Subgroups — C₈₉×#7464 #36385312

Order 22784.

76 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×#1373	C₂	1	2	yes	maximal
3	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
5	#364×C₈₉	—	2	2	no	—
6	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
7	M₃₂×C₈₉	—	4	2	no	—
8	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
11	C₃₂×C₈₉	—	2	1	no	—
12	C₃₂×C₈₉	C₈	1	2	yes	—
13	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
15	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
16	C₁₆×C₈₉	—	2	1	no	—
17	C₁₆×C₈₉	—	8	2	no	—
18	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₈₉	—	2	1	no	—
21	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
22	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₈₉	#108	1	1	yes	characteristic
24	C₈×C₈₉	SD₃₂	1	2	yes	—
25	C₂²×C₈₉	#368	1	1	yes	characteristic
26	C₄×C₈₉	—	4	1	no	—
27	C₄×C₈₉	#335	1	1	yes	characteristic
28	C₄×C₈₉	#367	1	1	yes	characteristic
29	#7464	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₈₉	—	2	1	no	—
31	C₂×C₈₉	#1372	1	1	yes	characteristic
32	#1397	C₂×C₈₉	1	1	yes	characteristic
33	#1373	C₂×C₈₉	1	2	yes	—
34	C₈₉	#7464	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃₂	C₄×C₈₉	1	1	yes	characteristic
36	C₂×C₃₂	C₄×C₈₉	1	1	yes	characteristic
37	#364	—	2	2	no	—
38	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
39	M₃₂	—	4	2	no	—
40	C₂×C₁₆	C₂×C₄×C₈₉	1	1	yes	characteristic
41	C₂×C₁₆	Q₈×C₈₉	1	1	yes	characteristic
42	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
43	C₃₂	—	2	1	no	—
44	C₃₂	C₈×C₈₉	1	2	yes	—
45	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
46	C₂×C₈	D₈×C₈₉	1	1	yes	characteristic
47	C₄²	Q₁₆×C₈₉	1	1	yes	characteristic
48	C₁₆	—	2	1	no	—
49	C₁₆	—	8	2	no	—
50	C₁₆	C₂×C₈×C₈₉	1	1	yes	characteristic
51	C₁₆	M₁₆×C₈₉	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	#107×C₈₉	1	1	yes	characteristic
54	C₈	C₄⋊C₈×C₈₉	1	1	yes	characteristic
55	C₈	#108×C₈₉	1	1	yes	characteristic
56	C₈	SD₃₂×C₈₉	1	2	yes	—
57	C₂²	#368×C₈₉	1	1	yes	characteristic
58	C₄	—	4	1	no	—
59	C₄	#335×C₈₉	1	1	yes	characteristic
60	C₄	#367×C₈₉	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	C₈₉×#1372	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₉	#367
commutator	C₁₆	C₂×C₈×C₈₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₈₉
socle	C₂×C₈₉	#1372
prime radical	C₂²×C₈₉	#368
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂×C₈×C₈₉ ◁ C₄×C₈×C₈₉ ◁ C₄×C₁₆×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
29. #7464	C₈₉	direct (N × H)
34. C₈₉	#7464	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×#1397	2
C₈₉×#1373	2
#7464	89

Subgroups — C89×#7464 #36385312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7464 #36385312