Subgroups — C₂×D₄×D₂₂₃ #20297340

Order 7136.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₂₂₃	C₂	1	2	yes	maximal, complemented
2	C₂×D₈₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₂₂₃	C₂	1	2	yes	maximal, complemented
4	D₄×D₂₂₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₂₂₃	—	2	8	no	—
8	C₂²×D₂₂₃	C₂²	1	1	yes	characteristic
9	C₂²×D₂₂₃	C₂²	1	8	yes	complemented
10	C₂²×D₂₂₃	C₂²	1	2	yes	complemented
11	D₈₉₂	C₂²	1	4	yes	complemented
12	C₂²⋊D₂₂₃	C₂²	1	8	yes	complemented
13	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic, complemented
14	C₄×D₂₂₃	C₂²	1	4	yes	complemented
15	C₂³×C₂₂₃	C₂²	1	2	yes	complemented
16	D₄×C₂₂₃	C₂²	1	4	yes	complemented
17	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₂₂₃	—	2	4	no	—
19	C₂²×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₂²×C₂₂₃	C₂³	1	4	yes	complemented
21	C₂×D₂₂₃	—	2	4	no	—
22	C₂×D₂₂₃	—	2	16	no	—
23	C₂×D₂₂₃	D₄	1	4	yes	direct product factor
24	C₂×D₂₂₃	C₂³	1	2	yes	—
25	C₂×D₂₂₃	C₂³	1	4	yes	complemented
26	C₄×C₂₂₃	C₂³	1	2	yes	complemented
27	Dic₂₂₃	C₂³	1	2	yes	complemented
28	C₂×C₂₂₃	—	2	4	no	—
29	C₂×C₂₂₃	C₂×D₄	1	2	yes	complemented
30	C₂×C₂₂₃	C₂⁴	1	1	yes	characteristic
31	D₂₂₃	—	2	4	no	—
32	D₂₂₃	C₂×D₄	1	4	yes	direct product factor
33	C₂₂₃	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
34	C₂²×D₄	—	223	1	no	maximal, Sylow
35	C₂⁴	—	223	2	no	—
36	C₂×D₄	—	223	2	no	—
37	C₂×D₄	—	223	8	no	—
38	C₂×D₄	—	223	1	no	—
39	C₂×D₄	D₂₂₃	1	1	yes	characteristic, direct product factor
40	C₂²×C₄	—	223	1	no	—
41	C₂³	—	446	8	no	—
42	C₂³	—	223	2	no	—
43	C₂³	—	223	8	no	—
44	C₂³	—	223	1	no	—
45	C₂³	C₂×D₂₂₃	1	2	yes	complemented
46	D₄	—	223	8	no	—
47	D₄	—	223	4	no	—
48	D₄	C₂×D₂₂₃	1	4	yes	direct product factor
49	C₂×C₄	—	223	1	no	—
50	C₂×C₄	—	223	4	no	—
51	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic, complemented
52	C₂²	—	2	4	no	—
53	C₂²	—	446	16	no	—
54	C₂²	—	446	4	no	—
55	C₂²	—	223	4	no	—
56	C₂²	—	223	4	no	—
57	C₂²	—	223	2	no	—
58	C₂²	C₂²×D₂₂₃	1	1	yes	characteristic
59	C₂²	C₂²×D₂₂₃	1	4	yes	complemented
60	C₄	—	223	2	no	—
61	C₄	C₂²×D₂₂₃	1	2	yes	complemented
62	C₂	—	2	4	no	—
63	C₂	—	446	4	no	—
64	C₂	—	223	4	no	—
65	C₂	D₄×D₂₂₃	1	2	yes	direct product factor
66	C₂	C₂³×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₂₃
commutator	C₂×C₂₂₃	C₂⁴
Frattini	C₂	C₂³×D₂₂₃
Fitting	C₂×D₄×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂³
hypercenter	C₂×D₄	D₂₂₃
hypocenter	C₂₂₃	C₂²×D₄
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₂₂₃	C₂	semidirect
2. C₂×D₈₉₂	C₂	semidirect
3. C₂×C₂²⋊D₂₂₃	C₂	semidirect
4. D₄×D₂₂₃	C₂	direct (N × H)
5. C₂×C₄×D₂₂₃	C₂	semidirect
6. C₂×D₄×C₂₂₃	C₂	semidirect
9. C₂²×D₂₂₃	C₂²	semidirect
10. C₂²×D₂₂₃	C₂²	semidirect
11. D₈₉₂	C₂²	semidirect
12. C₂²⋊D₂₂₃	C₂²	semidirect
13. C₂×Dic₂₂₃	C₂²	semidirect
14. C₄×D₂₂₃	C₂²	semidirect
15. C₂³×C₂₂₃	C₂²	semidirect
16. D₄×C₂₂₃	C₂²	semidirect
17. C₂×C₄×C₂₂₃	C₂²	semidirect
20. C₂²×C₂₂₃	C₂³	semidirect
23. C₂×D₂₂₃	D₄	direct (N × H)
25. C₂×D₂₂₃	C₂³	semidirect
26. C₄×C₂₂₃	C₂³	semidirect
27. Dic₂₂₃	C₂³	semidirect
29. C₂×C₂₂₃	C₂×D₄	semidirect
32. D₂₂₃	C₂×D₄	direct (N × H)
33. C₂₂₃	C₂²×D₄	semidirect
39. C₂×D₄	D₂₂₃	direct (N × H)
45. C₂³	C₂×D₂₂₃	semidirect
48. D₄	C₂×D₂₂₃	direct (N × H)
51. C₂×C₄	C₂×D₂₂₃	semidirect
59. C₂²	C₂²×D₂₂₃	semidirect
61. C₄	C₂²×D₂₂₃	semidirect
65. C₂	D₄×D₂₂₃	direct (N × H)

Subgroups — C2×D4×D223 #20297340

Conjugacy classes of subgroups

Complements

Subgroups — C₂×D₄×D₂₂₃ #20297340