Subgroups — C₂³×D₄×C₂₂₃ #28383292

Order 14272.

1362 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄×C₂₂₃	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₂₂₃	—	2	16	no	—
5	C₂⁴×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂⁴×C₂₂₃	C₂²	1	28	yes	complemented
7	C₂×D₄×C₂₂₃	C₂²	1	112	yes	direct product factor
8	C₂²×C₄×C₂₂₃	C₂²	1	14	yes	complemented
9	C₂³×C₂₂₃	—	2	112	no	—
10	C₂³×C₂₂₃	D₄	1	8	yes	direct product factor
11	C₂³×C₂₂₃	C₂³	1	7	yes	—
12	C₂³×C₂₂₃	C₂³	1	56	yes	complemented
13	D₄×C₂₂₃	C₂³	1	64	yes	direct product factor
14	C₂×C₄×C₂₂₃	C₂³	1	28	yes	complemented
15	C₂²×C₂₂₃	—	2	112	no	—
16	C₂²×C₂₂₃	C₂×D₄	1	28	yes	direct product factor
17	C₂²×C₂₂₃	C₂⁴	1	7	yes	—
18	C₂²×C₂₂₃	C₂⁴	1	16	yes	complemented
19	C₄×C₂₂₃	C₂⁴	1	8	yes	complemented
20	C₂×C₂₂₃	—	2	16	no	—
21	C₂×C₂₂₃	C₂²×D₄	1	14	yes	direct product factor
22	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
23	C₂₂₃	C₂³×D₄	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³×D₄	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵	C₂×C₂₂₃	1	2	yes	complemented
26	C₂²×D₄	C₂×C₂₂₃	1	28	yes	direct product factor
27	C₂³×C₄	C₂×C₂₂₃	1	1	yes	characteristic, complemented
28	C₂⁴	—	2	16	no	—
29	C₂⁴	C₂²×C₂₂₃	1	1	yes	characteristic
30	C₂⁴	C₂²×C₂₂₃	1	28	yes	complemented
31	C₂×D₄	C₂²×C₂₂₃	1	112	yes	direct product factor
32	C₂²×C₄	C₂²×C₂₂₃	1	14	yes	complemented
33	C₂³	—	2	112	no	—
34	C₂³	D₄×C₂₂₃	1	8	yes	direct product factor
35	C₂³	C₂³×C₂₂₃	1	7	yes	—
36	C₂³	C₂³×C₂₂₃	1	56	yes	complemented
37	D₄	C₂³×C₂₂₃	1	64	yes	direct product factor
38	C₂×C₄	C₂³×C₂₂₃	1	28	yes	complemented
39	C₂²	—	2	112	no	—
40	C₂²	C₂×D₄×C₂₂₃	1	28	yes	direct product factor
41	C₂²	C₂⁴×C₂₂₃	1	7	yes	—
42	C₂²	C₂⁴×C₂₂₃	1	16	yes	complemented
43	C₄	C₂⁴×C₂₂₃	1	8	yes	complemented
44	C₂	—	2	16	no	—
45	C₂	C₂²×D₄×C₂₂₃	1	14	yes	direct product factor
46	C₂	C₂⁵×C₂₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₂₂₃	C₂	semidirect
2. C₂²×D₄×C₂₂₃	C₂	direct (N × H)
3. C₂³×C₄×C₂₂₃	C₂	semidirect
6. C₂⁴×C₂₂₃	C₂²	semidirect
7. C₂×D₄×C₂₂₃	C₂²	direct (N × H)
8. C₂²×C₄×C₂₂₃	C₂²	semidirect
10. C₂³×C₂₂₃	D₄	direct (N × H)
12. C₂³×C₂₂₃	C₂³	semidirect
13. D₄×C₂₂₃	C₂³	direct (N × H)
14. C₂×C₄×C₂₂₃	C₂³	semidirect
16. C₂²×C₂₂₃	C₂×D₄	direct (N × H)
18. C₂²×C₂₂₃	C₂⁴	semidirect
19. C₄×C₂₂₃	C₂⁴	semidirect
21. C₂×C₂₂₃	C₂²×D₄	direct (N × H)
23. C₂₂₃	C₂³×D₄	direct (N × H)
24. C₂³×D₄	C₂₂₃	direct (N × H)
25. C₂⁵	C₂×C₂₂₃	semidirect
26. C₂²×D₄	C₂×C₂₂₃	direct (N × H)
27. C₂³×C₄	C₂×C₂₂₃	semidirect
30. C₂⁴	C₂²×C₂₂₃	semidirect
31. C₂×D₄	C₂²×C₂₂₃	direct (N × H)
32. C₂²×C₄	C₂²×C₂₂₃	semidirect
34. C₂³	D₄×C₂₂₃	direct (N × H)
36. C₂³	C₂³×C₂₂₃	semidirect
37. D₄	C₂³×C₂₂₃	direct (N × H)
38. C₂×C₄	C₂³×C₂₂₃	semidirect
40. C₂²	C₂×D₄×C₂₂₃	direct (N × H)
42. C₂²	C₂⁴×C₂₂₃	semidirect
43. C₄	C₂⁴×C₂₂₃	semidirect
45. C₂	C₂²×D₄×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
C₂⁵×C₂₂₃	2
C₂²×D₄×C₂₂₃	2
C₂³×C₄×C₂₂₃	2
C₂³×D₄	223

Subgroups — C23×D4×C223 #28383292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₄×C₂₂₃ #28383292