Subgroups — C₈₉×#7544 #36385392

Order 22784.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×M₁₂₈	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
5	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
6	C₆₄×C₈₉	—	2	2	no	—
7	C₂×C₁₆×C₈₉	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₈₉	C₈	1	1	yes	characteristic
10	C₃₂×C₈₉	—	2	1	no	—
11	C₃₂×C₈₉	D₄	1	1	yes	characteristic
12	C₃₂×C₈₉	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₈₉	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₈₉	C₁₆	1	1	yes	characteristic
16	C₁₆×C₈₉	—	2	1	no	—
17	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₈₉	—	2	1	no	—
20	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
21	C₈×C₈₉	C₃₂	1	2	yes	—
22	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₈₉	M₃₂	1	1	yes	characteristic
24	C₂²×C₈₉	M₆₄	1	1	yes	characteristic
25	C₄×C₈₉	—	2	2	no	—
26	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
27	C₄×C₈₉	C₂×C₃₂	1	1	yes	characteristic
28	#7544	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₈₉	—	2	1	no	—
30	C₂×C₈₉	C₄⋊C₃₂	1	1	yes	characteristic
31	M₁₂₈	C₂×C₈₉	1	2	yes	—
32	C₄×C₃₂	C₂×C₈₉	1	1	yes	characteristic
33	C₈₉	#7544	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₃₂	C₄×C₈₉	1	1	yes	characteristic
35	C₂×C₃₂	C₂²×C₈₉	1	1	yes	characteristic
36	C₄×C₁₆	C₄×C₈₉	1	1	yes	characteristic
37	C₆₄	—	2	2	no	—
38	C₂×C₁₆	C₈×C₈₉	1	1	yes	characteristic
39	C₂×C₁₆	C₂×C₄×C₈₉	1	1	yes	characteristic
40	C₄×C₈	C₈×C₈₉	1	1	yes	characteristic
41	C₃₂	—	2	1	no	—
42	C₃₂	D₄×C₈₉	1	1	yes	characteristic
43	C₃₂	Q₈×C₈₉	1	1	yes	characteristic
44	C₂×C₈	C₁₆×C₈₉	1	1	yes	characteristic
45	C₂×C₈	C₂×C₈×C₈₉	1	1	yes	characteristic
46	C₄²	C₁₆×C₈₉	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	C₄⋊C₄×C₈₉	1	1	yes	characteristic
49	C₁₆	M₁₆×C₈₉	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₂×C₁₆×C₈₉	1	1	yes	characteristic
52	C₈	C₃₂×C₈₉	1	2	yes	—
53	C₈	C₄⋊C₈×C₈₉	1	1	yes	characteristic
54	C₈	M₃₂×C₈₉	1	1	yes	characteristic
55	C₂²	M₆₄×C₈₉	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	C₄⋊C₁₆×C₈₉	1	1	yes	characteristic
58	C₄	C₂×C₃₂×C₈₉	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	C₈₉×C₄⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂×C₈₉	D₄
commutator	C₄	C₂×C₃₂×C₈₉
Frattini, cosocle	C₂×C₃₂	C₂²×C₈₉
socle	C₂×C₈₉	C₄⋊C₃₂
prime radical	C₂²×C₈₉	M₆₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂×C₈₉ ◁ C₂×C₃₂×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
28. #7544	C₈₉	direct (N × H)
33. C₈₉	#7544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×M₁₂₈	2
C₄×C₃₂×C₈₉	2
#7544	89

Subgroups — C89×#7544 #36385392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7544 #36385392