Subgroups — C₂⁶×#127 #72291409

Order 2048.

38972520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×#127	C₂	1	504	yes	maximal, direct product factor
2	C₂⁷×D₄	C₂	1	6	yes	maximal, complemented
3	C₂⁶×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁹	C₂²	1	4	yes	—
5	C₂⁶×D₄	—	2	768	no	—
6	C₂⁶×D₄	C₂²	1	1512	yes	complemented
7	C₂⁷×C₄	C₂²	1	3	yes	—
8	C₂⁴×#127	C₂²	1	41664	yes	direct product factor
9	C₂⁵×C₄²	C₂²	1	252	yes	complemented
10	C₂⁸	—	2	768	no	—
11	C₂⁸	C₂³	1	504	yes	—
12	C₂⁸	C₂³	1	1	yes	characteristic
13	C₂⁵×D₄	—	2	96768	no	—
14	C₂⁵×D₄	C₂³	1	62496	yes	complemented
15	C₂⁶×C₄	D₄	1	384	yes	complemented
16	C₂⁶×C₄	C₂³	1	378	yes	—
17	C₂³×#127	C₂³	1	714240	yes	direct product factor
18	C₂⁴×C₄²	C₂³	1	10416	yes	complemented
19	C₂⁷	—	2	48384	no	—
20	C₂⁷	—	4	16384	no	—
21	C₂⁷	C₂×D₄	1	192	yes	—
22	C₂⁷	C₂⁴	1	10416	yes	—
23	C₂⁷	C₂⁴	1	63	yes	—
24	C₂⁴×D₄	—	2	1999872	no	—
25	C₂⁴×D₄	C₂⁴	1	535680	yes	complemented
26	C₂⁵×C₄	C₂×D₄	1	24192	yes	complemented
27	C₂⁵×C₄	C₂⁴	1	7812	yes	—
28	C₂²×#127	C₂⁴	1	2666496	yes	direct product factor
29	C₂³×C₄²	C₂⁴	1	89280	yes	complemented
30	C₂⁶	—	2	499968	no	—
31	C₂⁶	—	4	516096	no	—
32	C₂⁶	#127	1	4096	yes	direct product factor
33	C₂⁶	C₂²×D₄	1	6048	yes	—
34	C₂⁶	C₂⁵	1	44640	yes	—
35	C₂⁶	C₂⁵	1	651	yes	—
36	C₂³×D₄	—	2	8570880	no	—
37	C₂³×D₄	C₂⁵	1	999936	yes	complemented
38	C₂⁴×C₄	C₂²×D₄	1	249984	yes	complemented
39	C₂⁴×C₄	C₂⁵	1	33480	yes	—
40	C₂×#127	C₂⁵	1	2064384	yes	direct product factor
41	C₂²×C₄²	C₂⁵	1	166656	yes	complemented
42	C₂⁵	—	2	1071360	no	—
43	C₂⁵	—	4	2666496	no	—
44	C₂⁵	C₂×#127	1	64512	yes	direct product factor
45	C₂⁵	C₂³×D₄	1	31248	yes	—
46	C₂⁵	C₂⁶	1	41664	yes	—
47	C₂⁵	C₂⁶	1	1395	yes	—
48	C₂²×D₄	—	2	7999488	no	—
49	C₂²×D₄	C₂⁶	1	387072	yes	complemented
50	C₂³×C₄	C₂³×D₄	1	535680	yes	complemented
51	C₂³×C₄	C₂⁶	1	31248	yes	—
52	#127	C₂⁶	1	262144	yes	direct product factor
53	C₂×C₄²	C₂⁶	1	64512	yes	complemented
54	C₂⁴	—	2	499968	no	—
55	C₂⁴	—	4	2856960	no	—
56	C₂⁴	C₂²×#127	1	166656	yes	direct product factor
57	C₂⁴	C₂⁴×D₄	1	33480	yes	—
58	C₂⁴	C₂⁷	1	8064	yes	—
59	C₂⁴	C₂⁷	1	651	yes	—
60	C₂×D₄	—	2	1548288	no	—
61	C₂×D₄	C₂⁷	1	24576	yes	complemented
62	C₂²×C₄	C₂⁴×D₄	1	249984	yes	complemented
63	C₂²×C₄	C₂⁷	1	6048	yes	—
64	C₄²	C₂⁷	1	4096	yes	complemented
65	C₂³	—	2	48384	no	—
66	C₂³	—	4	666624	no	—
67	C₂³	C₂³×#127	1	89280	yes	direct product factor
68	C₂³	C₂⁵×D₄	1	7812	yes	—
69	C₂³	C₂⁸	1	256	yes	—
70	C₂³	C₂⁸	1	63	yes	—
71	D₄	—	2	49152	no	—
72	C₂×C₄	C₂⁵×D₄	1	24192	yes	complemented
73	C₂×C₄	C₂⁸	1	192	yes	—
74	C₂²	—	2	768	no	—
75	C₂²	—	4	32256	no	—
76	C₂²	C₂⁴×#127	1	10416	yes	direct product factor
77	C₂²	C₂⁶×D₄	1	378	yes	—
78	C₂²	C₂⁹	1	1	yes	characteristic
79	C₄	C₂⁶×D₄	1	384	yes	complemented
80	C₂	—	4	256	no	—
81	C₂	C₂⁷×D₄	1	3	yes	—
82	C₂	C₂⁵×#127	1	252	yes	direct product factor

	subgroup	G/H
center, socle	C₂⁸	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁸ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁵×#127	C₂	direct (N × H)
2. C₂⁷×D₄	C₂	semidirect
3. C₂⁶×C₄²	C₂	semidirect
6. C₂⁶×D₄	C₂²	semidirect
8. C₂⁴×#127	C₂²	direct (N × H)
9. C₂⁵×C₄²	C₂²	semidirect
14. C₂⁵×D₄	C₂³	semidirect
15. C₂⁶×C₄	D₄	semidirect
17. C₂³×#127	C₂³	direct (N × H)
18. C₂⁴×C₄²	C₂³	semidirect
25. C₂⁴×D₄	C₂⁴	semidirect
26. C₂⁵×C₄	C₂×D₄	semidirect
28. C₂²×#127	C₂⁴	direct (N × H)
29. C₂³×C₄²	C₂⁴	semidirect
32. C₂⁶	#127	direct (N × H)
37. C₂³×D₄	C₂⁵	semidirect
38. C₂⁴×C₄	C₂²×D₄	semidirect
40. C₂×#127	C₂⁵	direct (N × H)
41. C₂²×C₄²	C₂⁵	semidirect
44. C₂⁵	C₂×#127	direct (N × H)
49. C₂²×D₄	C₂⁶	semidirect
50. C₂³×C₄	C₂³×D₄	semidirect
52. #127	C₂⁶	direct (N × H)
53. C₂×C₄²	C₂⁶	semidirect
56. C₂⁴	C₂²×#127	direct (N × H)
61. C₂×D₄	C₂⁷	semidirect
62. C₂²×C₄	C₂⁴×D₄	semidirect
64. C₄²	C₂⁷	semidirect
67. C₂³	C₂³×#127	direct (N × H)
72. C₂×C₄	C₂⁵×D₄	semidirect
76. C₂²	C₂⁴×#127	direct (N × H)
79. C₄	C₂⁶×D₄	semidirect
82. C₂	C₂⁵×#127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×#127	2
C₂⁷×D₄	2
C₂⁶×C₄²	2

Subgroups — C26×#127 #72291409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁶×#127 #72291409