Subgroups — C₂²×C₄×D₂₂₃ #20297338

Order 7136.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₂₂₃	C₂	semidirect
3. C₂×C₄×D₂₂₃	C₂	direct (N × H)
4. C₂²×C₄×C₂₂₃	C₂	semidirect
5. C₂²×D₂₂₃	C₄	direct (N × H)
7. C₂×Dic₂₂₃	C₂²	semidirect
8. C₄×D₂₂₃	C₂²	direct (N × H)
10. C₂×C₄×C₂₂₃	C₂²	semidirect
11. C₂²×C₂₂₃	C₂×C₄	semidirect
13. C₂×D₂₂₃	C₂×C₄	direct (N × H)
15. C₄×C₂₂₃	C₂³	semidirect
16. Dic₂₂₃	C₂³	semidirect
17. C₂×C₂₂₃	C₂²×C₄	semidirect
19. D₂₂₃	C₂²×C₄	direct (N × H)
20. C₂₂₃	C₂³×C₄	semidirect
25. C₂²×C₄	D₂₂₃	direct (N × H)
31. C₂×C₄	C₂×D₂₂₃	direct (N × H)
34. C₂²	C₄×D₂₂₃	direct (N × H)
37. C₄	C₂²×D₂₂₃	direct (N × H)
39. C₂	C₂×C₄×D₂₂₃	direct (N × H)

Subgroups — C22×C4×D223 #20297338

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×C₄×D₂₂₃ #20297338