Subgroups — C₂⁵×Dic₁₀₆ #28297972

Order 13568.

63326 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₀₆	C₂	1	124	yes	maximal, direct product factor
2	C₂⁵×Dic₅₃	C₂	1	2	yes	maximal
3	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₀₆	C₂²	1	2480	yes	direct product factor
5	C₂⁴×Dic₅₃	C₂²	1	124	yes	—
6	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₅₃	C₂²	1	62	yes	—
8	C₂²×Dic₁₀₆	C₂³	1	9920	yes	direct product factor
9	C₂³×Dic₅₃	C₂³	1	1240	yes	—
10	C₂⁵×C₅₃	Q₈	1	32	yes	complemented
11	C₂⁵×C₅₃	C₂³	1	31	yes	—
12	C₂³×C₄×C₅₃	C₂³	1	620	yes	—
13	C₂⁴×C₅₃	C₂×Q₈	1	496	yes	complemented
14	C₂⁴×C₅₃	C₂⁴	1	155	yes	—
15	C₂²×C₄×C₅₃	C₂⁴	1	1240	yes	—
16	C₂²×Dic₅₃	C₂⁴	1	2480	yes	—
17	C₂×Dic₁₀₆	C₂⁴	1	7936	yes	direct product factor
18	C₂³×C₅₃	C₂²×Q₈	1	1240	yes	complemented
19	C₂³×C₅₃	C₂⁵	1	155	yes	—
20	C₂×C₄×C₅₃	C₂⁵	1	496	yes	—
21	C₂×Dic₅₃	C₂⁵	1	992	yes	—
22	Dic₁₀₆	C₂⁵	1	1024	yes	direct product factor
23	C₂⁵×Q₈	—	53	1	no	maximal, Sylow
24	C₂²×C₅₃	C₂³×Q₈	1	620	yes	complemented
25	C₂²×C₅₃	C₂⁶	1	31	yes	—
26	C₄×C₅₃	C₂⁶	1	32	yes	—
27	Dic₅₃	C₂⁶	1	64	yes	—
28	C₂⁴×Q₈	—	53	124	no	—
29	C₂⁵×C₄	—	53	2	no	—
30	C₂⁵×C₄	D₅₃	1	1	yes	characteristic
31	C₂×C₅₃	C₂⁴×Q₈	1	62	yes	complemented
32	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
33	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
34	C₂³×Q₈	—	53	2480	no	—
35	C₂⁴×C₄	—	53	124	no	—
36	C₂⁴×C₄	C₂×D₅₃	1	62	yes	—
37	C₅₃	C₂⁵×Q₈	1	1	yes	characteristic, complemented, Sylow
38	C₂⁵	Dic₁₀₆	1	32	yes	direct product factor
39	C₂⁵	C₂²×D₅₃	1	31	yes	—
40	C₂²×Q₈	—	53	9920	no	—
41	C₂³×C₄	—	53	1240	no	—
42	C₂³×C₄	C₂²×D₅₃	1	620	yes	—
43	C₂⁴	C₂×Dic₁₀₆	1	496	yes	direct product factor
44	C₂⁴	C₂³×D₅₃	1	155	yes	—
45	C₂×Q₈	—	53	7936	no	—
46	C₂²×C₄	—	53	2480	no	—
47	C₂²×C₄	C₂³×D₅₃	1	1240	yes	—
48	C₂³	C₂²×Dic₁₀₆	1	1240	yes	direct product factor
49	C₂³	C₂⁴×D₅₃	1	155	yes	—
50	Q₈	—	53	1024	no	—
51	C₂×C₄	—	53	992	no	—
52	C₂×C₄	C₂⁴×D₅₃	1	496	yes	—
53	C₂²	C₂³×Dic₁₀₆	1	620	yes	direct product factor
54	C₂²	C₂⁵×D₅₃	1	31	yes	—
55	C₄	—	53	64	no	—
56	C₄	C₂⁵×D₅₃	1	32	yes	—
57	C₂	C₂⁴×Dic₁₀₆	1	62	yes	direct product factor
58	C₂	C₂⁶×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁶	C₂×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂⁵×C₄×C₅₃	C₂
socle, prime radical	C₂⁶×C₅₃	C₂²
hypercenter	C₂⁵×C₄	D₅₃
hypocenter	C₅₃	C₂⁵×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁶ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₁₀₆	C₂	direct (N × H)
4. C₂³×Dic₁₀₆	C₂²	direct (N × H)
8. C₂²×Dic₁₀₆	C₂³	direct (N × H)
10. C₂⁵×C₅₃	Q₈	semidirect
13. C₂⁴×C₅₃	C₂×Q₈	semidirect
17. C₂×Dic₁₀₆	C₂⁴	direct (N × H)
18. C₂³×C₅₃	C₂²×Q₈	semidirect
22. Dic₁₀₆	C₂⁵	direct (N × H)
24. C₂²×C₅₃	C₂³×Q₈	semidirect
31. C₂×C₅₃	C₂⁴×Q₈	semidirect
37. C₅₃	C₂⁵×Q₈	semidirect
38. C₂⁵	Dic₁₀₆	direct (N × H)
43. C₂⁴	C₂×Dic₁₀₆	direct (N × H)
48. C₂³	C₂²×Dic₁₀₆	direct (N × H)
53. C₂²	C₂³×Dic₁₀₆	direct (N × H)
57. C₂	C₂⁴×Dic₁₀₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₀₆	2
C₂⁵×Dic₅₃	2
C₂⁵×C₄×C₅₃	2
C₂⁵×Q₈	53

Subgroups — C25×Dic106 #28297972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×Dic₁₀₆ #28297972