Subgroups — C₂⁴×#126 #10585893

Order 512.

46530 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#126	C₂	1	120	yes	maximal, direct product factor
2	C₂⁴×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₂⁴×C₂²⋊C₄	C₂	1	3	yes	maximal
4	C₂⁴×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁷	C₂²	1	1	yes	characteristic
6	C₂⁵×C₄	C₂²	1	3	yes	—
7	C₂⁵×C₄	C₂²	1	3	yes	—
8	C₂²×#126	C₂²	1	2240	yes	direct product factor
9	C₂³×C₄⋊C₄	C₂²	1	180	yes	complemented
10	C₂³×C₂²⋊C₄	C₂²	1	180	yes	—
11	C₂³×C₄²	C₂²	1	60	yes	complemented
12	C₂⁶	—	2	48	no	—
13	C₂⁶	C₂³	1	30	yes	—
14	C₂⁶	C₂³	1	1	yes	characteristic
15	C₂⁴×C₄	—	2	48	no	—
16	C₂⁴×C₄	—	2	48	no	—
17	C₂⁴×C₄	C₂³	1	90	yes	—
18	C₂⁴×C₄	C₂³	1	90	yes	—
19	C₂×#126	C₂³	1	7680	yes	direct product factor
20	C₂²×C₄⋊C₄	C₂³	1	1680	yes	complemented
21	C₂²×C₂²⋊C₄	C₂³	1	1680	yes	—
22	C₂²×C₄²	C₂³	1	560	yes	complemented
23	C₂⁵	—	2	720	no	—
24	C₂⁵	—	4	256	no	—
25	C₂⁵	#41	1	48	yes	—
26	C₂⁵	C₂⁴	1	140	yes	—
27	C₂⁵	C₂⁴	1	15	yes	—
28	C₂³×C₄	—	2	720	no	—
29	C₂³×C₄	—	2	720	no	—
30	C₂³×C₄	C₂⁴	1	420	yes	—
31	C₂³×C₄	C₂⁴	1	420	yes	—
32	#126	C₂⁴	1	4096	yes	direct product factor
33	C₂×C₄⋊C₄	C₂⁴	1	2880	yes	complemented
34	C₂×C₂²⋊C₄	C₂⁴	1	2880	yes	—
35	C₂×C₄²	C₂⁴	1	960	yes	complemented
36	C₂⁴	—	2	1680	no	—
37	C₂⁴	—	4	1920	no	—
38	C₂⁴	#126	1	256	yes	direct product factor
39	C₂⁴	C₂×#41	1	360	yes	—
40	C₂⁴	C₂⁵	1	120	yes	—
41	C₂⁴	C₂⁵	1	35	yes	—
42	C₂²×C₄	—	2	1680	no	—
43	C₂²×C₄	—	2	1680	no	—
44	C₂²×C₄	C₂⁵	1	360	yes	—
45	C₂²×C₄	C₂⁵	1	360	yes	—
46	C₄⋊C₄	C₂⁵	1	768	yes	complemented
47	C₂²⋊C₄	C₂⁵	1	768	yes	—
48	C₄²	C₂⁵	1	256	yes	complemented
49	C₂³	—	2	720	no	—
50	C₂³	—	4	2240	no	—
51	C₂³	C₂×#126	1	960	yes	direct product factor
52	C₂³	C₂²×#41	1	420	yes	—
53	C₂³	C₂⁶	1	16	yes	—
54	C₂³	C₂⁶	1	15	yes	—
55	C₂×C₄	—	2	720	no	—
56	C₂×C₄	—	2	720	no	—
57	C₂×C₄	C₂⁶	1	48	yes	—
58	C₂×C₄	C₂⁶	1	48	yes	—
59	C₂²	—	2	48	no	—
60	C₂²	—	4	480	no	—
61	C₂²	C₂²×#126	1	560	yes	direct product factor
62	C₂²	C₂³×#41	1	90	yes	—
63	C₂²	C₂⁷	1	1	yes	characteristic
64	C₄	—	2	48	no	—
65	C₄	—	2	48	no	—
66	C₂	—	4	16	no	—
67	C₂	C₂³×#126	1	60	yes	direct product factor
68	C₂	C₂⁴×#41	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁶	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×#126	C₂	direct (N × H)
2. C₂⁴×C₄⋊C₄	C₂	semidirect
4. C₂⁴×C₄²	C₂	semidirect
8. C₂²×#126	C₂²	direct (N × H)
9. C₂³×C₄⋊C₄	C₂²	semidirect
11. C₂³×C₄²	C₂²	semidirect
19. C₂×#126	C₂³	direct (N × H)
20. C₂²×C₄⋊C₄	C₂³	semidirect
22. C₂²×C₄²	C₂³	semidirect
32. #126	C₂⁴	direct (N × H)
33. C₂×C₄⋊C₄	C₂⁴	semidirect
35. C₂×C₄²	C₂⁴	semidirect
38. C₂⁴	#126	direct (N × H)
46. C₄⋊C₄	C₂⁵	semidirect
48. C₄²	C₂⁵	semidirect
51. C₂³	C₂×#126	direct (N × H)
61. C₂²	C₂²×#126	direct (N × H)
67. C₂	C₂³×#126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#126	2
C₂⁴×C₄⋊C₄	2
C₂⁴×C₂²⋊C₄	2
C₂⁴×C₄²	2

Subgroups — C24×#126 #10585893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#126 #10585893