Subgroups — C₂²×C₃×Dic₂₆ #11929270

Order 1248.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₁₃	C₂	1	2	yes	maximal
3	C₂×C₃×Dic₂₆	C₂	1	12	yes	maximal, direct product factor
4	C₂²×Dic₂₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄×C₁₃	C₂²	1	6	yes	—
7	C₂×C₃×Dic₁₃	C₂²	1	12	yes	—
8	C₃×Dic₂₆	C₂²	1	16	yes	direct product factor
9	C₂²×C₄×C₁₃	C₂×C₃	1	1	yes	characteristic
10	C₂²×Dic₁₃	C₂×C₃	1	2	yes	—
11	C₂×Dic₂₆	C₂×C₃	1	12	yes	direct product factor
12	C₂²×C₃×C₁₃	Q₈	1	4	yes	complemented
13	C₂²×C₃×C₁₃	C₂³	1	3	yes	—
14	C₃×C₄×C₁₃	C₂³	1	4	yes	—
15	C₃×Dic₁₃	C₂³	1	8	yes	—
16	C₂³×C₁₃	C₂²×C₃	1	1	yes	characteristic
17	C₂×C₄×C₁₃	C₂²×C₃	1	6	yes	—
18	C₂×Dic₁₃	C₂²×C₃	1	12	yes	—
19	Dic₂₆	C₂²×C₃	1	16	yes	direct product factor
20	C₂²×C₃×Q₈	—	13	1	no	maximal, Hall
21	C₂×C₃×C₁₃	C₂×Q₈	1	6	yes	complemented
22	C₂×C₃×C₁₃	C₂⁴	1	1	yes	characteristic
23	C₂²×C₁₃	C₃×Q₈	1	4	yes	complemented
24	C₂²×C₁₃	C₂³×C₃	1	3	yes	—
25	C₄×C₁₃	C₂³×C₃	1	4	yes	—
26	Dic₁₃	C₂³×C₃	1	8	yes	—
27	C₂×C₃×Q₈	—	13	12	no	—
28	C₂²×C₃×C₄	—	13	2	no	—
29	C₂²×C₃×C₄	D₁₃	1	1	yes	characteristic
30	C₃×C₁₃	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
31	C₂²×Q₈	—	13	1	no	Sylow
32	C₂×C₁₃	C₂×C₃×Q₈	1	6	yes	complemented
33	C₂×C₁₃	C₂⁴×C₃	1	1	yes	characteristic
34	C₂³×C₃	C₂×D₁₃	1	1	yes	characteristic
35	C₃×Q₈	—	13	16	no	—
36	C₂×C₃×C₄	—	13	12	no	—
37	C₂×C₃×C₄	C₂×D₁₃	1	6	yes	—
38	C₂×Q₈	—	13	12	no	—
39	C₂²×C₄	—	13	2	no	—
40	C₂²×C₄	C₃×D₁₃	1	1	yes	characteristic
41	C₁₃	C₂²×C₃×Q₈	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₃	Dic₂₆	1	4	yes	direct product factor
43	C₂²×C₃	C₂²×D₁₃	1	3	yes	—
44	C₃×C₄	—	13	8	no	—
45	C₃×C₄	C₂²×D₁₃	1	4	yes	—
46	C₂³	C₂×C₃×D₁₃	1	1	yes	characteristic
47	Q₈	—	13	16	no	—
48	C₂×C₄	—	13	12	no	—
49	C₂×C₄	C₂×C₃×D₁₃	1	6	yes	—
50	C₂×C₃	C₂×Dic₂₆	1	6	yes	direct product factor
51	C₂×C₃	C₂³×D₁₃	1	1	yes	characteristic
52	C₂²	C₃×Dic₂₆	1	4	yes	direct product factor
53	C₂²	C₂²×C₃×D₁₃	1	3	yes	—
54	C₄	—	13	8	no	—
55	C₄	C₂²×C₃×D₁₃	1	4	yes	—
56	C₃	C₂²×Dic₂₆	1	1	yes	characteristic, direct product factor, Sylow
57	C₂	C₂×C₃×Dic₂₆	1	6	yes	direct product factor
58	C₂	C₂³×C₃×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₁₃
Fitting	C₂²×C₃×C₄×C₁₃	C₂
socle, prime radical	C₂³×C₃×C₁₃	C₂²
hypercenter	C₂²×C₃×C₄	D₁₃
hypocenter	C₁₃	C₂²×C₃×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×Dic₂₆	C₂	direct (N × H)
4. C₂²×Dic₂₆	C₃	direct (N × H)
8. C₃×Dic₂₆	C₂²	direct (N × H)
11. C₂×Dic₂₆	C₂×C₃	direct (N × H)
12. C₂²×C₃×C₁₃	Q₈	semidirect
19. Dic₂₆	C₂²×C₃	direct (N × H)
21. C₂×C₃×C₁₃	C₂×Q₈	semidirect
23. C₂²×C₁₃	C₃×Q₈	semidirect
30. C₃×C₁₃	C₂²×Q₈	semidirect
32. C₂×C₁₃	C₂×C₃×Q₈	semidirect
41. C₁₃	C₂²×C₃×Q₈	semidirect
42. C₂²×C₃	Dic₂₆	direct (N × H)
50. C₂×C₃	C₂×Dic₂₆	direct (N × H)
52. C₂²	C₃×Dic₂₆	direct (N × H)
56. C₃	C₂²×Dic₂₆	direct (N × H)
57. C₂	C₂×C₃×Dic₂₆	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×C₃×C₄×C₁₃	2
C₂²×C₃×Dic₁₃	2
C₂×C₃×Dic₂₆	2
C₂²×Dic₂₆	3
C₂²×C₃×Q₈	13

Subgroups — C22×C3×Dic26 #11929270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×Dic₂₆ #11929270