Subgroups — C₂³×C₄×C₈×C₈₉ #36437898

Order 22784.

4638 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₈₉	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₈₉	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₈₉	C₄	1	2	yes	—
5	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₈₉	C₄	1	32	yes	direct product factor
7	C₂³×C₈×C₈₉	C₂²	1	28	yes	—
8	C₂²×C₄²×C₈₉	C₄	1	14	yes	—
9	C₂²×C₄²×C₈₉	C₂²	1	14	yes	—
10	C₂×C₄×C₈×C₈₉	C₂²	1	112	yes	direct product factor
11	C₂⁵×C₈₉	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₄×C₈₉	C₈	1	32	yes	direct product factor
13	C₂³×C₄×C₈₉	C₂×C₄	1	7	yes	—
14	C₂³×C₄×C₈₉	C₂×C₄	1	16	yes	—
15	C₂³×C₄×C₈₉	C₂×C₄	1	28	yes	—
16	C₂³×C₄×C₈₉	C₂³	1	7	yes	—
17	C₂²×C₈×C₈₉	C₂×C₄	1	224	yes	direct product factor
18	C₂²×C₈×C₈₉	C₂³	1	56	yes	—
19	C₂×C₄²×C₈₉	C₂×C₄	1	84	yes	—
20	C₂×C₄²×C₈₉	C₂³	1	28	yes	—
21	C₄×C₈×C₈₉	C₂³	1	64	yes	direct product factor
22	C₂⁴×C₈₉	C₄²	1	8	yes	—
23	C₂⁴×C₈₉	C₂×C₈	1	16	yes	—
24	C₂⁴×C₈₉	C₂²×C₄	1	7	yes	—
25	C₂²×C₄×C₈₉	C₄²	1	56	yes	—
26	C₂²×C₄×C₈₉	C₂×C₈	1	224	yes	direct product factor
27	C₂²×C₄×C₈₉	C₂²×C₄	1	21	yes	—
28	C₂²×C₄×C₈₉	C₂²×C₄	1	56	yes	—
29	C₂²×C₄×C₈₉	C₂²×C₄	1	56	yes	—
30	C₂²×C₄×C₈₉	C₂⁴	1	7	yes	—
31	C₂×C₈×C₈₉	C₂²×C₄	1	224	yes	direct product factor
32	C₂×C₈×C₈₉	C₂⁴	1	16	yes	—
33	C₄²×C₈₉	C₂²×C₄	1	56	yes	—
34	C₄²×C₈₉	C₂⁴	1	8	yes	—
35	C₂³×C₈₉	C₄×C₈	1	64	yes	direct product factor
36	C₂³×C₈₉	C₂×C₄²	1	28	yes	—
37	C₂³×C₈₉	C₂²×C₈	1	56	yes	—
38	C₂³×C₈₉	C₂³×C₄	1	7	yes	—
39	C₂×C₄×C₈₉	C₂×C₄²	1	84	yes	—
40	C₂×C₄×C₈₉	C₂²×C₈	1	224	yes	direct product factor
41	C₂×C₄×C₈₉	C₂³×C₄	1	7	yes	—
42	C₂×C₄×C₈₉	C₂³×C₄	1	16	yes	—
43	C₂×C₄×C₈₉	C₂³×C₄	1	28	yes	—
44	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
45	C₈×C₈₉	C₂³×C₄	1	32	yes	direct product factor
46	C₂²×C₈₉	C₂×C₄×C₈	1	112	yes	direct product factor
47	C₂²×C₈₉	C₂²×C₄²	1	14	yes	—
48	C₂²×C₈₉	C₂³×C₈	1	28	yes	—
49	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
50	C₄×C₈₉	C₂²×C₄²	1	14	yes	—
51	C₄×C₈₉	C₂³×C₈	1	32	yes	direct product factor
52	C₄×C₈₉	C₂⁴×C₄	1	2	yes	—
53	C₂³×C₄×C₈	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₈₉	C₂²×C₄×C₈	1	28	yes	direct product factor
55	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
56	C₂×C₈₉	C₂⁴×C₈	1	2	yes	—
57	C₂⁴×C₈	C₂×C₈₉	1	2	yes	—
58	C₂³×C₄²	C₂×C₈₉	1	1	yes	characteristic
59	C₂²×C₄×C₈	C₂×C₈₉	1	28	yes	direct product factor
60	C₈₉	C₂³×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
61	C₂⁴×C₄	C₄×C₈₉	1	2	yes	—
62	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
63	C₂³×C₈	C₄×C₈₉	1	32	yes	direct product factor
64	C₂³×C₈	C₂²×C₈₉	1	28	yes	—
65	C₂²×C₄²	C₄×C₈₉	1	14	yes	—
66	C₂²×C₄²	C₂²×C₈₉	1	14	yes	—
67	C₂×C₄×C₈	C₂²×C₈₉	1	112	yes	direct product factor
68	C₂⁵	C₂×C₄×C₈₉	1	1	yes	characteristic
69	C₂³×C₄	C₈×C₈₉	1	32	yes	direct product factor
70	C₂³×C₄	C₂×C₄×C₈₉	1	7	yes	—
71	C₂³×C₄	C₂×C₄×C₈₉	1	16	yes	—
72	C₂³×C₄	C₂×C₄×C₈₉	1	28	yes	—
73	C₂³×C₄	C₂³×C₈₉	1	7	yes	—
74	C₂²×C₈	C₂×C₄×C₈₉	1	224	yes	direct product factor
75	C₂²×C₈	C₂³×C₈₉	1	56	yes	—
76	C₂×C₄²	C₂×C₄×C₈₉	1	84	yes	—
77	C₂×C₄²	C₂³×C₈₉	1	28	yes	—
78	C₄×C₈	C₂³×C₈₉	1	64	yes	direct product factor
79	C₂⁴	C₄²×C₈₉	1	8	yes	—
80	C₂⁴	C₂×C₈×C₈₉	1	16	yes	—
81	C₂⁴	C₂²×C₄×C₈₉	1	7	yes	—
82	C₂²×C₄	C₄²×C₈₉	1	56	yes	—
83	C₂²×C₄	C₂×C₈×C₈₉	1	224	yes	direct product factor
84	C₂²×C₄	C₂²×C₄×C₈₉	1	21	yes	—
85	C₂²×C₄	C₂²×C₄×C₈₉	1	56	yes	—
86	C₂²×C₄	C₂²×C₄×C₈₉	1	56	yes	—
87	C₂²×C₄	C₂⁴×C₈₉	1	7	yes	—
88	C₂×C₈	C₂²×C₄×C₈₉	1	224	yes	direct product factor
89	C₂×C₈	C₂⁴×C₈₉	1	16	yes	—
90	C₄²	C₂²×C₄×C₈₉	1	56	yes	—
91	C₄²	C₂⁴×C₈₉	1	8	yes	—
92	C₂³	C₄×C₈×C₈₉	1	64	yes	direct product factor
93	C₂³	C₂×C₄²×C₈₉	1	28	yes	—
94	C₂³	C₂²×C₈×C₈₉	1	56	yes	—
95	C₂³	C₂³×C₄×C₈₉	1	7	yes	—
96	C₂×C₄	C₂×C₄²×C₈₉	1	84	yes	—
97	C₂×C₄	C₂²×C₈×C₈₉	1	224	yes	direct product factor
98	C₂×C₄	C₂³×C₄×C₈₉	1	7	yes	—
99	C₂×C₄	C₂³×C₄×C₈₉	1	16	yes	—
100	C₂×C₄	C₂³×C₄×C₈₉	1	28	yes	—
101	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
102	C₈	C₂³×C₄×C₈₉	1	32	yes	direct product factor
103	C₂²	C₂×C₄×C₈×C₈₉	1	112	yes	direct product factor
104	C₂²	C₂²×C₄²×C₈₉	1	14	yes	—
105	C₂²	C₂³×C₈×C₈₉	1	28	yes	—
106	C₂²	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
107	C₄	C₂²×C₄²×C₈₉	1	14	yes	—
108	C₄	C₂³×C₈×C₈₉	1	32	yes	direct product factor
109	C₄	C₂⁴×C₄×C₈₉	1	2	yes	—
110	C₂	C₂²×C₄×C₈×C₈₉	1	28	yes	direct product factor
111	C₂	C₂³×C₄²×C₈₉	1	1	yes	characteristic
112	C₂	C₂⁴×C₈×C₈₉	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle, prime radical	C₂⁵×C₈₉	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×C₈×C₈₉	C₂	direct (N × H)
6. C₂³×C₈×C₈₉	C₄	direct (N × H)
10. C₂×C₄×C₈×C₈₉	C₂²	direct (N × H)
12. C₂³×C₄×C₈₉	C₈	direct (N × H)
17. C₂²×C₈×C₈₉	C₂×C₄	direct (N × H)
21. C₄×C₈×C₈₉	C₂³	direct (N × H)
26. C₂²×C₄×C₈₉	C₂×C₈	direct (N × H)
31. C₂×C₈×C₈₉	C₂²×C₄	direct (N × H)
35. C₂³×C₈₉	C₄×C₈	direct (N × H)
40. C₂×C₄×C₈₉	C₂²×C₈	direct (N × H)
45. C₈×C₈₉	C₂³×C₄	direct (N × H)
46. C₂²×C₈₉	C₂×C₄×C₈	direct (N × H)
51. C₄×C₈₉	C₂³×C₈	direct (N × H)
53. C₂³×C₄×C₈	C₈₉	direct (N × H)
54. C₂×C₈₉	C₂²×C₄×C₈	direct (N × H)
59. C₂²×C₄×C₈	C₂×C₈₉	direct (N × H)
60. C₈₉	C₂³×C₄×C₈	direct (N × H)
63. C₂³×C₈	C₄×C₈₉	direct (N × H)
67. C₂×C₄×C₈	C₂²×C₈₉	direct (N × H)
69. C₂³×C₄	C₈×C₈₉	direct (N × H)
74. C₂²×C₈	C₂×C₄×C₈₉	direct (N × H)
78. C₄×C₈	C₂³×C₈₉	direct (N × H)
83. C₂²×C₄	C₂×C₈×C₈₉	direct (N × H)
88. C₂×C₈	C₂²×C₄×C₈₉	direct (N × H)
92. C₂³	C₄×C₈×C₈₉	direct (N × H)
97. C₂×C₄	C₂²×C₈×C₈₉	direct (N × H)
102. C₈	C₂³×C₄×C₈₉	direct (N × H)
103. C₂²	C₂×C₄×C₈×C₈₉	direct (N × H)
108. C₄	C₂³×C₈×C₈₉	direct (N × H)
110. C₂	C₂²×C₄×C₈×C₈₉	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×C₈×C₈₉	2
C₂³×C₄²×C₈₉	2
C₂²×C₄×C₈×C₈₉	2
C₂³×C₄×C₈	89

Subgroups — C23×C4×C8×C89 #36437898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×C₈×C₈₉ #36437898