Subgroups — C₂⁴×#120 #10585886

Order 512.

281810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁸	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁵×D₄	C₂	1	3	yes	maximal, complemented
3	C₂³×#120	C₂	1	120	yes	maximal, direct product factor
4	C₂⁴×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
5	C₂⁷	—	2	96	no	—
6	C₂⁷	C₂²	1	3	yes	—
7	C₂⁷	C₂²	1	60	yes	complemented
8	C₂⁷	C₂²	1	1	yes	characteristic, complemented
9	C₂⁴×D₄	—	2	96	no	—
10	C₂⁴×D₄	C₂²	1	180	yes	complemented
11	C₂⁵×C₄	C₂²	1	3	yes	complemented
12	C₂²×#120	C₂²	1	2240	yes	direct product factor
13	C₂³×C₂²⋊C₄	C₂²	1	180	yes	complemented
14	C₂⁶	—	2	2048	no	—
15	C₂⁶	—	2	96	no	—
16	C₂⁶	—	2	2880	no	—
17	C₂⁶	—	2	48	no	—
18	C₂⁶	D₄	1	96	yes	complemented
19	C₂⁶	C₂³	1	90	yes	—
20	C₂⁶	C₂³	1	1	yes	characteristic
21	C₂⁶	C₂³	1	560	yes	complemented
22	C₂⁶	C₂³	1	30	yes	complemented
23	C₂³×D₄	—	2	2880	no	—
24	C₂³×D₄	C₂³	1	1680	yes	complemented
25	C₂⁴×C₄	—	2	48	no	—
26	C₂⁴×C₄	C₂³	1	90	yes	complemented
27	C₂×#120	C₂³	1	7680	yes	direct product factor
28	C₂²×C₂²⋊C₄	C₂³	1	1680	yes	complemented
29	C₂⁵	—	2	30720	no	—
30	C₂⁵	—	2	1536	no	—
31	C₂⁵	—	2	1440	no	—
32	C₂⁵	—	2	13440	no	—
33	C₂⁵	—	2	720	no	—
34	C₂⁵	—	4	256	no	—
35	C₂⁵	C₂×D₄	1	48	yes	—
36	C₂⁵	C₂×D₄	1	1440	yes	complemented
37	C₂⁵	C₂⁴	1	420	yes	—
38	C₂⁵	C₂⁴	1	15	yes	—
39	C₂⁵	C₂⁴	1	960	yes	complemented
40	C₂⁵	C₂⁴	1	140	yes	complemented
41	C₂²×D₄	—	2	13440	no	—
42	C₂²×D₄	C₂⁴	1	2880	yes	complemented
43	C₂³×C₄	—	2	720	no	—
44	C₂³×C₄	C₂⁴	1	420	yes	complemented
45	#120	C₂⁴	1	4096	yes	direct product factor
46	C₂×C₂²⋊C₄	C₂⁴	1	2880	yes	complemented
47	C₂⁴	—	2	71680	no	—
48	C₂⁴	—	2	11520	no	—
49	C₂⁴	—	2	3360	no	—
50	C₂⁴	—	2	11520	no	—
51	C₂⁴	—	2	1680	no	—
52	C₂⁴	—	4	1920	no	—
53	C₂⁴	#120	1	256	yes	direct product factor
54	C₂⁴	C₂²×D₄	1	360	yes	—
55	C₂⁴	C₂²×D₄	1	3360	yes	complemented
56	C₂⁴	C₂⁵	1	360	yes	—
57	C₂⁴	C₂⁵	1	35	yes	—
58	C₂⁴	C₂⁵	1	256	yes	complemented
59	C₂⁴	C₂⁵	1	120	yes	complemented
60	C₂×D₄	—	2	11520	no	—
61	C₂×D₄	C₂⁵	1	768	yes	complemented
62	C₂²×C₄	—	2	1680	no	—
63	C₂²×C₄	C₂⁵	1	360	yes	complemented
64	C₂²⋊C₄	C₂⁵	1	768	yes	complemented
65	C₂³	—	2	30720	no	—
66	C₂³	—	2	1440	no	—
67	C₂³	—	2	1536	no	—
68	C₂³	—	2	13440	no	—
69	C₂³	—	2	720	no	—
70	C₂³	—	4	2240	no	—
71	C₂³	C₂×#120	1	960	yes	direct product factor
72	C₂³	C₂³×D₄	1	420	yes	—
73	C₂³	C₂³×D₄	1	1440	yes	complemented
74	C₂³	C₂⁶	1	48	yes	—
75	C₂³	C₂⁶	1	15	yes	—
76	C₂³	C₂⁶	1	16	yes	complemented
77	D₄	—	2	1536	no	—
78	C₂×C₄	—	2	720	no	—
79	C₂×C₄	C₂⁶	1	48	yes	complemented
80	C₂²	—	2	2048	no	—
81	C₂²	—	2	96	no	—
82	C₂²	—	2	2880	no	—
83	C₂²	—	2	48	no	—
84	C₂²	—	4	480	no	—
85	C₂²	C₂²×#120	1	560	yes	direct product factor
86	C₂²	C₂⁴×D₄	1	90	yes	—
87	C₂²	C₂⁴×D₄	1	96	yes	complemented
88	C₂²	C₂⁷	1	1	yes	characteristic
89	C₄	—	2	48	no	—
90	C₂	—	2	96	no	—
91	C₂	—	4	16	no	—
92	C₂	C₂³×#120	1	60	yes	direct product factor
93	C₂	C₂⁵×D₄	1	3	yes	—

	subgroup	G/H
center, socle	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. C₂⁸	C₂	semidirect
2. C₂⁵×D₄	C₂	semidirect
3. C₂³×#120	C₂	direct (N × H)
4. C₂⁴×C₂²⋊C₄	C₂	semidirect
7. C₂⁷	C₂²	semidirect
8. C₂⁷	C₂²	semidirect
10. C₂⁴×D₄	C₂²	semidirect
11. C₂⁵×C₄	C₂²	semidirect
12. C₂²×#120	C₂²	direct (N × H)
13. C₂³×C₂²⋊C₄	C₂²	semidirect
18. C₂⁶	D₄	semidirect
21. C₂⁶	C₂³	semidirect
22. C₂⁶	C₂³	semidirect
24. C₂³×D₄	C₂³	semidirect
26. C₂⁴×C₄	C₂³	semidirect
27. C₂×#120	C₂³	direct (N × H)
28. C₂²×C₂²⋊C₄	C₂³	semidirect
36. C₂⁵	C₂×D₄	semidirect
39. C₂⁵	C₂⁴	semidirect
40. C₂⁵	C₂⁴	semidirect
42. C₂²×D₄	C₂⁴	semidirect
44. C₂³×C₄	C₂⁴	semidirect
45. #120	C₂⁴	direct (N × H)
46. C₂×C₂²⋊C₄	C₂⁴	semidirect
53. C₂⁴	#120	direct (N × H)
55. C₂⁴	C₂²×D₄	semidirect
58. C₂⁴	C₂⁵	semidirect
59. C₂⁴	C₂⁵	semidirect
61. C₂×D₄	C₂⁵	semidirect
63. C₂²×C₄	C₂⁵	semidirect
64. C₂²⋊C₄	C₂⁵	semidirect
71. C₂³	C₂×#120	direct (N × H)
73. C₂³	C₂³×D₄	semidirect
76. C₂³	C₂⁶	semidirect
79. C₂×C₄	C₂⁶	semidirect
85. C₂²	C₂²×#120	direct (N × H)
87. C₂²	C₂⁴×D₄	semidirect
92. C₂	C₂³×#120	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁸	2
C₂⁵×D₄	2
C₂³×#120	2
C₂⁴×C₂²⋊C₄	2

Subgroups — C24×#120 #10585886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#120 #10585886