Subgroups — C₂⁴×Q₁₆×C₆₁ #29566512

Order 15616.

11166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₆₁	C₂	1	2	yes	maximal
2	C₂³×Q₁₆×C₆₁	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×Q₈×C₆₁	—	2	30	no	—
5	C₂³×Q₈×C₆₁	C₂²	1	60	yes	—
6	C₂⁴×C₄×C₆₁	—	2	2	no	—
7	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
8	C₂²×Q₁₆×C₆₁	C₂²	1	560	yes	direct product factor
9	C₂³×C₈×C₆₁	C₂²	1	30	yes	—
10	C₂⁵×C₆₁	D₄	1	1	yes	characteristic
11	C₂²×Q₈×C₆₁	—	2	420	no	—
12	C₂²×Q₈×C₆₁	C₂³	1	280	yes	—
13	C₂³×C₄×C₆₁	—	2	60	no	—
14	C₂³×C₄×C₆₁	D₄	1	15	yes	—
15	C₂³×C₄×C₆₁	C₂³	1	15	yes	—
16	C₂×Q₁₆×C₆₁	C₂³	1	960	yes	direct product factor
17	C₂²×C₈×C₆₁	C₂³	1	140	yes	—
18	C₂⁴×C₆₁	Q₁₆	1	16	yes	direct product factor
19	C₂⁴×C₆₁	C₂×D₄	1	15	yes	—
20	C₂×Q₈×C₆₁	—	2	840	no	—
21	C₂×Q₈×C₆₁	C₂⁴	1	240	yes	—
22	C₂²×C₄×C₆₁	—	2	280	no	—
23	C₂²×C₄×C₆₁	C₂×D₄	1	105	yes	—
24	C₂²×C₄×C₆₁	C₂⁴	1	35	yes	—
25	Q₁₆×C₆₁	C₂⁴	1	256	yes	direct product factor
26	C₂×C₈×C₆₁	C₂⁴	1	120	yes	—
27	C₂³×C₆₁	C₂×Q₁₆	1	120	yes	direct product factor
28	C₂³×C₆₁	C₂²×D₄	1	35	yes	—
29	Q₈×C₆₁	—	2	240	no	—
30	Q₈×C₆₁	C₂⁵	1	32	yes	—
31	C₂×C₄×C₆₁	—	2	240	no	—
32	C₂×C₄×C₆₁	C₂²×D₄	1	105	yes	—
33	C₂×C₄×C₆₁	C₂⁵	1	15	yes	—
34	C₈×C₆₁	C₂⁵	1	16	yes	—
35	C₂⁴×Q₁₆	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₆₁	C₂²×Q₁₆	1	140	yes	direct product factor
37	C₂²×C₆₁	C₂³×D₄	1	15	yes	—
38	C₄×C₆₁	—	2	32	no	—
39	C₄×C₆₁	C₂³×D₄	1	15	yes	—
40	C₄×C₆₁	C₂⁶	1	1	yes	characteristic
41	C₂⁴×Q₈	C₂×C₆₁	1	2	yes	—
42	C₂³×Q₁₆	C₂×C₆₁	1	60	yes	direct product factor
43	C₂⁴×C₈	C₂×C₆₁	1	1	yes	characteristic
44	C₂×C₆₁	C₂³×Q₁₆	1	30	yes	direct product factor
45	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
46	C₂³×Q₈	—	2	30	no	—
47	C₂³×Q₈	C₂²×C₆₁	1	60	yes	—
48	C₂⁴×C₄	—	2	2	no	—
49	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
50	C₂²×Q₁₆	C₂²×C₆₁	1	560	yes	direct product factor
51	C₂³×C₈	C₂²×C₆₁	1	30	yes	—
52	C₆₁	C₂⁴×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
53	C₂⁵	D₄×C₆₁	1	1	yes	characteristic
54	C₂²×Q₈	—	2	420	no	—
55	C₂²×Q₈	C₂³×C₆₁	1	280	yes	—
56	C₂³×C₄	—	2	60	no	—
57	C₂³×C₄	D₄×C₆₁	1	15	yes	—
58	C₂³×C₄	C₂³×C₆₁	1	15	yes	—
59	C₂×Q₁₆	C₂³×C₆₁	1	960	yes	direct product factor
60	C₂²×C₈	C₂³×C₆₁	1	140	yes	—
61	C₂⁴	Q₁₆×C₆₁	1	16	yes	direct product factor
62	C₂⁴	C₂×D₄×C₆₁	1	15	yes	—
63	C₂×Q₈	—	2	840	no	—
64	C₂×Q₈	C₂⁴×C₆₁	1	240	yes	—
65	C₂²×C₄	—	2	280	no	—
66	C₂²×C₄	C₂×D₄×C₆₁	1	105	yes	—
67	C₂²×C₄	C₂⁴×C₆₁	1	35	yes	—
68	Q₁₆	C₂⁴×C₆₁	1	256	yes	direct product factor
69	C₂×C₈	C₂⁴×C₆₁	1	120	yes	—
70	C₂³	C₂×Q₁₆×C₆₁	1	120	yes	direct product factor
71	C₂³	C₂²×D₄×C₆₁	1	35	yes	—
72	Q₈	—	2	240	no	—
73	Q₈	C₂⁵×C₆₁	1	32	yes	—
74	C₂×C₄	—	2	240	no	—
75	C₂×C₄	C₂²×D₄×C₆₁	1	105	yes	—
76	C₂×C₄	C₂⁵×C₆₁	1	15	yes	—
77	C₈	C₂⁵×C₆₁	1	16	yes	—
78	C₂²	C₂²×Q₁₆×C₆₁	1	140	yes	direct product factor
79	C₂²	C₂³×D₄×C₆₁	1	15	yes	—
80	C₄	—	2	32	no	—
81	C₄	C₂³×D₄×C₆₁	1	15	yes	—
82	C₄	C₂⁶×C₆₁	1	1	yes	characteristic
83	C₂	C₂³×Q₁₆×C₆₁	1	30	yes	direct product factor
84	C₂	C₂⁴×D₄×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₆₁	D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₆₁
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₆₁ ◁ C₂⁴×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₁₆×C₆₁	C₂	direct (N × H)
8. C₂²×Q₁₆×C₆₁	C₂²	direct (N × H)
16. C₂×Q₁₆×C₆₁	C₂³	direct (N × H)
18. C₂⁴×C₆₁	Q₁₆	direct (N × H)
25. Q₁₆×C₆₁	C₂⁴	direct (N × H)
27. C₂³×C₆₁	C₂×Q₁₆	direct (N × H)
35. C₂⁴×Q₁₆	C₆₁	direct (N × H)
36. C₂²×C₆₁	C₂²×Q₁₆	direct (N × H)
42. C₂³×Q₁₆	C₂×C₆₁	direct (N × H)
44. C₂×C₆₁	C₂³×Q₁₆	direct (N × H)
50. C₂²×Q₁₆	C₂²×C₆₁	direct (N × H)
52. C₆₁	C₂⁴×Q₁₆	direct (N × H)
59. C₂×Q₁₆	C₂³×C₆₁	direct (N × H)
61. C₂⁴	Q₁₆×C₆₁	direct (N × H)
68. Q₁₆	C₂⁴×C₆₁	direct (N × H)
70. C₂³	C₂×Q₁₆×C₆₁	direct (N × H)
78. C₂²	C₂²×Q₁₆×C₆₁	direct (N × H)
83. C₂	C₂³×Q₁₆×C₆₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₆₁	2
C₂³×Q₁₆×C₆₁	2
C₂⁴×C₈×C₆₁	2
C₂⁴×Q₁₆	61

Subgroups — C24×Q16×C61 #29566512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×Q₁₆×C₆₁ #29566512