Subgroups — C₈₉×#8813 #36386661

Order 22784.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#1844	C₂	1	6	yes	maximal
2	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#106×C₈₉	—	2	12	no	—
4	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
5	#384×C₈₉	C₂²	1	4	yes	—
6	C₈²×C₈₉	C₄	1	4	yes	complemented
7	C₂²×C₈×C₈₉	D₄	1	6	yes	—
8	C₂×C₄⋊C₄×C₈₉	—	2	12	no	—
9	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
10	#106×C₈₉	—	4	24	no	—
11	C₄×C₈×C₈₉	C₂×C₄	1	6	yes	—
12	C₄×C₈×C₈₉	Q₈	1	6	yes	—
13	C₂²×C₄×C₈₉	—	4	4	no	—
14	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
15	C₂×C₈×C₈₉	C₄⋊C₄	1	12	yes	—
16	C₂×C₈×C₈₉	SD₁₆	1	24	yes	—
17	C₄⋊C₄×C₈₉	—	4	24	no	—
18	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
19	C₄²×C₈₉	C₂×Q₈	1	3	yes	—
20	C₂³×C₈₉	#127	1	1	yes	characteristic
21	C₂×C₄×C₈₉	—	8	12	no	—
22	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
23	C₂×C₄×C₈₉	C₄⋊Q₈	1	3	yes	—
24	C₂×C₄×C₈₉	C₂×SD₁₆	1	12	yes	—
25	C₈×C₈₉	#106	1	24	yes	complemented
26	C₂²×C₈₉	#384	1	1	yes	characteristic
27	C₂²×C₈₉	#492	1	6	yes	—
28	C₄×C₈₉	—	16	4	no	—
29	C₄×C₈₉	C₂×#106	1	6	yes	—
30	C₄×C₈₉	#498	1	6	yes	—
31	#8813	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₈₉	#1705	1	4	yes	—
33	C₂×C₈₉	#1844	1	3	yes	—
34	#1844	C₂×C₈₉	1	6	yes	—
35	C₂×C₈²	C₂×C₈₉	1	1	yes	characteristic
36	C₈₉	#8813	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×#106	—	2	12	no	—
38	C₂×C₄×C₈	C₂²×C₈₉	1	3	yes	—
39	#384	C₂²×C₈₉	1	4	yes	—
40	C₈²	C₄×C₈₉	1	4	yes	complemented
41	C₂²×C₈	D₄×C₈₉	1	6	yes	—
42	C₂×C₄⋊C₄	—	2	12	no	—
43	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
44	#106	—	4	24	no	—
45	C₄×C₈	C₂×C₄×C₈₉	1	6	yes	—
46	C₄×C₈	Q₈×C₈₉	1	6	yes	—
47	C₂²×C₄	—	4	4	no	—
48	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
49	C₂×C₈	C₄⋊C₄×C₈₉	1	12	yes	—
50	C₂×C₈	SD₁₆×C₈₉	1	24	yes	—
51	C₄⋊C₄	—	4	24	no	—
52	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
53	C₄²	C₂×Q₈×C₈₉	1	3	yes	—
54	C₂³	#127×C₈₉	1	1	yes	characteristic
55	C₂×C₄	—	8	12	no	—
56	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	3	yes	—
57	C₂×C₄	C₄⋊Q₈×C₈₉	1	3	yes	—
58	C₂×C₄	C₂×SD₁₆×C₈₉	1	12	yes	—
59	C₈	#106×C₈₉	1	24	yes	complemented
60	C₂²	#384×C₈₉	1	1	yes	characteristic
61	C₂²	#492×C₈₉	1	6	yes	—
62	C₄	—	16	4	no	—
63	C₄	C₂×#106×C₈₉	1	6	yes	—
64	C₄	#498×C₈₉	1	6	yes	—
65	C₂	C₈₉×#1705	1	4	yes	—
66	C₂	C₈₉×#1844	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	#127
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₈²×C₈₉	C₄	semidirect
25. C₈×C₈₉	#106	semidirect
31. #8813	C₈₉	direct (N × H)
36. C₈₉	#8813	direct (N × H)
40. C₈²	C₄×C₈₉	semidirect
59. C₈	#106×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#1844	2
C₂×C₈²×C₈₉	2
#8813	89

Subgroups — C89×#8813 #36386661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#8813 #36386661