Subgroups — C₂⁴×Q₈×C₂₂₃ #43252604

Order 28544.

6264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₂₂₃	C₂	1	60	yes	maximal, direct product factor
2	C₂⁴×C₄×C₂₂₃	C₂	1	3	yes	maximal
3	C₂⁵×C₂₂₃	C₂²	1	1	yes	characteristic
4	C₂²×Q₈×C₂₂₃	C₂²	1	560	yes	direct product factor
5	C₂³×C₄×C₂₂₃	C₂²	1	90	yes	—
6	C₂⁴×C₂₂₃	Q₈	1	16	yes	direct product factor
7	C₂⁴×C₂₂₃	C₂³	1	15	yes	—
8	C₂×Q₈×C₂₂₃	C₂³	1	960	yes	direct product factor
9	C₂²×C₄×C₂₂₃	C₂³	1	420	yes	—
10	C₂³×C₂₂₃	C₂×Q₈	1	120	yes	direct product factor
11	C₂³×C₂₂₃	C₂⁴	1	35	yes	—
12	Q₈×C₂₂₃	C₂⁴	1	256	yes	direct product factor
13	C₂×C₄×C₂₂₃	C₂⁴	1	360	yes	—
14	C₂²×C₂₂₃	C₂²×Q₈	1	140	yes	direct product factor
15	C₂²×C₂₂₃	C₂⁵	1	15	yes	—
16	C₄×C₂₂₃	C₂⁵	1	48	yes	—
17	C₂×C₂₂₃	C₂³×Q₈	1	30	yes	direct product factor
18	C₂×C₂₂₃	C₂⁶	1	1	yes	characteristic
19	C₂₂₃	C₂⁴×Q₈	1	1	yes	characteristic, direct product factor, Sylow
20	C₂⁴×Q₈	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×Q₈	C₂×C₂₂₃	1	60	yes	direct product factor
22	C₂⁴×C₄	C₂×C₂₂₃	1	3	yes	—
23	C₂⁵	C₂²×C₂₂₃	1	1	yes	characteristic
24	C₂²×Q₈	C₂²×C₂₂₃	1	560	yes	direct product factor
25	C₂³×C₄	C₂²×C₂₂₃	1	90	yes	—
26	C₂⁴	Q₈×C₂₂₃	1	16	yes	direct product factor
27	C₂⁴	C₂³×C₂₂₃	1	15	yes	—
28	C₂×Q₈	C₂³×C₂₂₃	1	960	yes	direct product factor
29	C₂²×C₄	C₂³×C₂₂₃	1	420	yes	—
30	C₂³	C₂×Q₈×C₂₂₃	1	120	yes	direct product factor
31	C₂³	C₂⁴×C₂₂₃	1	35	yes	—
32	Q₈	C₂⁴×C₂₂₃	1	256	yes	direct product factor
33	C₂×C₄	C₂⁴×C₂₂₃	1	360	yes	—
34	C₂²	C₂²×Q₈×C₂₂₃	1	140	yes	direct product factor
35	C₂²	C₂⁵×C₂₂₃	1	15	yes	—
36	C₄	C₂⁵×C₂₂₃	1	48	yes	—
37	C₂	C₂³×Q₈×C₂₂₃	1	30	yes	direct product factor
38	C₂	C₂⁶×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₂₂₃	C₂²
commutator, Frattini, cosocle	C₂	C₂⁶×C₂₂₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×C₂₂₃	C₂	direct (N × H)
4. C₂²×Q₈×C₂₂₃	C₂²	direct (N × H)
6. C₂⁴×C₂₂₃	Q₈	direct (N × H)
8. C₂×Q₈×C₂₂₃	C₂³	direct (N × H)
10. C₂³×C₂₂₃	C₂×Q₈	direct (N × H)
12. Q₈×C₂₂₃	C₂⁴	direct (N × H)
14. C₂²×C₂₂₃	C₂²×Q₈	direct (N × H)
17. C₂×C₂₂₃	C₂³×Q₈	direct (N × H)
19. C₂₂₃	C₂⁴×Q₈	direct (N × H)
20. C₂⁴×Q₈	C₂₂₃	direct (N × H)
21. C₂³×Q₈	C₂×C₂₂₃	direct (N × H)
24. C₂²×Q₈	C₂²×C₂₂₃	direct (N × H)
26. C₂⁴	Q₈×C₂₂₃	direct (N × H)
28. C₂×Q₈	C₂³×C₂₂₃	direct (N × H)
30. C₂³	C₂×Q₈×C₂₂₃	direct (N × H)
32. Q₈	C₂⁴×C₂₂₃	direct (N × H)
34. C₂²	C₂²×Q₈×C₂₂₃	direct (N × H)
37. C₂	C₂³×Q₈×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
C₂⁴×C₄×C₂₂₃	2
C₂⁴×Q₈	223

Subgroups — C24×Q8×C223 #43252604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×Q₈×C₂₂₃ #43252604