Subgroups — C₂⁷×D₈ #72314419

Order 2048.

24015976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₈	C₂	1	508	yes	maximal, direct product factor
2	C₂⁷×D₄	C₂	1	2	yes	maximal, complemented
3	C₂⁷×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁹	—	2	2	no	—
5	C₂⁶×D₄	—	2	254	no	—
6	C₂⁶×D₄	C₂²	1	508	yes	complemented
7	C₂⁷×C₄	C₂²	1	1	yes	characteristic
8	C₂⁵×D₈	C₂²	1	42672	yes	direct product factor
9	C₂⁶×C₈	C₂²	1	254	yes	complemented
10	C₂⁸	—	2	508	no	—
11	C₂⁸	—	4	256	no	—
12	C₂⁸	D₄	1	1	yes	characteristic
13	C₂⁵×D₄	—	2	32004	no	—
14	C₂⁵×D₄	C₂³	1	21336	yes	complemented
15	C₂⁶×C₄	D₄	1	127	yes	—
16	C₂⁶×C₄	C₂³	1	127	yes	—
17	C₂⁴×D₈	C₂³	1	755904	yes	direct product factor
18	C₂⁵×C₈	C₂³	1	10668	yes	complemented
19	C₂⁷	—	2	21336	no	—
20	C₂⁷	—	4	32512	no	—
21	C₂⁷	D₈	1	128	yes	direct product factor
22	C₂⁷	C₂×D₄	1	127	yes	—
23	C₂⁴×D₄	—	2	661416	no	—
24	C₂⁴×D₄	C₂⁴	1	188976	yes	complemented
25	C₂⁵×C₄	C₂×D₄	1	8001	yes	—
26	C₂⁵×C₄	C₂⁴	1	2667	yes	—
27	C₂³×D₈	C₂⁴	1	3023616	yes	direct product factor
28	C₂⁴×C₈	C₂⁴	1	94488	yes	complemented
29	C₂⁶	—	2	188976	no	—
30	C₂⁶	—	4	682752	no	—
31	C₂⁶	C₂×D₈	1	8128	yes	direct product factor
32	C₂⁶	C₂²×D₄	1	2667	yes	—
33	C₂³×D₄	—	2	2834640	no	—
34	C₂³×D₄	C₂⁵	1	377952	yes	complemented
35	C₂⁴×C₄	C₂²×D₄	1	82677	yes	—
36	C₂⁴×C₄	C₂⁵	1	11811	yes	—
37	C₂²×D₈	C₂⁵	1	2731008	yes	direct product factor
38	C₂³×C₈	C₂⁵	1	188976	yes	complemented
39	C₂⁵	—	2	377952	no	—
40	C₂⁵	—	4	3023616	no	—
41	C₂⁵	C₂²×D₈	1	85344	yes	direct product factor
42	C₂⁵	C₂³×D₄	1	11811	yes	—
43	C₂²×D₄	—	2	2645664	no	—
44	C₂²×D₄	C₂⁶	1	170688	yes	complemented
45	C₂³×C₄	C₂³×D₄	1	177165	yes	—
46	C₂³×C₄	C₂⁶	1	11811	yes	—
47	C₂×D₈	C₂⁶	1	520192	yes	direct product factor
48	C₂²×C₈	C₂⁶	1	85344	yes	complemented
49	C₂⁴	—	2	170688	no	—
50	C₂⁴	—	4	3023616	no	—
51	C₂⁴	C₂³×D₈	1	188976	yes	direct product factor
52	C₂⁴	C₂⁴×D₄	1	11811	yes	—
53	C₂×D₄	—	2	512064	no	—
54	C₂×D₄	C₂⁷	1	16256	yes	complemented
55	C₂²×C₄	C₂⁴×D₄	1	82677	yes	—
56	C₂²×C₄	C₂⁷	1	2667	yes	—
57	D₈	C₂⁷	1	16384	yes	direct product factor
58	C₂×C₈	C₂⁷	1	8128	yes	complemented
59	C₂³	—	2	16256	no	—
60	C₂³	—	4	682752	no	—
61	C₂³	C₂⁴×D₈	1	94488	yes	direct product factor
62	C₂³	C₂⁵×D₄	1	2667	yes	—
63	D₄	—	2	16256	no	—
64	D₄	C₂⁸	1	256	yes	complemented
65	C₂×C₄	C₂⁵×D₄	1	8001	yes	—
66	C₂×C₄	C₂⁸	1	127	yes	—
67	C₈	C₂⁸	1	128	yes	complemented
68	C₂²	—	2	256	no	—
69	C₂²	—	4	32512	no	—
70	C₂²	C₂⁵×D₈	1	10668	yes	direct product factor
71	C₂²	C₂⁶×D₄	1	127	yes	—
72	C₄	C₂⁶×D₄	1	127	yes	—
73	C₄	C₂⁹	1	1	yes	characteristic
74	C₂	—	4	256	no	—
75	C₂	C₂⁷×D₄	1	1	yes	characteristic
76	C₂	C₂⁶×D₈	1	254	yes	direct product factor

	subgroup	G/H
center, socle	C₂⁸	D₄
commutator, Frattini, cosocle	C₄	C₂⁹
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁸ ◁ C₂⁷×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₈	C₂	direct (N × H)
2. C₂⁷×D₄	C₂	semidirect
3. C₂⁷×C₈	C₂	semidirect
6. C₂⁶×D₄	C₂²	semidirect
8. C₂⁵×D₈	C₂²	direct (N × H)
9. C₂⁶×C₈	C₂²	semidirect
14. C₂⁵×D₄	C₂³	semidirect
17. C₂⁴×D₈	C₂³	direct (N × H)
18. C₂⁵×C₈	C₂³	semidirect
21. C₂⁷	D₈	direct (N × H)
24. C₂⁴×D₄	C₂⁴	semidirect
27. C₂³×D₈	C₂⁴	direct (N × H)
28. C₂⁴×C₈	C₂⁴	semidirect
31. C₂⁶	C₂×D₈	direct (N × H)
34. C₂³×D₄	C₂⁵	semidirect
37. C₂²×D₈	C₂⁵	direct (N × H)
38. C₂³×C₈	C₂⁵	semidirect
41. C₂⁵	C₂²×D₈	direct (N × H)
44. C₂²×D₄	C₂⁶	semidirect
47. C₂×D₈	C₂⁶	direct (N × H)
48. C₂²×C₈	C₂⁶	semidirect
51. C₂⁴	C₂³×D₈	direct (N × H)
54. C₂×D₄	C₂⁷	semidirect
57. D₈	C₂⁷	direct (N × H)
58. C₂×C₈	C₂⁷	semidirect
61. C₂³	C₂⁴×D₈	direct (N × H)
64. D₄	C₂⁸	semidirect
67. C₈	C₂⁸	semidirect
70. C₂²	C₂⁵×D₈	direct (N × H)
76. C₂	C₂⁶×D₈	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×D₈	2
C₂⁷×D₄	2
C₂⁷×C₈	2

Subgroups — C27×D8 #72314419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁷×D₈ #72314419