Subgroups — C₂⁶×M₁₆ #72313894

Order 1024.

1012373 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×M₁₆	C₂	1	252	yes	maximal, direct product factor
3	C₂⁶×C₈	C₂	1	2	yes	maximal, complemented
4	C₂⁸	C₄	1	1	yes	characteristic
5	C₂⁶×C₄	C₄	1	1	yes	characteristic
6	C₂⁶×C₄	C₄	1	126	yes	—
7	C₂⁶×C₄	C₂²	1	126	yes	—
8	C₂⁶×C₄	C₂²	1	1	yes	characteristic
9	C₂⁴×M₁₆	C₂²	1	10416	yes	direct product factor
10	C₂⁵×C₈	C₂²	1	252	yes	complemented
11	C₂⁷	—	2	64	no	—
12	C₂⁷	C₂×C₄	1	126	yes	—
13	C₂⁷	C₂×C₄	1	1	yes	characteristic
14	C₂⁵×C₄	C₂×C₄	1	7812	yes	—
15	C₂⁵×C₄	C₂×C₄	1	126	yes	—
16	C₂⁵×C₄	C₂×C₄	1	63	yes	—
17	C₂⁵×C₄	C₂³	1	63	yes	—
18	C₂⁵×C₄	C₂³	1	2604	yes	—
19	C₂³×M₁₆	C₂³	1	89280	yes	direct product factor
20	C₂⁴×C₈	C₂³	1	5208	yes	complemented
21	C₂⁶	—	2	4032	no	—
22	C₂⁶	M₁₆	1	64	yes	direct product factor
23	C₂⁶	C₂²×C₄	1	2604	yes	—
24	C₂⁶	C₂²×C₄	1	63	yes	—
25	C₂⁴×C₄	C₂²×C₄	1	78120	yes	—
26	C₂⁴×C₄	C₂²×C₄	1	2604	yes	—
27	C₂⁴×C₄	C₂²×C₄	1	1953	yes	—
28	C₂⁴×C₄	C₂⁴	1	651	yes	—
29	C₂⁴×C₄	C₂⁴	1	11160	yes	—
30	C₂²×M₁₆	C₂⁴	1	166656	yes	direct product factor
31	C₂³×C₈	C₂⁴	1	22320	yes	complemented
32	C₂⁵	—	2	41664	no	—
33	C₂⁵	C₂×M₁₆	1	2016	yes	direct product factor
34	C₂⁵	C₂³×C₄	1	11160	yes	—
35	C₂⁵	C₂³×C₄	1	651	yes	—
36	C₂³×C₄	C₂³×C₄	1	156240	yes	—
37	C₂³×C₄	C₂³×C₄	1	9765	yes	—
38	C₂³×C₄	C₂³×C₄	1	11160	yes	—
39	C₂³×C₄	C₂⁵	1	1395	yes	—
40	C₂³×C₄	C₂⁵	1	10416	yes	—
41	C₂×M₁₆	C₂⁵	1	64512	yes	direct product factor
42	C₂²×C₈	C₂⁵	1	20832	yes	complemented
43	C₂⁴	—	2	89280	no	—
44	C₂⁴	C₂²×M₁₆	1	10416	yes	direct product factor
45	C₂⁴	C₂⁴×C₄	1	10416	yes	—
46	C₂⁴	C₂⁴×C₄	1	1395	yes	—
47	C₂²×C₄	C₂⁴×C₄	1	62496	yes	—
48	C₂²×C₄	C₂⁴×C₄	1	9765	yes	—
49	C₂²×C₄	C₂⁴×C₄	1	10416	yes	—
50	C₂²×C₄	C₂⁶	1	651	yes	—
51	C₂²×C₄	C₂⁶	1	2016	yes	—
52	M₁₆	C₂⁶	1	4096	yes	direct product factor
53	C₂×C₈	C₂⁶	1	4032	yes	complemented
54	C₂³	—	2	41664	no	—
55	C₂³	C₂³×M₁₆	1	11160	yes	direct product factor
56	C₂³	C₂⁵×C₄	1	2016	yes	—
57	C₂³	C₂⁵×C₄	1	651	yes	—
58	C₂×C₄	C₂⁵×C₄	1	4032	yes	—
59	C₂×C₄	C₂⁵×C₄	1	1953	yes	—
60	C₂×C₄	C₂⁵×C₄	1	2016	yes	—
61	C₂×C₄	C₂⁷	1	63	yes	—
62	C₂×C₄	C₂⁷	1	64	yes	—
63	C₈	C₂⁷	1	128	yes	complemented
64	C₂²	—	2	4032	no	—
65	C₂²	C₂⁴×M₁₆	1	2604	yes	direct product factor
66	C₂²	C₂⁶×C₄	1	64	yes	—
67	C₂²	C₂⁶×C₄	1	63	yes	—
68	C₄	C₂⁶×C₄	1	63	yes	—
69	C₄	C₂⁶×C₄	1	64	yes	—
70	C₄	C₂⁸	1	1	yes	characteristic
71	C₂	—	2	64	no	—
72	C₂	C₂⁵×M₁₆	1	126	yes	direct product factor
73	C₂	C₂⁷×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁶×C₄	C₂²
commutator	C₂	C₂⁷×C₄
Frattini, cosocle	C₄	C₂⁸
socle	C₂⁷	C₂×C₄
prime radical	C₂⁸	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂⁵×M₁₆	C₂	direct (N × H)
3. C₂⁶×C₈	C₂	semidirect
9. C₂⁴×M₁₆	C₂²	direct (N × H)
10. C₂⁵×C₈	C₂²	semidirect
19. C₂³×M₁₆	C₂³	direct (N × H)
20. C₂⁴×C₈	C₂³	semidirect
22. C₂⁶	M₁₆	direct (N × H)
30. C₂²×M₁₆	C₂⁴	direct (N × H)
31. C₂³×C₈	C₂⁴	semidirect
33. C₂⁵	C₂×M₁₆	direct (N × H)
41. C₂×M₁₆	C₂⁵	direct (N × H)
42. C₂²×C₈	C₂⁵	semidirect
44. C₂⁴	C₂²×M₁₆	direct (N × H)
52. M₁₆	C₂⁶	direct (N × H)
53. C₂×C₈	C₂⁶	semidirect
55. C₂³	C₂³×M₁₆	direct (N × H)
63. C₈	C₂⁷	semidirect
65. C₂²	C₂⁴×M₁₆	direct (N × H)
72. C₂	C₂⁵×M₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁷×C₄	2
C₂⁵×M₁₆	2
C₂⁶×C₈	2

Subgroups — C26×M16 #72313894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁶×M₁₆ #72313894