Subgroups — D₄×D₁₂₃ #14520252

Order 1968.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₃	C₂	1	2	yes	maximal, complemented
2	C₃×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#11758315	C₂	1	2	yes	maximal, complemented
4	D₄₉₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₁₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×D₄₁	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₄₁	C₂²	1	2	yes	complemented
8	C₂×D₁₂₃	—	2	4	no	—
9	C₂×D₁₂₃	C₂²	1	1	yes	characteristic
10	C₂×D₁₂₃	C₂²	1	2	yes	complemented
11	C₃×C₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
12	Dic₁₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄₁	—	3	2	no	—
14	D₄×C₄₁	D₃	1	1	yes	characteristic, complemented
15	C₂²⋊D₄₁	—	3	2	no	—
16	D₁₆₄	—	3	1	no	—
17	C₄×D₄₁	—	3	1	no	—
18	C₂×C₃×C₄₁	—	2	2	no	—
19	C₂×C₃×C₄₁	C₂³	1	1	yes	characteristic
20	D₁₂₃	—	2	2	no	—
21	D₁₂₃	D₄	1	2	yes	direct product factor
22	C₂²×C₄₁	C₂×D₃	1	2	yes	complemented
23	C₂×D₄₁	—	3	1	no	—
24	C₂×D₄₁	—	3	2	no	—
25	C₂×D₄₁	—	6	4	no	—
26	C₄×C₄₁	C₂×D₃	1	1	yes	characteristic, complemented
27	Dic₄₁	—	3	1	no	—
28	C₃×C₄₁	C₂×D₄	1	1	yes	characteristic, complemented, Hall
29	C₂×C₄₁	—	2	2	no	—
30	C₂×C₄₁	C₂²×D₃	1	1	yes	characteristic
31	D₄₁	—	3	2	no	—
32	D₄₁	—	6	2	no	—
33	D₃×D₄	—	41	1	no	maximal, Hall
34	C₄₁	D₃×D₄	1	1	yes	characteristic, complemented, Sylow
35	C₂²×D₃	—	41	2	no	—
36	C₃×D₄	D₄₁	1	1	yes	characteristic, complemented
37	#67	—	41	2	no	—
38	D₁₂	—	41	1	no	—
39	C₄×D₃	—	41	1	no	—
40	C₂×D₄	—	123	1	no	Sylow
41	C₂²×C₃	C₂×D₄₁	1	2	yes	complemented
42	C₂×D₃	—	41	1	no	—
43	C₂×D₃	—	41	2	no	—
44	C₂×D₃	—	82	4	no	—
45	C₃×C₄	C₂×D₄₁	1	1	yes	characteristic, complemented
46	Dic₃	—	41	1	no	—
47	C₂³	—	123	2	no	—
48	D₄	—	123	2	no	—
49	D₄	—	123	1	no	—
50	D₄	D₁₂₃	1	1	yes	characteristic, direct product factor
51	C₂×C₄	—	123	1	no	—
52	C₂×C₃	—	2	2	no	—
53	C₂×C₃	C₂²×D₄₁	1	1	yes	characteristic
54	D₃	—	41	2	no	—
55	D₃	—	82	2	no	—
56	C₂²	—	123	1	no	—
57	C₂²	—	123	2	no	—
58	C₂²	—	246	4	no	—
59	C₂²	C₂×D₁₂₃	1	2	yes	complemented
60	C₄	—	123	1	no	—
61	C₄	C₂×D₁₂₃	1	1	yes	characteristic, complemented
62	C₃	D₄×D₄₁	1	1	yes	characteristic, complemented, Sylow
63	C₂	—	2	2	no	—
64	C₂	—	123	2	no	—
65	C₂	—	246	2	no	—
66	C₂	C₂²×D₁₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₁₂₃
commutator, socle, cosocle	C₂×C₃×C₄₁	C₂³
Fitting	C₃×D₄×C₄₁	C₂
hypercenter	D₄	D₁₂₃
hypocenter	C₃×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₃×C₄₁ ▷ C₃×C₄₁
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₂₃	C₂	semidirect
2. C₃×D₄×C₄₁	C₂	semidirect
3. #11758315	C₂	semidirect
4. D₄₉₂	C₂	semidirect
5. C₄×D₁₂₃	C₂	semidirect
7. C₂²×C₃×C₄₁	C₂²	semidirect
10. C₂×D₁₂₃	C₂²	semidirect
11. C₃×C₄×C₄₁	C₂²	semidirect
12. Dic₁₂₃	C₂²	semidirect
14. D₄×C₄₁	D₃	semidirect
21. D₁₂₃	D₄	direct (N × H)
22. C₂²×C₄₁	C₂×D₃	semidirect
26. C₄×C₄₁	C₂×D₃	semidirect
28. C₃×C₄₁	C₂×D₄	semidirect
34. C₄₁	D₃×D₄	semidirect
36. C₃×D₄	D₄₁	semidirect
41. C₂²×C₃	C₂×D₄₁	semidirect
45. C₃×C₄	C₂×D₄₁	semidirect
50. D₄	D₁₂₃	direct (N × H)
59. C₂²	C₂×D₁₂₃	semidirect
61. C₄	C₂×D₁₂₃	semidirect
62. C₃	D₄×D₄₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₁₂₃	2
C₃×D₄×C₄₁	2
#11758315	2
D₄₉₂	2
C₄×D₁₂₃	2
D₄×D₄₁	3
D₃×D₄	41

Subgroups — D4×D123 #14520252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×D₁₂₃ #14520252