	1	21	yes	—
24	C₂×C₄×C₂₂₃	C₂⁴	1	7	yes	—
25	C₈×C₂₂₃	C₂⁴	1	8	yes	—
26	C₂²×C₂₂₃	C₂×Q₁₆	1	28	yes	direct product factor
27	C₂²×C₂₂₃	C₂²×D₄	1	7	yes	—
28	C₄×C₂₂₃	—	2	16	no	—
29	C₄×C₂₂₃	C₂²×D₄	1	7	yes	—
30	C₄×C₂₂₃	C₂⁵	1	1	yes	characteristic
31	C₂×C₂₂₃	C₂²×Q₁₆	1	14	yes	direct product factor
32	C₂×C₂₂₃	C₂³×D₄	1	1	yes	characteristic
33	C₂₂₃	C₂³×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
34	C₂³×Q₁₆	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×Q₈	C₂×C₂₂₃	1	2	yes	—
36	C₂²×Q₁₆	C₂×C₂₂₃	1	28	yes	direct product factor
37	C₂³×C₈	C₂×C₂₂₃	1	1	yes	characteristic
38	C₂²×Q₈	—	2	14	no	—
39	C₂²×Q₈	C₂²×C₂₂₃	1	28	yes	—
40	C₂³×C₄	—	2	2	no	—
41	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
42	C₂×Q₁₆	C₂²×C₂₂₃	1	112	yes	direct product factor
43	C₂²×C₈	C₂²×C₂₂₃	1	14	yes	—
44	C₂⁴	D₄×C₂₂₃	1	1	yes	characteristic
45	C₂×Q₈	—	2	84	no	—
46	C₂×Q₈	C₂³×C₂₂₃	1	56	yes	—
47	C₂²×C₄	—	2	28	no	—
48	C₂²×C₄	D₄×C₂₂₃	1	7	yes	—
49	C₂²×C₄	C₂³×C₂₂₃	1	7	yes	—
50	Q₁₆	C₂³×C₂₂₃	1	64	yes	direct product factor
51	C₂×C₈	C₂³×C₂₂₃	1	28	yes	—
52	C₂³	Q₁₆×C₂₂₃	1	8	yes	direct product factor
53	C₂³	C₂×D₄×C₂₂₃	1	7	yes	—
54	Q₈	—	2	56	no	—
55	Q₈	C₂⁴×C₂₂₃	1	16	yes	—
56	C₂×C₄	—	2	56	no	—
57	C₂×C₄	C₂×D₄×C₂₂₃	1	21	yes	—
58	C₂×C₄	C₂⁴×C₂₂₃	1	7	yes	—
59	C₈	C₂⁴×C₂₂₃	1	8	yes	—
60	C₂²	C₂×Q₁₆×C₂₂₃	1	28	yes	direct product factor
61	C₂²	C₂²×D₄×C₂₂₃	1	7	yes	—
62	C₄	—	2	16	no	—
63	C₄	C₂²×D₄×C₂₂₃	1	7	yes	—
64	C₄	C₂⁵×C₂₂₃	1	1	yes	characteristic
65	C₂	C₂²×Q₁₆×C₂₂₃	1	14	yes	direct product factor
66	C₂	C₂³×D₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₂₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₂₂₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₂₃ ◁ C₂³×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₁₆×C₂₂₃	C₂	direct (N × H)
8. C₂×Q₁₆×C₂₂₃	C₂²	direct (N × H)
16. Q₁₆×C₂₂₃	C₂³	direct (N × H)
18. C₂³×C₂₂₃	Q₁₆	direct (N × H)
26. C₂²×C₂₂₃	C₂×Q₁₆	direct (N × H)
31. C₂×C₂₂₃	C₂²×Q₁₆	direct (N × H)
33. C₂₂₃	C₂³×Q₁₆	direct (N × H)
34. C₂³×Q₁₆	C₂₂₃	direct (N × H)
36. C₂²×Q₁₆	C₂×C₂₂₃	direct (N × H)
42. C₂×Q₁₆	C₂²×C₂₂₃	direct (N × H)
50. Q₁₆	C₂³×C₂₂₃	direct (N × H)
52. C₂³	Q₁₆×C₂₂₃	direct (N × H)
60. C₂²	C₂×Q₁₆×C₂₂₃	direct (N × H)
65. C₂	C₂²×Q₁₆×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
C₂³×Q₈×C₂₂₃	2
C₂²×Q₁₆×C₂₂₃	2
C₂³×C₈×C₂₂₃	2
C₂³×Q₁₆	223

Subgroups — C23×Q16×C223 #43252591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Q₁₆×C₂₂₃ #43252591