Subgroups — C₈₉×#7456 #36385304

Order 22784.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#1422	C₂	1	2	yes	maximal
2	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₈₉	—	2	2	no	—
4	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
6	C₈²×C₈₉	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₈₉	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₈₉	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
10	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
11	C₄×C₈×C₈₉	—	2	1	no	—
12	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₈₉	—	4	2	no	—
14	C₂×C₈×C₈₉	—	2	2	no	—
15	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
18	C₁₆×C₈₉	—	2	1	no	—
19	C₁₆×C₈₉	—	2	1	no	—
20	C₁₆×C₈₉	C₁₆	1	2	yes	—
21	C₁₆×C₈₉	SD₁₆	1	2	yes	—
22	C₂×C₄×C₈₉	—	2	1	no	—
23	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₈₉	—	4	2	no	—
25	C₈×C₈₉	#106	1	1	yes	characteristic
26	C₈×C₈₉	#108	1	1	yes	characteristic
27	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₈₉	M₃₂	1	1	yes	characteristic
29	C₂²×C₈₉	#364	1	1	yes	characteristic
30	C₄×C₈₉	—	2	2	no	—
31	C₄×C₈₉	#334	1	1	yes	characteristic
32	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
33	#7456	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₈₉	—	2	1	no	—
35	C₂×C₈₉	#1367	1	1	yes	characteristic
36	#1422	C₂×C₈₉	1	2	yes	—
37	#1311	C₂×C₈₉	1	1	yes	characteristic
38	C₈₉	#7456	1	1	yes	characteristic, direct product factor, Sylow
39	M₆₄	—	2	2	no	—
40	C₄×C₁₆	C₄×C₈₉	1	1	yes	characteristic
41	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
42	C₈²	C₄×C₈₉	1	1	yes	characteristic
43	C₂×C₁₆	C₈×C₈₉	1	1	yes	characteristic
44	C₂×C₁₆	C₈×C₈₉	1	1	yes	characteristic
45	C₂×C₁₆	D₄×C₈₉	1	1	yes	characteristic
46	C₂×C₁₆	Q₈×C₈₉	1	1	yes	characteristic
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
49	C₃₂	—	4	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
52	C₂×C₈	C₂×C₈×C₈₉	1	1	yes	characteristic
53	C₄²	M₁₆×C₈₉	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	2	1	no	—
56	C₁₆	C₁₆×C₈₉	1	2	yes	—
57	C₁₆	SD₁₆×C₈₉	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊C₈×C₈₉	1	1	yes	characteristic
60	C₈	—	4	2	no	—
61	C₈	#106×C₈₉	1	1	yes	characteristic
62	C₈	#108×C₈₉	1	1	yes	characteristic
63	C₈	C₂×C₁₆×C₈₉	1	1	yes	characteristic
64	C₈	M₃₂×C₈₉	1	1	yes	characteristic
65	C₂²	#364×C₈₉	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#334×C₈₉	1	1	yes	characteristic
68	C₄	C₄⋊C₁₆×C₈₉	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	C₈₉×#1367	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₈₉	#106
commutator	C₈	C₂×C₁₆×C₈₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₈₉
socle	C₂×C₈₉	#1367
prime radical	C₂²×C₈₉	#364
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₈₉ ◁ C₂×C₁₆×C₈₉ ◁ C₄×C₁₆×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
33. #7456	C₈₉	direct (N × H)
38. C₈₉	#7456	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×#1422	2
C₈₉×#1311	2
#7456	89

Subgroups — C89×#7456 #36385304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7456 #36385304