Subgroups — C₂³×2₋¹⁺⁴×C₆₁ #29566532

Order 15616.

59806 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₆₁	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#41×C₆₁	C₂	1	10	yes	maximal, complemented
3	C₂⁴×Q₈×C₆₁	C₂	1	5	yes	maximal, complemented
4	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	1792	yes	direct product factor
5	C₂²×#41×C₆₁	C₂²	1	560	yes	complemented
6	C₂³×Q₈×C₆₁	C₂²	1	10	yes	—
7	C₂³×Q₈×C₆₁	C₂²	1	280	yes	complemented
8	C₂³×D₄×C₆₁	C₂²	1	10	yes	—
9	C₂⁴×C₄×C₆₁	C₂²	1	15	yes	—
10	C₂⁵×C₆₁	C₂³	1	5	yes	—
11	2₋¹⁺⁴×C₆₁	C₂³	1	4096	yes	direct product factor
12	C₂×#41×C₆₁	C₂³	1	4480	yes	complemented
13	C₂²×Q₈×C₆₁	C₂³	1	280	yes	—
14	C₂²×Q₈×C₆₁	C₂³	1	2240	yes	complemented
15	C₂²×D₄×C₆₁	C₂³	1	280	yes	—
16	C₂³×C₄×C₆₁	C₂³	1	420	yes	—
17	C₂³×C₄×C₆₁	C₂³	1	10	yes	—
18	C₂⁴×C₆₁	—	2	40	no	—
19	C₂⁴×C₆₁	C₂⁴	1	70	yes	—
20	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
21	#41×C₆₁	C₂⁴	1	5120	yes	complemented
22	C₂×Q₈×C₆₁	C₂⁴	1	1120	yes	—
23	C₂×Q₈×C₆₁	C₂⁴	1	2560	yes	complemented
24	C₂×D₄×C₆₁	C₂⁴	1	1120	yes	—
25	C₂²×C₄×C₆₁	C₂⁴	1	140	yes	—
26	C₂²×C₄×C₆₁	C₂⁴	1	1680	yes	—
27	C₂³×C₆₁	—	2	280	no	—
28	C₂³×C₆₁	2₋¹⁺⁴	1	8	yes	direct product factor
29	C₂³×C₆₁	C₂⁵	1	140	yes	—
30	C₂³×C₆₁	C₂⁵	1	7	yes	—
31	Q₈×C₆₁	C₂⁵	1	640	yes	—
32	D₄×C₆₁	C₂⁵	1	640	yes	—
33	C₂×C₄×C₆₁	C₂⁵	1	280	yes	—
34	C₂×C₄×C₆₁	C₂⁵	1	960	yes	—
35	C₂³×2₋¹⁺⁴	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₆₁	—	2	280	no	—
37	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	28	yes	direct product factor
38	C₂²×C₆₁	C₂⁶	1	40	yes	—
39	C₂²×C₆₁	C₂⁶	1	7	yes	—
40	C₄×C₆₁	C₂⁶	1	80	yes	—
41	C₂²×2₋¹⁺⁴	C₂×C₆₁	1	112	yes	direct product factor
42	C₂³×#41	C₂×C₆₁	1	10	yes	complemented
43	C₂⁴×Q₈	C₂×C₆₁	1	5	yes	complemented
44	C₂×C₆₁	—	2	40	no	—
45	C₂×C₆₁	C₂²×2₋¹⁺⁴	1	14	yes	direct product factor
46	C₂×C₆₁	C₂⁷	1	1	yes	characteristic
47	C₂×2₋¹⁺⁴	C₂²×C₆₁	1	1792	yes	direct product factor
48	C₂²×#41	C₂²×C₆₁	1	560	yes	complemented
49	C₂³×Q₈	C₂²×C₆₁	1	10	yes	—
50	C₂³×Q₈	C₂²×C₆₁	1	280	yes	complemented
51	C₂³×D₄	C₂²×C₆₁	1	10	yes	—
52	C₂⁴×C₄	C₂²×C₆₁	1	15	yes	—
53	C₆₁	C₂³×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
54	C₂⁵	C₂³×C₆₁	1	5	yes	—
55	2₋¹⁺⁴	C₂³×C₆₁	1	4096	yes	direct product factor
56	C₂×#41	C₂³×C₆₁	1	4480	yes	complemented
57	C₂²×Q₈	C₂³×C₆₁	1	280	yes	—
58	C₂²×Q₈	C₂³×C₆₁	1	2240	yes	complemented
59	C₂²×D₄	C₂³×C₆₁	1	280	yes	—
60	C₂³×C₄	C₂³×C₆₁	1	10	yes	—
61	C₂³×C₄	C₂³×C₆₁	1	420	yes	—
62	C₂⁴	—	2	40	no	—
63	C₂⁴	C₂⁴×C₆₁	1	70	yes	—
64	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
65	#41	C₂⁴×C₆₁	1	5120	yes	complemented
66	C₂×Q₈	C₂⁴×C₆₁	1	1120	yes	—
67	C₂×Q₈	C₂⁴×C₆₁	1	2560	yes	complemented
68	C₂×D₄	C₂⁴×C₆₁	1	1120	yes	—
69	C₂²×C₄	C₂⁴×C₆₁	1	140	yes	—
70	C₂²×C₄	C₂⁴×C₆₁	1	1680	yes	—
71	C₂³	—	2	280	no	—
72	C₂³	2₋¹⁺⁴×C₆₁	1	8	yes	direct product factor
73	C₂³	C₂⁵×C₆₁	1	140	yes	—
74	C₂³	C₂⁵×C₆₁	1	7	yes	—
75	Q₈	C₂⁵×C₆₁	1	640	yes	—
76	D₄	C₂⁵×C₆₁	1	640	yes	—
77	C₂×C₄	C₂⁵×C₆₁	1	280	yes	—
78	C₂×C₄	C₂⁵×C₆₁	1	960	yes	—
79	C₂²	—	2	280	no	—
80	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	28	yes	direct product factor
81	C₂²	C₂⁶×C₆₁	1	40	yes	—
82	C₂²	C₂⁶×C₆₁	1	7	yes	—
83	C₄	C₂⁶×C₆₁	1	80	yes	—
84	C₂	—	2	40	no	—
85	C₂	C₂²×2₋¹⁺⁴×C₆₁	1	14	yes	direct product factor
86	C₂	C₂⁷×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁷×C₆₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×2₋¹⁺⁴×C₆₁	C₂	direct (N × H)
2. C₂³×#41×C₆₁	C₂	semidirect
3. C₂⁴×Q₈×C₆₁	C₂	semidirect
4. C₂×2₋¹⁺⁴×C₆₁	C₂²	direct (N × H)
5. C₂²×#41×C₆₁	C₂²	semidirect
7. C₂³×Q₈×C₆₁	C₂²	semidirect
11. 2₋¹⁺⁴×C₆₁	C₂³	direct (N × H)
12. C₂×#41×C₆₁	C₂³	semidirect
14. C₂²×Q₈×C₆₁	C₂³	semidirect
21. #41×C₆₁	C₂⁴	semidirect
23. C₂×Q₈×C₆₁	C₂⁴	semidirect
28. C₂³×C₆₁	2₋¹⁺⁴	direct (N × H)
35. C₂³×2₋¹⁺⁴	C₆₁	direct (N × H)
37. C₂²×C₆₁	C₂×2₋¹⁺⁴	direct (N × H)
41. C₂²×2₋¹⁺⁴	C₂×C₆₁	direct (N × H)
42. C₂³×#41	C₂×C₆₁	semidirect
43. C₂⁴×Q₈	C₂×C₆₁	semidirect
45. C₂×C₆₁	C₂²×2₋¹⁺⁴	direct (N × H)
47. C₂×2₋¹⁺⁴	C₂²×C₆₁	direct (N × H)
48. C₂²×#41	C₂²×C₆₁	semidirect
50. C₂³×Q₈	C₂²×C₆₁	semidirect
53. C₆₁	C₂³×2₋¹⁺⁴	direct (N × H)
55. 2₋¹⁺⁴	C₂³×C₆₁	direct (N × H)
56. C₂×#41	C₂³×C₆₁	semidirect
58. C₂²×Q₈	C₂³×C₆₁	semidirect
65. #41	C₂⁴×C₆₁	semidirect
67. C₂×Q₈	C₂⁴×C₆₁	semidirect
72. C₂³	2₋¹⁺⁴×C₆₁	direct (N × H)
80. C₂²	C₂×2₋¹⁺⁴×C₆₁	direct (N × H)
85. C₂	C₂²×2₋¹⁺⁴×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
C₂²×2₋¹⁺⁴×C₆₁	2
C₂³×#41×C₆₁	2
C₂⁴×Q₈×C₆₁	2
C₂³×2₋¹⁺⁴	61

Subgroups — C23×2−1+4×C61 #29566532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×2₋¹⁺⁴×C₆₁ #29566532