Subgroups — C₂×C₃×D₁₀₄ #11928793

Order 1248.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₁₀₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₁₀₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×D₁₃	—	2	2	no	—
6	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₃×D₅₂	—	2	2	no	—
8	C₃×D₅₂	C₂²	1	4	yes	complemented
9	C₃×C₈×C₁₃	C₂²	1	2	yes	complemented
10	C₂×D₅₂	C₂×C₃	1	2	yes	complemented
11	C₂×C₈×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
12	D₁₀₄	C₂×C₃	1	4	yes	direct product factor
13	C₂²×C₃×C₁₃	D₄	1	1	yes	characteristic
14	C₂×C₃×D₁₃	—	2	4	no	—
15	C₂×C₃×D₁₃	—	4	4	no	—
16	C₃×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₃×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×D₁₃	—	2	2	no	—
19	C₂×C₄×C₁₃	C₂²×C₃	1	1	yes	characteristic
20	D₅₂	—	2	2	no	—
21	D₅₂	C₂²×C₃	1	4	yes	complemented
22	C₈×C₁₃	C₂²×C₃	1	2	yes	complemented
23	C₂×C₃×D₈	—	13	1	no	maximal, Hall
24	C₂×C₃×C₁₃	D₈	1	2	yes	complemented
25	C₂×C₃×C₁₃	C₂×D₄	1	1	yes	characteristic
26	C₃×D₁₃	—	4	4	no	—
27	C₂²×C₁₃	C₃×D₄	1	1	yes	characteristic
28	C₂×D₁₃	—	2	4	no	—
29	C₂×D₁₃	—	4	4	no	—
30	C₄×C₁₃	C₃×D₄	1	1	yes	characteristic
31	C₄×C₁₃	C₂³×C₃	1	1	yes	characteristic
32	C₂×C₃×D₄	—	13	2	no	—
33	C₃×D₈	—	13	4	no	—
34	C₂×C₃×C₈	D₁₃	1	1	yes	characteristic, complemented
35	C₃×C₁₃	C₂×D₈	1	1	yes	characteristic, complemented, Hall
36	C₂×D₈	—	13	1	no	Sylow
37	C₂×C₁₃	C₃×D₈	1	2	yes	complemented
38	C₂×C₁₃	C₂×C₃×D₄	1	1	yes	characteristic
39	D₁₃	—	4	4	no	—
40	C₂³×C₃	—	26	2	no	—
41	C₃×D₄	—	13	4	no	—
42	C₃×D₄	—	26	2	no	—
43	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
44	C₃×C₈	C₂×D₁₃	1	2	yes	complemented
45	C₂×D₄	—	13	2	no	—
46	D₈	—	13	4	no	—
47	C₂×C₈	C₃×D₁₃	1	1	yes	characteristic, complemented
48	C₁₃	C₂×C₃×D₈	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₃	—	26	4	no	—
50	C₂²×C₃	—	52	4	no	—
51	C₂²×C₃	D₅₂	1	1	yes	characteristic
52	C₃×C₄	D₅₂	1	1	yes	characteristic
53	C₃×C₄	C₂²×D₁₃	1	1	yes	characteristic
54	C₂³	—	26	2	no	—
55	D₄	—	13	4	no	—
56	D₄	—	26	2	no	—
57	C₂×C₄	C₂×C₃×D₁₃	1	1	yes	characteristic
58	C₈	C₂×C₃×D₁₃	1	2	yes	complemented
59	C₂×C₃	—	52	4	no	—
60	C₂×C₃	D₁₀₄	1	2	yes	direct product factor
61	C₂×C₃	C₂×D₅₂	1	1	yes	characteristic
62	C₂²	—	26	4	no	—
63	C₂²	—	52	4	no	—
64	C₂²	C₃×D₅₂	1	1	yes	characteristic
65	C₄	C₃×D₅₂	1	1	yes	characteristic
66	C₄	C₂²×C₃×D₁₃	1	1	yes	characteristic
67	C₃	C₂×D₁₀₄	1	1	yes	characteristic, direct product factor, Sylow
68	C₂	—	52	4	no	—
69	C₂	C₃×D₁₀₄	1	2	yes	direct product factor
70	C₂	C₂×C₃×D₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₅₂
commutator, cosocle	C₄×C₁₃	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₁₃
Fitting	C₂×C₃×C₈×C₁₃	C₂
socle	C₂²×C₃×C₁₃	D₄
hypercenter	C₂×C₃×C₈	D₁₃
hypocenter	C₁₃	C₂×C₃×D₈
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₅₂	C₂	semidirect
2. C₂×C₃×C₈×C₁₃	C₂	semidirect
3. C₃×D₁₀₄	C₂	direct (N × H)
4. C₂×D₁₀₄	C₃	direct (N × H)
8. C₃×D₅₂	C₂²	semidirect
9. C₃×C₈×C₁₃	C₂²	semidirect
10. C₂×D₅₂	C₂×C₃	semidirect
11. C₂×C₈×C₁₃	C₂×C₃	semidirect
12. D₁₀₄	C₂×C₃	direct (N × H)
21. D₅₂	C₂²×C₃	semidirect
22. C₈×C₁₃	C₂²×C₃	semidirect
24. C₂×C₃×C₁₃	D₈	semidirect
34. C₂×C₃×C₈	D₁₃	semidirect
35. C₃×C₁₃	C₂×D₈	semidirect
37. C₂×C₁₃	C₃×D₈	semidirect
44. C₃×C₈	C₂×D₁₃	semidirect
47. C₂×C₈	C₃×D₁₃	semidirect
48. C₁₃	C₂×C₃×D₈	semidirect
58. C₈	C₂×C₃×D₁₃	semidirect
60. C₂×C₃	D₁₀₄	direct (N × H)
67. C₃	C₂×D₁₀₄	direct (N × H)
69. C₂	C₃×D₁₀₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₅₂	2
C₂×C₃×C₈×C₁₃	2
C₃×D₁₀₄	2
C₂×D₁₀₄	3
C₂×C₃×D₈	13

Subgroups — C2×C3×D104 #11928793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₁₀₄ #11928793