Subgroups — C₂×C₄×D₁₂₃ #14520249

Order 1968.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₄	D₁₂₃
commutator, hypocenter	C₃×C₄₁	C₂²×C₄
Frattini	C₂	C₂²×D₁₂₃
Fitting	C₂×C₃×C₄×C₄₁	C₂
socle	C₂²×C₃×C₄₁	C₂²
prime radical	C₂²×D₁₂₃	C₂
cosocle	C₂×C₃×C₄₁	C₂³
LCS	G ▷ C₃×C₄₁
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₄₁	C₂	semidirect
3. C₂×Dic₁₂₃	C₂	semidirect
4. C₄×D₁₂₃	C₂	direct (N × H)
7. C₂×D₁₂₃	C₄	direct (N × H)
9. C₃×C₄×C₄₁	C₂²	semidirect
10. Dic₁₂₃	C₂²	semidirect
12. C₂×C₄×C₄₁	D₃	semidirect
15. C₂×C₃×C₄₁	C₂×C₄	semidirect
17. D₁₂₃	C₂×C₄	direct (N × H)
21. C₄×C₄₁	C₂×D₃	semidirect
23. C₃×C₄₁	C₂²×C₄	semidirect
24. C₂×C₄₁	C₄×D₃	semidirect
28. C₄₁	C₂×C₄×D₃	semidirect
30. C₂×C₃×C₄	D₄₁	semidirect
37. C₃×C₄	C₂×D₄₁	semidirect
42. C₂×C₄	D₁₂₃	direct (N × H)
43. C₂×C₃	C₄×D₄₁	semidirect
50. C₄	C₂×D₁₂₃	direct (N × H)
51. C₃	C₂×C₄×D₄₁	semidirect
53. C₂	C₄×D₁₂₃	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×D₁₂₃	2
C₂×C₃×C₄×C₄₁	2
C₂×Dic₁₂₃	2
C₄×D₁₂₃	2
C₂×C₄×D₄₁	3
C₂×C₄×D₃	41

Subgroups — C2×C4×D123 #14520249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₁₂₃ #14520249