Subgroups — C₂²×Q₈×D₂₂₃ #28384403

Order 14272.

850 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₄₆	C₂	1	3	yes	maximal, complemented
2	C₂×Q₈×D₂₂₃	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄×D₂₂₃	C₂	1	3	yes	maximal
4	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₂₂₃	C₂²	1	1	yes	characteristic
6	C₂×Dic₄₄₆	C₂²	1	36	yes	complemented
7	Q₈×D₂₂₃	C₂²	1	64	yes	direct product factor
8	C₂²×Dic₂₂₃	C₂²	1	3	yes	—
9	C₂×C₄×D₂₂₃	C₂²	1	36	yes	—
10	C₂×Q₈×C₂₂₃	C₂²	1	12	yes	complemented
11	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
12	C₂²×D₂₂₃	Q₈	1	8	yes	direct product factor
13	C₂²×D₂₂₃	C₂³	1	6	yes	—
14	Dic₄₄₆	C₂³	1	48	yes	complemented
15	C₂×Dic₂₂₃	C₂³	1	18	yes	—
16	C₄×D₂₂₃	C₂³	1	48	yes	—
17	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
18	Q₈×C₂₂₃	C₂³	1	16	yes	complemented
19	C₂×C₄×C₂₂₃	C₂³	1	18	yes	—
20	C₂²×C₂₂₃	C₂×Q₈	1	4	yes	complemented
21	C₂²×C₂₂₃	C₂⁴	1	3	yes	—
22	C₂×D₂₂₃	C₂×Q₈	1	24	yes	direct product factor
23	C₂×D₂₂₃	C₂⁴	1	4	yes	—
24	C₄×C₂₂₃	C₂⁴	1	12	yes	—
25	Dic₂₂₃	C₂⁴	1	12	yes	—
26	C₂×C₂₂₃	C₂²×Q₈	1	6	yes	complemented
27	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
28	D₂₂₃	C₂²×Q₈	1	8	yes	direct product factor
29	C₂₂₃	C₂³×Q₈	1	1	yes	characteristic, complemented, Sylow
30	C₂³×Q₈	—	223	1	no	maximal, Sylow
31	C₂²×Q₈	—	223	24	no	—
32	C₂²×Q₈	—	223	3	no	—
33	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic, direct product factor
34	C₂³×C₄	—	223	3	no	—
35	C₂⁴	—	223	1	no	—
36	C₂×Q₈	—	223	64	no	—
37	C₂×Q₈	—	223	36	no	—
38	C₂×Q₈	C₂×D₂₂₃	1	12	yes	direct product factor
39	C₂²×C₄	—	223	3	no	—
40	C₂²×C₄	—	223	36	no	—
41	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
42	C₂³	—	223	8	no	—
43	C₂³	—	223	6	no	—
44	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
45	Q₈	—	223	48	no	—
46	Q₈	C₂²×D₂₂₃	1	16	yes	direct product factor
47	C₂×C₄	—	223	18	no	—
48	C₂×C₄	—	223	48	no	—
49	C₂×C₄	C₂²×D₂₂₃	1	18	yes	—
50	C₂²	—	223	24	no	—
51	C₂²	—	223	4	no	—
52	C₂²	Q₈×D₂₂₃	1	4	yes	direct product factor
53	C₂²	C₂³×D₂₂₃	1	3	yes	—
54	C₄	—	223	12	no	—
55	C₄	C₂³×D₂₂₃	1	12	yes	—
56	C₂	—	223	8	no	—
57	C₂	C₂×Q₈×D₂₂₃	1	6	yes	direct product factor
58	C₂	C₂⁴×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₂₃
commutator, cosocle	C₂×C₂₂₃	C₂⁵
Frattini	C₂	C₂⁴×D₂₂₃
Fitting	C₂²×Q₈×C₂₂₃	C₂
socle	C₂³×C₂₂₃	C₂³
prime radical	C₂³×D₂₂₃	C₂²
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	C₂³×Q₈
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₄₄₆	C₂	semidirect
2. C₂×Q₈×D₂₂₃	C₂	direct (N × H)
4. C₂²×Q₈×C₂₂₃	C₂	semidirect
6. C₂×Dic₄₄₆	C₂²	semidirect
7. Q₈×D₂₂₃	C₂²	direct (N × H)
10. C₂×Q₈×C₂₂₃	C₂²	semidirect
12. C₂²×D₂₂₃	Q₈	direct (N × H)
14. Dic₄₄₆	C₂³	semidirect
18. Q₈×C₂₂₃	C₂³	semidirect
20. C₂²×C₂₂₃	C₂×Q₈	semidirect
22. C₂×D₂₂₃	C₂×Q₈	direct (N × H)
26. C₂×C₂₂₃	C₂²×Q₈	semidirect
28. D₂₂₃	C₂²×Q₈	direct (N × H)
29. C₂₂₃	C₂³×Q₈	semidirect
33. C₂²×Q₈	D₂₂₃	direct (N × H)
38. C₂×Q₈	C₂×D₂₂₃	direct (N × H)
46. Q₈	C₂²×D₂₂₃	direct (N × H)
52. C₂²	Q₈×D₂₂₃	direct (N × H)
57. C₂	C₂×Q₈×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×Dic₄₄₆	2
C₂×Q₈×D₂₂₃	2
C₂²×C₄×D₂₂₃	2
C₂²×Q₈×C₂₂₃	2
C₂³×Q₈	223

Subgroups — C22×Q8×D223 #28384403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×D₂₂₃ #28384403