Subgroups — C₂⁴×C₈×C₂₂₃ #43252584

Order 28544.

1976 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	C₂³×C₈×C₂₂₃	C₂	1	30	yes	maximal, direct product factor
3	C₂⁵×C₂₂₃	C₄	1	1	yes	characteristic
4	C₂³×C₄×C₂₂₃	C₄	1	15	yes	—
5	C₂³×C₄×C₂₂₃	C₂²	1	15	yes	—
6	C₂²×C₈×C₂₂₃	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₂₂₃	C₈	1	16	yes	direct product factor
8	C₂⁴×C₂₂₃	C₂×C₄	1	15	yes	—
9	C₂²×C₄×C₂₂₃	C₂×C₄	1	105	yes	—
10	C₂²×C₄×C₂₂₃	C₂³	1	35	yes	—
11	C₂×C₈×C₂₂₃	C₂³	1	120	yes	direct product factor
12	C₂³×C₂₂₃	C₂×C₈	1	120	yes	direct product factor
13	C₂³×C₂₂₃	C₂²×C₄	1	35	yes	—
14	C₂×C₄×C₂₂₃	C₂²×C₄	1	105	yes	—
15	C₂×C₄×C₂₂₃	C₂⁴	1	15	yes	—
16	C₈×C₂₂₃	C₂⁴	1	16	yes	direct product factor
17	C₂²×C₂₂₃	C₂²×C₈	1	140	yes	direct product factor
18	C₂²×C₂₂₃	C₂³×C₄	1	15	yes	—
19	C₄×C₂₂₃	C₂³×C₄	1	15	yes	—
20	C₄×C₂₂₃	C₂⁵	1	1	yes	characteristic
21	C₂×C₂₂₃	C₂³×C₈	1	30	yes	direct product factor
22	C₂×C₂₂₃	C₂⁴×C₄	1	1	yes	characteristic
23	C₂₂₃	C₂⁴×C₈	1	1	yes	characteristic, direct product factor, Sylow
24	C₂⁴×C₈	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁴×C₄	C₂×C₂₂₃	1	1	yes	characteristic
26	C₂³×C₈	C₂×C₂₂₃	1	30	yes	direct product factor
27	C₂⁵	C₄×C₂₂₃	1	1	yes	characteristic
28	C₂³×C₄	C₄×C₂₂₃	1	15	yes	—
29	C₂³×C₄	C₂²×C₂₂₃	1	15	yes	—
30	C₂²×C₈	C₂²×C₂₂₃	1	140	yes	direct product factor
31	C₂⁴	C₈×C₂₂₃	1	16	yes	direct product factor
32	C₂⁴	C₂×C₄×C₂₂₃	1	15	yes	—
33	C₂²×C₄	C₂×C₄×C₂₂₃	1	105	yes	—
34	C₂²×C₄	C₂³×C₂₂₃	1	35	yes	—
35	C₂×C₈	C₂³×C₂₂₃	1	120	yes	direct product factor
36	C₂³	C₂×C₈×C₂₂₃	1	120	yes	direct product factor
37	C₂³	C₂²×C₄×C₂₂₃	1	35	yes	—
38	C₂×C₄	C₂²×C₄×C₂₂₃	1	105	yes	—
39	C₂×C₄	C₂⁴×C₂₂₃	1	15	yes	—
40	C₈	C₂⁴×C₂₂₃	1	16	yes	direct product factor
41	C₂²	C₂²×C₈×C₂₂₃	1	140	yes	direct product factor
42	C₂²	C₂³×C₄×C₂₂₃	1	15	yes	—
43	C₄	C₂³×C₄×C₂₂₃	1	15	yes	—
44	C₄	C₂⁵×C₂₂₃	1	1	yes	characteristic
45	C₂	C₂³×C₈×C₂₂₃	1	30	yes	direct product factor
46	C₂	C₂⁴×C₄×C₂₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂³×C₈×C₂₂₃	C₂	direct (N × H)
6. C₂²×C₈×C₂₂₃	C₂²	direct (N × H)
7. C₂⁴×C₂₂₃	C₈	direct (N × H)
11. C₂×C₈×C₂₂₃	C₂³	direct (N × H)
12. C₂³×C₂₂₃	C₂×C₈	direct (N × H)
16. C₈×C₂₂₃	C₂⁴	direct (N × H)
17. C₂²×C₂₂₃	C₂²×C₈	direct (N × H)
21. C₂×C₂₂₃	C₂³×C₈	direct (N × H)
23. C₂₂₃	C₂⁴×C₈	direct (N × H)
24. C₂⁴×C₈	C₂₂₃	direct (N × H)
26. C₂³×C₈	C₂×C₂₂₃	direct (N × H)
30. C₂²×C₈	C₂²×C₂₂₃	direct (N × H)
31. C₂⁴	C₈×C₂₂₃	direct (N × H)
35. C₂×C₈	C₂³×C₂₂₃	direct (N × H)
36. C₂³	C₂×C₈×C₂₂₃	direct (N × H)
40. C₈	C₂⁴×C₂₂₃	direct (N × H)
41. C₂²	C₂²×C₈×C₂₂₃	direct (N × H)
45. C₂	C₂³×C₈×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
C₂³×C₈×C₂₂₃	2
C₂⁴×C₈	223

Subgroups — C24×C8×C223 #43252584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₈×C₂₂₃ #43252584