Subgroups — C₂⁶×H(5,𝔽₂) #72291317

Order 2048.

241541800 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×#41	C₂	1	6	yes	maximal, complemented
2	C₂⁷×D₄	C₂	1	9	yes	maximal, complemented
3	C₂⁵×H(5,𝔽₂)	C₂	1	1008	yes	maximal, direct product factor
4	C₂⁹	C₂²	1	6	yes	complemented
5	C₂⁴×H(5,𝔽₂)	C₂²	1	166656	yes	direct product factor
6	C₂⁵×#41	C₂²	1	3024	yes	complemented
7	C₂⁶×Q₈	C₂²	1	2	yes	complemented
8	C₂⁶×D₄	C₂²	1	18	yes	complemented
9	C₂⁶×D₄	C₂²	1	4536	yes	complemented
10	C₂⁷×C₄	C₂²	1	9	yes	complemented
11	C₂⁸	—	4	384	no	—
12	C₂⁸	C₂³	1	9	yes	—
13	C₂⁸	C₂³	1	1512	yes	complemented
14	C₂³×H(5,𝔽₂)	C₂³	1	5713920	yes	direct product factor
15	C₂⁴×#41	C₂³	1	249984	yes	complemented
16	C₂⁵×Q₈	C₂³	1	504	yes	complemented
17	C₂⁵×D₄	C₂³	1	4536	yes	complemented
18	C₂⁵×D₄	C₂³	1	374976	yes	complemented
19	C₂⁶×C₄	C₂³	1	6	yes	—
20	C₂⁶×C₄	C₂³	1	2268	yes	complemented
21	C₂⁷	—	2	576	no	—
22	C₂⁷	—	4	48384	no	—
23	C₂⁷	C₂⁴	1	1134	yes	—
24	C₂⁷	C₂⁴	1	1	yes	characteristic
25	C₂⁷	C₂⁴	1	62496	yes	complemented
26	C₂²×H(5,𝔽₂)	C₂⁴	1	42663936	yes	direct product factor
27	C₂³×#41	C₂⁴	1	4285440	yes	complemented
28	C₂⁴×Q₈	C₂⁴	1	20832	yes	complemented
29	C₂⁴×D₄	C₂⁴	1	187488	yes	complemented
30	C₂⁴×D₄	C₂⁴	1	6428160	yes	complemented
31	C₂⁵×C₄	C₂⁴	1	756	yes	—
32	C₂⁵×C₄	C₂⁴	1	93744	yes	complemented
33	C₂⁶	—	2	36288	no	—
34	C₂⁶	—	4	999936	no	—
35	C₂⁶	H(5,𝔽₂)	1	64	yes	direct product factor
36	C₂⁶	C₂⁵	1	23436	yes	—
37	C₂⁶	C₂⁵	1	63	yes	—
38	C₂⁶	C₂⁵	1	535680	yes	complemented
39	C₂×H(5,𝔽₂)	C₂⁵	1	66060288	yes	direct product factor
40	C₂²×#41	C₂⁵	1	15998976	yes	complemented
41	C₂³×Q₈	C₂⁵	1	178560	yes	complemented
42	C₂³×D₄	C₂⁵	1	1607040	yes	complemented
43	C₂³×D₄	C₂⁵	1	23998464	yes	complemented
44	C₂⁴×C₄	C₂⁵	1	15624	yes	—
45	C₂⁴×C₄	C₂⁵	1	803520	yes	complemented
46	C₂⁵	—	2	374976	no	—
47	C₂⁵	—	4	4285440	no	—
48	C₂⁵	C₂×H(5,𝔽₂)	1	2016	yes	direct product factor
49	C₂⁵	C₂⁶	1	100440	yes	—
50	C₂⁵	C₂⁶	1	651	yes	—
51	C₂⁵	C₂⁶	1	999936	yes	complemented
52	H(5,𝔽₂)	C₂⁶	1	16777216	yes	direct product factor
53	C₂×#41	C₂⁶	1	12386304	yes	complemented
54	C₂²×Q₈	C₂⁶	1	333312	yes	complemented
55	C₂²×D₄	C₂⁶	1	2999808	yes	complemented
56	C₂²×D₄	C₂⁶	1	18579456	yes	complemented
57	C₂³×C₄	C₂⁶	1	66960	yes	—
58	C₂³×C₄	C₂⁶	1	1499904	yes	complemented
59	C₂⁴	—	2	803520	no	—
60	C₂⁴	—	4	3999744	no	—
61	C₂⁴	C₂²×H(5,𝔽₂)	1	10416	yes	direct product factor
62	C₂⁴	C₂⁷	1	93744	yes	—
63	C₂⁴	C₂⁷	1	1395	yes	—
64	C₂⁴	C₂⁷	1	387072	yes	complemented
65	#41	C₂⁷	1	1572864	yes	complemented
66	C₂×Q₈	C₂⁷	1	129024	yes	complemented
67	C₂×D₄	C₂⁷	1	1161216	yes	complemented
68	C₂×D₄	C₂⁷	1	2359296	yes	complemented
69	C₂²×C₄	C₂⁷	1	62496	yes	—
70	C₂²×C₄	C₂⁷	1	580608	yes	complemented
71	C₂³	—	2	374976	no	—
72	C₂³	—	4	774144	no	—
73	C₂³	C₂³×H(5,𝔽₂)	1	11160	yes	direct product factor
74	C₂³	C₂⁸	1	18144	yes	—
75	C₂³	C₂⁸	1	651	yes	—
76	C₂³	C₂⁸	1	24576	yes	complemented
77	Q₈	C₂⁸	1	8192	yes	complemented
78	D₄	C₂⁸	1	73728	yes	complemented
79	C₂×C₄	C₂⁸	1	12096	yes	—
80	C₂×C₄	C₂⁸	1	36864	yes	complemented
81	C₂²	—	2	36288	no	—
82	C₂²	—	4	24576	no	—
83	C₂²	C₂⁴×H(5,𝔽₂)	1	2604	yes	direct product factor
84	C₂²	C₂⁹	1	576	yes	—
85	C₂²	C₂⁹	1	63	yes	—
86	C₄	C₂⁹	1	384	yes	—
87	C₂	—	2	576	no	—
88	C₂	C₂¹⁰	1	1	yes	characteristic
89	C₂	C₂⁵×H(5,𝔽₂)	1	126	yes	direct product factor

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂	C₂¹⁰
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁶×#41	C₂	semidirect
2. C₂⁷×D₄	C₂	semidirect
3. C₂⁵×H(5,𝔽₂)	C₂	direct (N × H)
4. C₂⁹	C₂²	semidirect
5. C₂⁴×H(5,𝔽₂)	C₂²	direct (N × H)
6. C₂⁵×#41	C₂²	semidirect
7. C₂⁶×Q₈	C₂²	semidirect
8. C₂⁶×D₄	C₂²	semidirect
9. C₂⁶×D₄	C₂²	semidirect
10. C₂⁷×C₄	C₂²	semidirect
13. C₂⁸	C₂³	semidirect
14. C₂³×H(5,𝔽₂)	C₂³	direct (N × H)
15. C₂⁴×#41	C₂³	semidirect
16. C₂⁵×Q₈	C₂³	semidirect
17. C₂⁵×D₄	C₂³	semidirect
18. C₂⁵×D₄	C₂³	semidirect
20. C₂⁶×C₄	C₂³	semidirect
25. C₂⁷	C₂⁴	semidirect
26. C₂²×H(5,𝔽₂)	C₂⁴	direct (N × H)
27. C₂³×#41	C₂⁴	semidirect
28. C₂⁴×Q₈	C₂⁴	semidirect
29. C₂⁴×D₄	C₂⁴	semidirect
30. C₂⁴×D₄	C₂⁴	semidirect
32. C₂⁵×C₄	C₂⁴	semidirect
35. C₂⁶	H(5,𝔽₂)	direct (N × H)
38. C₂⁶	C₂⁵	semidirect
39. C₂×H(5,𝔽₂)	C₂⁵	direct (N × H)
40. C₂²×#41	C₂⁵	semidirect
41. C₂³×Q₈	C₂⁵	semidirect
42. C₂³×D₄	C₂⁵	semidirect
43. C₂³×D₄	C₂⁵	semidirect
45. C₂⁴×C₄	C₂⁵	semidirect
48. C₂⁵	C₂×H(5,𝔽₂)	direct (N × H)
51. C₂⁵	C₂⁶	semidirect
52. H(5,𝔽₂)	C₂⁶	direct (N × H)
53. C₂×#41	C₂⁶	semidirect
54. C₂²×Q₈	C₂⁶	semidirect
55. C₂²×D₄	C₂⁶	semidirect
56. C₂²×D₄	C₂⁶	semidirect
58. C₂³×C₄	C₂⁶	semidirect
61. C₂⁴	C₂²×H(5,𝔽₂)	direct (N × H)
64. C₂⁴	C₂⁷	semidirect
65. #41	C₂⁷	semidirect
66. C₂×Q₈	C₂⁷	semidirect
67. C₂×D₄	C₂⁷	semidirect
68. C₂×D₄	C₂⁷	semidirect
70. C₂²×C₄	C₂⁷	semidirect
73. C₂³	C₂³×H(5,𝔽₂)	direct (N × H)
76. C₂³	C₂⁸	semidirect
77. Q₈	C₂⁸	semidirect
78. D₄	C₂⁸	semidirect
80. C₂×C₄	C₂⁸	semidirect
83. C₂²	C₂⁴×H(5,𝔽₂)	direct (N × H)
89. C₂	C₂⁵×H(5,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×#41	2
C₂⁷×D₄	2
C₂⁵×H(5,𝔽₂)	2

Subgroups — C26×H(5,𝔽2) #72291317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁶×H(5,𝔽₂) #72291317