Subgroups — C₂⁴×Dic₂₂₆ #28405369

Order 14464.

6264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₂₆	C₂	1	60	yes	maximal, direct product factor
2	C₂⁴×Dic₁₁₃	C₂	1	2	yes	maximal
3	C₂⁴×C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₂₆	C₂²	1	560	yes	direct product factor
5	C₂³×Dic₁₁₃	C₂²	1	60	yes	—
6	C₂⁵×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₁₁₃	C₂²	1	30	yes	—
8	C₂×Dic₂₂₆	C₂³	1	960	yes	direct product factor
9	C₂²×Dic₁₁₃	C₂³	1	280	yes	—
10	C₂⁴×C₁₁₃	Q₈	1	16	yes	complemented
11	C₂⁴×C₁₁₃	C₂³	1	15	yes	—
12	C₂²×C₄×C₁₁₃	C₂³	1	140	yes	—
13	C₂³×C₁₁₃	C₂×Q₈	1	120	yes	complemented
14	C₂³×C₁₁₃	C₂⁴	1	35	yes	—
15	C₂×C₄×C₁₁₃	C₂⁴	1	120	yes	—
16	C₂×Dic₁₁₃	C₂⁴	1	240	yes	—
17	Dic₂₂₆	C₂⁴	1	256	yes	direct product factor
18	C₂²×C₁₁₃	C₂²×Q₈	1	140	yes	complemented
19	C₂²×C₁₁₃	C₂⁵	1	15	yes	—
20	C₄×C₁₁₃	C₂⁵	1	16	yes	—
21	Dic₁₁₃	C₂⁵	1	32	yes	—
22	C₂×C₁₁₃	C₂³×Q₈	1	30	yes	complemented
23	C₂×C₁₁₃	C₂⁶	1	1	yes	characteristic
24	C₂⁴×Q₈	—	113	1	no	maximal, Sylow
25	C₁₁₃	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
26	C₂³×Q₈	—	113	60	no	—
27	C₂⁴×C₄	—	113	2	no	—
28	C₂⁴×C₄	D₁₁₃	1	1	yes	characteristic
29	C₂⁵	C₂×D₁₁₃	1	1	yes	characteristic
30	C₂²×Q₈	—	113	560	no	—
31	C₂³×C₄	—	113	60	no	—
32	C₂³×C₄	C₂×D₁₁₃	1	30	yes	—
33	C₂⁴	Dic₂₂₆	1	16	yes	direct product factor
34	C₂⁴	C₂²×D₁₁₃	1	15	yes	—
35	C₂×Q₈	—	113	960	no	—
36	C₂²×C₄	—	113	280	no	—
37	C₂²×C₄	C₂²×D₁₁₃	1	140	yes	—
38	C₂³	C₂×Dic₂₂₆	1	120	yes	direct product factor
39	C₂³	C₂³×D₁₁₃	1	35	yes	—
40	Q₈	—	113	256	no	—
41	C₂×C₄	—	113	240	no	—
42	C₂×C₄	C₂³×D₁₁₃	1	120	yes	—
43	C₂²	C₂²×Dic₂₂₆	1	140	yes	direct product factor
44	C₂²	C₂⁴×D₁₁₃	1	15	yes	—
45	C₄	—	113	32	no	—
46	C₄	C₂⁴×D₁₁₃	1	16	yes	—
47	C₂	C₂³×Dic₂₂₆	1	30	yes	direct product factor
48	C₂	C₂⁵×D₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₁₁₃
commutator, cosocle	C₂×C₁₁₃	C₂⁶
Frattini	C₂	C₂⁵×D₁₁₃
Fitting	C₂⁴×C₄×C₁₁₃	C₂
socle, prime radical	C₂⁵×C₁₁₃	C₂²
hypercenter	C₂⁴×C₄	D₁₁₃
hypocenter	C₁₁₃	C₂⁴×Q₈
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₂₆	C₂	direct (N × H)
4. C₂²×Dic₂₂₆	C₂²	direct (N × H)
8. C₂×Dic₂₂₆	C₂³	direct (N × H)
10. C₂⁴×C₁₁₃	Q₈	semidirect
13. C₂³×C₁₁₃	C₂×Q₈	semidirect
17. Dic₂₂₆	C₂⁴	direct (N × H)
18. C₂²×C₁₁₃	C₂²×Q₈	semidirect
22. C₂×C₁₁₃	C₂³×Q₈	semidirect
25. C₁₁₃	C₂⁴×Q₈	semidirect
33. C₂⁴	Dic₂₂₆	direct (N × H)
38. C₂³	C₂×Dic₂₂₆	direct (N × H)
43. C₂²	C₂²×Dic₂₂₆	direct (N × H)
47. C₂	C₂³×Dic₂₂₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₂₆	2
C₂⁴×Dic₁₁₃	2
C₂⁴×C₄×C₁₁₃	2
C₂⁴×Q₈	113

Subgroups — C24×Dic226 #28405369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×Dic₂₂₆ #28405369