Subgroups — C₂⁴×#125 #10585892

Order 512.

40546 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#125	C₂	1	120	yes	maximal, direct product factor
2	C₂⁴×C₄⋊C₄	C₂	1	4	yes	maximal
3	C₂⁴×C₄⋊C₄	C₂	1	2	yes	maximal
4	C₂⁴×C₄²	C₂	1	1	yes	maximal, characteristic
5	C₂⁵×C₄	C₂²	1	2	yes	—
6	C₂⁵×C₄	C₂²	1	4	yes	—
7	C₂⁵×C₄	C₂²	1	1	yes	characteristic
8	C₂²×#125	C₂²	1	2240	yes	direct product factor
9	C₂³×C₄⋊C₄	C₂²	1	240	yes	—
10	C₂³×C₄⋊C₄	C₂²	1	120	yes	—
11	C₂³×C₄²	C₂²	1	60	yes	—
12	C₂⁶	C₂³	1	1	yes	characteristic
13	C₂⁴×C₄	—	2	32	no	—
14	C₂⁴×C₄	—	2	64	no	—
15	C₂⁴×C₄	Q₈	1	32	yes	—
16	C₂⁴×C₄	C₂³	1	60	yes	—
17	C₂⁴×C₄	C₂³	1	120	yes	—
18	C₂⁴×C₄	C₂³	1	30	yes	—
19	C₂×#125	C₂³	1	7680	yes	direct product factor
20	C₂²×C₄⋊C₄	C₂³	1	2240	yes	—
21	C₂²×C₄⋊C₄	C₂³	1	1120	yes	—
22	C₂²×C₄²	C₂³	1	560	yes	—
23	C₂⁵	C₂×Q₈	1	16	yes	—
24	C₂⁵	#41	1	32	yes	—
25	C₂⁵	C₂⁴	1	15	yes	—
26	C₂³×C₄	—	2	480	no	—
27	C₂³×C₄	—	2	960	no	—
28	C₂³×C₄	C₂×Q₈	1	480	yes	—
29	C₂³×C₄	C₂⁴	1	560	yes	—
30	C₂³×C₄	C₂⁴	1	280	yes	—
31	C₂³×C₄	C₂⁴	1	140	yes	—
32	#125	C₂⁴	1	4096	yes	direct product factor
33	C₂×C₄⋊C₄	C₂⁴	1	3840	yes	—
34	C₂×C₄⋊C₄	C₂⁴	1	1920	yes	—
35	C₂×C₄²	C₂⁴	1	960	yes	—
36	C₂⁴	#125	1	256	yes	direct product factor
37	C₂⁴	C₂²×Q₈	1	120	yes	—
38	C₂⁴	C₂×#41	1	240	yes	—
39	C₂⁴	C₂⁵	1	35	yes	—
40	C₂²×C₄	—	2	2240	no	—
41	C₂²×C₄	—	2	1120	no	—
42	C₂²×C₄	C₂²×Q₈	1	1120	yes	—
43	C₂²×C₄	C₂⁵	1	240	yes	—
44	C₂²×C₄	C₂⁵	1	480	yes	—
45	C₂²×C₄	C₂⁵	1	120	yes	—
46	C₄⋊C₄	C₂⁵	1	1024	yes	—
47	C₄⋊C₄	C₂⁵	1	512	yes	—
48	C₄²	C₂⁵	1	256	yes	—
49	C₂³	C₂×#125	1	960	yes	direct product factor
50	C₂³	C₂³×Q₈	1	140	yes	—
51	C₂³	C₂²×#41	1	280	yes	—
52	C₂³	C₂⁶	1	15	yes	—
53	C₂×C₄	—	2	480	no	—
54	C₂×C₄	—	2	960	no	—
55	C₂×C₄	C₂³×Q₈	1	480	yes	—
56	C₂×C₄	C₂⁶	1	32	yes	—
57	C₂×C₄	C₂⁶	1	64	yes	—
58	C₂×C₄	C₂⁶	1	16	yes	—
59	C₂²	C₂²×#125	1	560	yes	direct product factor
60	C₂²	C₂⁴×Q₈	1	30	yes	—
61	C₂²	C₂³×#41	1	60	yes	—
62	C₂²	C₂⁷	1	1	yes	characteristic
63	C₄	—	2	32	no	—
64	C₄	—	2	64	no	—
65	C₄	C₂⁴×Q₈	1	32	yes	—
66	C₂	C₂³×#125	1	60	yes	direct product factor
67	C₂	C₂⁵×Q₈	1	1	yes	characteristic
68	C₂	C₂⁴×#41	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁶	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁷
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×#125	C₂	direct (N × H)
8. C₂²×#125	C₂²	direct (N × H)
19. C₂×#125	C₂³	direct (N × H)
32. #125	C₂⁴	direct (N × H)
36. C₂⁴	#125	direct (N × H)
49. C₂³	C₂×#125	direct (N × H)
59. C₂²	C₂²×#125	direct (N × H)
66. C₂	C₂³×#125	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×#125	2
C₂⁴×C₄⋊C₄	2
C₂⁴×C₄²	2

Subgroups — C24×#125 #10585892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#125 #10585892