Subgroups — C₂²×C₄×Dic₂₂₃ #28384025

Order 14272.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₂₃	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₂₃	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₂₃	C₄	1	16	yes	direct product factor
5	C₂²×Dic₂₂₃	C₂²	1	12	yes	—
6	C₄×Dic₂₂₃	C₂²	1	16	yes	direct product factor
7	C₂⁴×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₂₂₃	C₄	1	8	yes	complemented
9	C₂²×C₄×C₂₂₃	C₂²	1	6	yes	—
10	C₂×Dic₂₂₃	C₂×C₄	1	48	yes	direct product factor
11	C₂×Dic₂₂₃	C₂³	1	8	yes	—
12	C₂³×C₂₂₃	C₂×C₄	1	8	yes	—
13	C₂³×C₂₂₃	C₂×C₄	1	4	yes	—
14	C₂³×C₂₂₃	C₂³	1	3	yes	—
15	C₂×C₄×C₂₂₃	C₂×C₄	1	24	yes	complemented
16	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
17	C₂²×C₂₂₃	C₄²	1	16	yes	complemented
18	C₂²×C₂₂₃	C₂²×C₄	1	12	yes	—
19	C₂²×C₂₂₃	C₂²×C₄	1	6	yes	—
20	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
21	C₄×C₂₂₃	C₂²×C₄	1	8	yes	complemented
22	Dic₂₂₃	C₂²×C₄	1	16	yes	direct product factor
23	C₂×C₂₂₃	C₂×C₄²	1	12	yes	complemented
24	C₂×C₂₂₃	C₂³×C₄	1	2	yes	—
25	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
26	C₂₂₃	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
27	C₂²×C₄²	—	223	1	no	maximal, Sylow
28	C₂³×C₄	—	223	2	no	—
29	C₂³×C₄	D₂₂₃	1	1	yes	characteristic
30	C₂×C₄²	—	223	12	no	—
31	C₂⁴	C₂×D₂₂₃	1	1	yes	characteristic
32	C₂²×C₄	—	223	16	no	—
33	C₂²×C₄	—	223	12	no	—
34	C₂²×C₄	Dic₂₂₃	1	8	yes	direct product factor
35	C₂²×C₄	C₂×D₂₂₃	1	6	yes	—
36	C₄²	—	223	16	no	—
37	C₂³	C₄×D₂₂₃	1	8	yes	—
38	C₂³	C₂×Dic₂₂₃	1	4	yes	—
39	C₂³	C₂²×D₂₂₃	1	3	yes	—
40	C₂×C₄	—	223	48	no	—
41	C₂×C₄	—	223	8	no	—
42	C₂×C₄	C₂×Dic₂₂₃	1	24	yes	direct product factor
43	C₂×C₄	C₂²×D₂₂₃	1	4	yes	—
44	C₂²	C₄×Dic₂₂₃	1	16	yes	direct product factor
45	C₂²	C₂×C₄×D₂₂₃	1	12	yes	—
46	C₂²	C₂²×Dic₂₂₃	1	6	yes	—
47	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
48	C₄	—	223	16	no	—
49	C₄	C₂²×Dic₂₂₃	1	8	yes	direct product factor
50	C₂	C₂×C₄×Dic₂₂₃	1	12	yes	direct product factor
51	C₂	C₂²×C₄×D₂₂₃	1	2	yes	—
52	C₂	C₂³×Dic₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₂₂₃
commutator, hypocenter	C₂₂₃	C₂²×C₄²
Frattini	C₂²	C₂³×D₂₂₃
Fitting	C₂³×C₄×C₂₂₃	C₂
socle, prime radical	C₂⁴×C₂₂₃	C₂²
cosocle	C₂²×C₂₂₃	C₂⁴
LCS	G ▷ C₂₂₃
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₂₂₃	C₂	direct (N × H)
4. C₂²×Dic₂₂₃	C₄	direct (N × H)
6. C₄×Dic₂₂₃	C₂²	direct (N × H)
8. C₂²×C₄×C₂₂₃	C₄	semidirect
10. C₂×Dic₂₂₃	C₂×C₄	direct (N × H)
15. C₂×C₄×C₂₂₃	C₂×C₄	semidirect
17. C₂²×C₂₂₃	C₄²	semidirect
21. C₄×C₂₂₃	C₂²×C₄	semidirect
22. Dic₂₂₃	C₂²×C₄	direct (N × H)
23. C₂×C₂₂₃	C₂×C₄²	semidirect
26. C₂₂₃	C₂²×C₄²	semidirect
34. C₂²×C₄	Dic₂₂₃	direct (N × H)
42. C₂×C₄	C₂×Dic₂₂₃	direct (N × H)
44. C₂²	C₄×Dic₂₂₃	direct (N × H)
49. C₄	C₂²×Dic₂₂₃	direct (N × H)
50. C₂	C₂×C₄×Dic₂₂₃	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×Dic₂₂₃	2
C₂×C₄×Dic₂₂₃	2
C₂³×C₄×C₂₂₃	2
C₂²×C₄²	223

Subgroups — C22×C4×Dic223 #28384025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×Dic₂₂₃ #28384025