Subgroups — C₂³×C₄²×C₂₂₃ #43252433

Order 28544.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₂₂₃	C₂	1	3	yes	maximal
2	C₂²×C₄²×C₂₂₃	C₂	1	28	yes	maximal, direct product factor
3	C₂⁵×C₂₂₃	C₂²	1	1	yes	characteristic
4	C₂³×C₄×C₂₂₃	C₄	1	48	yes	direct product factor
5	C₂³×C₄×C₂₂₃	C₂²	1	42	yes	—
6	C₂×C₄²×C₂₂₃	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₂₂₃	C₂×C₄	1	24	yes	—
8	C₂⁴×C₂₂₃	C₂³	1	7	yes	—
9	C₂²×C₄×C₂₂₃	C₂×C₄	1	336	yes	direct product factor
10	C₂²×C₄×C₂₂₃	C₂³	1	84	yes	—
11	C₄²×C₂₂₃	C₂³	1	64	yes	direct product factor
12	C₂³×C₂₂₃	C₄²	1	64	yes	direct product factor
13	C₂³×C₂₂₃	C₂²×C₄	1	84	yes	—
14	C₂³×C₂₂₃	C₂⁴	1	7	yes	—
15	C₂×C₄×C₂₂₃	C₂²×C₄	1	336	yes	direct product factor
16	C₂×C₄×C₂₂₃	C₂⁴	1	24	yes	—
17	C₂²×C₂₂₃	C₂×C₄²	1	112	yes	direct product factor
18	C₂²×C₂₂₃	C₂³×C₄	1	42	yes	—
19	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
20	C₄×C₂₂₃	C₂³×C₄	1	48	yes	direct product factor
21	C₂×C₂₂₃	C₂²×C₄²	1	28	yes	direct product factor
22	C₂×C₂₂₃	C₂⁴×C₄	1	3	yes	—
23	C₂₂₃	C₂³×C₄²	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³×C₄²	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁴×C₄	C₂×C₂₂₃	1	3	yes	—
26	C₂²×C₄²	C₂×C₂₂₃	1	28	yes	direct product factor
27	C₂⁵	C₂²×C₂₂₃	1	1	yes	characteristic
28	C₂³×C₄	C₄×C₂₂₃	1	48	yes	direct product factor
29	C₂³×C₄	C₂²×C₂₂₃	1	42	yes	—
30	C₂×C₄²	C₂²×C₂₂₃	1	112	yes	direct product factor
31	C₂⁴	C₂×C₄×C₂₂₃	1	24	yes	—
32	C₂⁴	C₂³×C₂₂₃	1	7	yes	—
33	C₂²×C₄	C₂×C₄×C₂₂₃	1	336	yes	direct product factor
34	C₂²×C₄	C₂³×C₂₂₃	1	84	yes	—
35	C₄²	C₂³×C₂₂₃	1	64	yes	direct product factor
36	C₂³	C₄²×C₂₂₃	1	64	yes	direct product factor
37	C₂³	C₂²×C₄×C₂₂₃	1	84	yes	—
38	C₂³	C₂⁴×C₂₂₃	1	7	yes	—
39	C₂×C₄	C₂²×C₄×C₂₂₃	1	336	yes	direct product factor
40	C₂×C₄	C₂⁴×C₂₂₃	1	24	yes	—
41	C₂²	C₂×C₄²×C₂₂₃	1	112	yes	direct product factor
42	C₂²	C₂³×C₄×C₂₂₃	1	42	yes	—
43	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
44	C₄	C₂³×C₄×C₂₂₃	1	48	yes	direct product factor
45	C₂	C₂²×C₄²×C₂₂₃	1	28	yes	direct product factor
46	C₂	C₂⁴×C₄×C₂₂₃	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄²×C₂₂₃	C₂	direct (N × H)
4. C₂³×C₄×C₂₂₃	C₄	direct (N × H)
6. C₂×C₄²×C₂₂₃	C₂²	direct (N × H)
9. C₂²×C₄×C₂₂₃	C₂×C₄	direct (N × H)
11. C₄²×C₂₂₃	C₂³	direct (N × H)
12. C₂³×C₂₂₃	C₄²	direct (N × H)
15. C₂×C₄×C₂₂₃	C₂²×C₄	direct (N × H)
17. C₂²×C₂₂₃	C₂×C₄²	direct (N × H)
20. C₄×C₂₂₃	C₂³×C₄	direct (N × H)
21. C₂×C₂₂₃	C₂²×C₄²	direct (N × H)
23. C₂₂₃	C₂³×C₄²	direct (N × H)
24. C₂³×C₄²	C₂₂₃	direct (N × H)
26. C₂²×C₄²	C₂×C₂₂₃	direct (N × H)
28. C₂³×C₄	C₄×C₂₂₃	direct (N × H)
30. C₂×C₄²	C₂²×C₂₂₃	direct (N × H)
33. C₂²×C₄	C₂×C₄×C₂₂₃	direct (N × H)
35. C₄²	C₂³×C₂₂₃	direct (N × H)
36. C₂³	C₄²×C₂₂₃	direct (N × H)
39. C₂×C₄	C₂²×C₄×C₂₂₃	direct (N × H)
41. C₂²	C₂×C₄²×C₂₂₃	direct (N × H)
44. C₄	C₂³×C₄×C₂₂₃	direct (N × H)
45. C₂	C₂²×C₄²×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×C₄²×C₂₂₃	2
C₂³×C₄²	223

Subgroups — C23×C42×C223 #43252433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄²×C₂₂₃ #43252433