Subgroups — C₂³×C₄×D₂₂₃ #28384398

Order 14272.

1362 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×D₂₂₃	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₂₂₃	C₄	1	16	yes	direct product factor
6	C₂³×D₂₂₃	C₂²	1	14	yes	—
7	C₂²×Dic₂₂₃	C₂²	1	14	yes	complemented
8	C₂×C₄×D₂₂₃	C₂²	1	112	yes	direct product factor
9	C₂⁴×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₂₃	C₂²	1	14	yes	complemented
11	C₂²×D₂₂₃	C₂×C₄	1	112	yes	direct product factor
12	C₂²×D₂₂₃	C₂³	1	28	yes	—
13	C₂×Dic₂₂₃	C₂³	1	28	yes	complemented
14	C₄×D₂₂₃	C₂³	1	64	yes	direct product factor
15	C₂³×C₂₂₃	C₂×C₄	1	8	yes	complemented
16	C₂³×C₂₂₃	C₂³	1	7	yes	—
17	C₂×C₄×C₂₂₃	C₂³	1	28	yes	complemented
18	C₂²×C₂₂₃	C₂²×C₄	1	28	yes	complemented
19	C₂²×C₂₂₃	C₂⁴	1	7	yes	—
20	C₂×D₂₂₃	C₂²×C₄	1	112	yes	direct product factor
21	C₂×D₂₂₃	C₂⁴	1	8	yes	—
22	C₄×C₂₂₃	C₂⁴	1	8	yes	complemented
23	Dic₂₂₃	C₂⁴	1	8	yes	complemented
24	C₂×C₂₂₃	C₂³×C₄	1	14	yes	complemented
25	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
26	D₂₂₃	C₂³×C₄	1	16	yes	direct product factor
27	C₂₂₃	C₂⁴×C₄	1	1	yes	characteristic, complemented, Sylow
28	C₂⁴×C₄	—	223	1	no	maximal, Sylow
29	C₂⁵	—	223	1	no	—
30	C₂³×C₄	—	223	1	no	—
31	C₂³×C₄	—	223	28	no	—
32	C₂³×C₄	D₂₂₃	1	1	yes	characteristic, direct product factor
33	C₂⁴	—	223	16	no	—
34	C₂⁴	—	223	14	no	—
35	C₂⁴	C₂×D₂₂₃	1	1	yes	characteristic
36	C₂²×C₄	—	223	14	no	—
37	C₂²×C₄	—	223	112	no	—
38	C₂²×C₄	C₂×D₂₂₃	1	14	yes	direct product factor
39	C₂³	—	223	112	no	—
40	C₂³	—	223	28	no	—
41	C₂³	C₄×D₂₂₃	1	8	yes	direct product factor
42	C₂³	C₂²×D₂₂₃	1	7	yes	—
43	C₂×C₄	—	223	28	no	—
44	C₂×C₄	—	223	64	no	—
45	C₂×C₄	C₂²×D₂₂₃	1	28	yes	direct product factor
46	C₂²	—	223	112	no	—
47	C₂²	—	223	8	no	—
48	C₂²	C₂×C₄×D₂₂₃	1	28	yes	direct product factor
49	C₂²	C₂³×D₂₂₃	1	7	yes	—
50	C₄	—	223	8	no	—
51	C₄	C₂³×D₂₂₃	1	8	yes	direct product factor
52	C₂	—	223	16	no	—
53	C₂	C₂²×C₄×D₂₂₃	1	14	yes	direct product factor
54	C₂	C₂⁴×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₂₂₃
commutator, hypocenter	C₂₂₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×D₂₂₃
Fitting	C₂³×C₄×C₂₂₃	C₂
socle	C₂⁴×C₂₂₃	C₂²
prime radical	C₂⁴×D₂₂₃	C₂
cosocle	C₂×C₂₂₃	C₂⁵
LCS	G ▷ C₂₂₃
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×Dic₂₂₃	C₂	semidirect
3. C₂²×C₄×D₂₂₃	C₂	direct (N × H)
4. C₂³×C₄×C₂₂₃	C₂	semidirect
5. C₂³×D₂₂₃	C₄	direct (N × H)
7. C₂²×Dic₂₂₃	C₂²	semidirect
8. C₂×C₄×D₂₂₃	C₂²	direct (N × H)
10. C₂²×C₄×C₂₂₃	C₂²	semidirect
11. C₂²×D₂₂₃	C₂×C₄	direct (N × H)
13. C₂×Dic₂₂₃	C₂³	semidirect
14. C₄×D₂₂₃	C₂³	direct (N × H)
15. C₂³×C₂₂₃	C₂×C₄	semidirect
17. C₂×C₄×C₂₂₃	C₂³	semidirect
18. C₂²×C₂₂₃	C₂²×C₄	semidirect
20. C₂×D₂₂₃	C₂²×C₄	direct (N × H)
22. C₄×C₂₂₃	C₂⁴	semidirect
23. Dic₂₂₃	C₂⁴	semidirect
24. C₂×C₂₂₃	C₂³×C₄	semidirect
26. D₂₂₃	C₂³×C₄	direct (N × H)
27. C₂₂₃	C₂⁴×C₄	semidirect
32. C₂³×C₄	D₂₂₃	direct (N × H)
38. C₂²×C₄	C₂×D₂₂₃	direct (N × H)
41. C₂³	C₄×D₂₂₃	direct (N × H)
45. C₂×C₄	C₂²×D₂₂₃	direct (N × H)
48. C₂²	C₂×C₄×D₂₂₃	direct (N × H)
51. C₄	C₂³×D₂₂₃	direct (N × H)
53. C₂	C₂²×C₄×D₂₂₃	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×D₂₂₃	2
C₂³×Dic₂₂₃	2
C₂²×C₄×D₂₂₃	2
C₂³×C₄×C₂₂₃	2
C₂⁴×C₄	223

Subgroups — C23×C4×D223 #28384398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×D₂₂₃ #28384398