Subgroups — C₈₉×#7463 #36385311

Order 22784.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#1397	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1372	C₂	1	2	yes	maximal
3	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
6	#335×C₈₉	—	2	2	no	—
7	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
8	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
9	C₄⋊C₈×C₈₉	—	4	2	no	—
10	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
11	C₃₂×C₈₉	—	2	1	no	—
12	C₃₂×C₈₉	C₈	1	2	yes	complemented
13	C₂×C₈×C₈₉	—	8	2	no	—
14	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
16	C₄²×C₈₉	D₈	1	1	yes	characteristic
17	C₁₆×C₈₉	—	2	1	no	—
18	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
21	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
22	C₈×C₈₉	—	2	1	no	—
23	C₈×C₈₉	—	16	2	no	—
24	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₈₉	#108	1	1	yes	characteristic
26	C₂²×C₈₉	#367	1	1	yes	characteristic
27	C₄×C₈₉	—	2	2	no	—
28	C₄×C₈₉	#335	1	1	yes	characteristic
29	C₄×C₈₉	#368	1	1	yes	characteristic
30	#7463	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₈₉	#1372	1	1	yes	characteristic
32	C₂×C₈₉	#1426	1	2	yes	—
33	#1397	C₂×C₈₉	1	1	yes	characteristic
34	#1372	C₂×C₈₉	1	2	yes	—
35	C₈₉	#7463	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃₂	C₄×C₈₉	1	1	yes	characteristic
37	C₂×C₃₂	C₄×C₈₉	1	1	yes	characteristic
38	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
39	#335	—	2	2	no	—
40	C₂×C₁₆	C₂×C₄×C₈₉	1	1	yes	characteristic
41	C₂×C₁₆	Q₈×C₈₉	1	1	yes	characteristic
42	C₄⋊C₈	—	4	2	no	—
43	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	C₈×C₈₉	1	2	yes	complemented
46	C₂×C₈	—	8	2	no	—
47	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
48	C₂×C₈	Q₁₆×C₈₉	1	1	yes	characteristic
49	C₄²	D₈×C₈₉	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₂×C₈×C₈₉	1	1	yes	characteristic
52	C₁₆	M₁₆×C₈₉	1	1	yes	characteristic
53	C₂×C₄	#107×C₈₉	1	1	yes	characteristic
54	C₂×C₄	SD₃₂×C₈₉	1	2	yes	—
55	C₈	—	2	1	no	—
56	C₈	—	16	2	no	—
57	C₈	C₄⋊C₈×C₈₉	1	1	yes	characteristic
58	C₈	#108×C₈₉	1	1	yes	characteristic
59	C₂²	#367×C₈₉	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#335×C₈₉	1	1	yes	characteristic
62	C₄	#368×C₈₉	1	1	yes	characteristic
63	C₂	C₈₉×#1372	1	1	yes	characteristic
64	C₂	C₈₉×#1426	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₈₉	#367
commutator	C₁₆	C₂×C₈×C₈₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₈₉
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ C₄×C₈×C₈₉ ◁ C₄×C₁₆×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₈₉	C₈	semidirect
30. #7463	C₈₉	direct (N × H)
35. C₈₉	#7463	direct (N × H)
45. C₃₂	C₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#1397	2
C₈₉×#1372	2
#7463	89

Subgroups — C89×#7463 #36385311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7463 #36385311