Subgroups — C₂²×C₈×D₂₂₃ #28384336

Order 14272.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₂₂₃	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₂₂₃	C₄	1	1	yes	characteristic
6	C₂²×Dic₂₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₂₃	C₄	1	6	yes	—
8	C₂×C₄×D₂₂₃	C₂²	1	6	yes	—
9	C₂×C₂₂₃⋊C₈	C₂²	1	6	yes	complemented
10	C₈×D₂₂₃	C₂²	1	16	yes	direct product factor
11	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	C₂²	1	6	yes	complemented
13	C₂²×D₂₂₃	C₈	1	8	yes	direct product factor
14	C₂²×D₂₂₃	C₂×C₄	1	6	yes	—
15	C₂×Dic₂₂₃	C₂×C₄	1	6	yes	—
16	C₄×D₂₂₃	C₂×C₄	1	12	yes	—
17	C₄×D₂₂₃	C₂³	1	4	yes	—
18	C₂₂₃⋊C₈	C₂³	1	4	yes	complemented
19	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
21	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
22	C₈×C₂₂₃	C₂³	1	4	yes	complemented
23	C₂²×C₂₂₃	C₂×C₈	1	4	yes	complemented
24	C₂²×C₂₂₃	C₂²×C₄	1	3	yes	—
25	C₂×D₂₂₃	C₂×C₈	1	24	yes	direct product factor
26	C₂×D₂₂₃	C₂²×C₄	1	4	yes	—
27	C₄×C₂₂₃	C₂²×C₄	1	3	yes	—
28	C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	Dic₂₂₃	C₂²×C₄	1	4	yes	—
30	C₂×C₂₂₃	C₂²×C₈	1	6	yes	complemented
31	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
32	D₂₂₃	C₂²×C₈	1	8	yes	direct product factor
33	C₂₂₃	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
34	C₂³×C₈	—	223	1	no	maximal, Sylow
35	C₂³×C₄	—	223	1	no	—
36	C₂²×C₈	—	223	1	no	—
37	C₂²×C₈	—	223	12	no	—
38	C₂²×C₈	D₂₂₃	1	1	yes	characteristic, direct product factor
39	C₂⁴	—	223	1	no	—
40	C₂²×C₄	—	223	1	no	—
41	C₂²×C₄	—	223	6	no	—
42	C₂²×C₄	—	223	6	no	—
43	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
44	C₂×C₈	—	223	6	no	—
45	C₂×C₈	—	223	16	no	—
46	C₂×C₈	C₂×D₂₂₃	1	6	yes	direct product factor
47	C₂³	—	223	8	no	—
48	C₂³	—	223	6	no	—
49	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
50	C₂×C₄	—	223	6	no	—
51	C₂×C₄	—	223	12	no	—
52	C₂×C₄	—	223	4	no	—
53	C₂×C₄	C₄×D₂₂₃	1	3	yes	—
54	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
55	C₈	—	223	4	no	—
56	C₈	C₂²×D₂₂₃	1	4	yes	direct product factor
57	C₂²	—	223	24	no	—
58	C₂²	—	223	4	no	—
59	C₂²	C₈×D₂₂₃	1	4	yes	direct product factor
60	C₂²	C₂×C₄×D₂₂₃	1	3	yes	—
61	C₄	—	223	4	no	—
62	C₄	C₂×C₄×D₂₂₃	1	3	yes	—
63	C₄	C₂³×D₂₂₃	1	1	yes	characteristic
64	C₂	—	223	8	no	—
65	C₂	C₂×C₈×D₂₂₃	1	6	yes	direct product factor
66	C₂	C₂²×C₄×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₂₂₃
commutator, hypocenter	C₂₂₃	C₂³×C₈
Frattini	C₄	C₂³×D₂₂₃
Fitting	C₂²×C₈×C₂₂₃	C₂
socle	C₂³×C₂₂₃	C₂×C₄
prime radical	C₂³×D₂₂₃	C₄
cosocle	C₄×C₂₂₃	C₂⁴
LCS	G ▷ C₂₂₃
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₂₃⋊C₈	C₂	semidirect
3. C₂×C₈×D₂₂₃	C₂	direct (N × H)
4. C₂²×C₈×C₂₂₃	C₂	semidirect
9. C₂×C₂₂₃⋊C₈	C₂²	semidirect
10. C₈×D₂₂₃	C₂²	direct (N × H)
12. C₂×C₈×C₂₂₃	C₂²	semidirect
13. C₂²×D₂₂₃	C₈	direct (N × H)
18. C₂₂₃⋊C₈	C₂³	semidirect
22. C₈×C₂₂₃	C₂³	semidirect
23. C₂²×C₂₂₃	C₂×C₈	semidirect
25. C₂×D₂₂₃	C₂×C₈	direct (N × H)
30. C₂×C₂₂₃	C₂²×C₈	semidirect
32. D₂₂₃	C₂²×C₈	direct (N × H)
33. C₂₂₃	C₂³×C₈	semidirect
38. C₂²×C₈	D₂₂₃	direct (N × H)
46. C₂×C₈	C₂×D₂₂₃	direct (N × H)
56. C₈	C₂²×D₂₂₃	direct (N × H)
59. C₂²	C₈×D₂₂₃	direct (N × H)
65. C₂	C₂×C₈×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
C₂²×C₄×D₂₂₃	2
C₂²×C₂₂₃⋊C₈	2
C₂×C₈×D₂₂₃	2
C₂²×C₈×C₂₂₃	2
C₂³×C₈	223

Subgroups — C22×C8×D223 #28384336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×D₂₂₃ #28384336