Subgroups — #36442098

Order 22784.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018320	C₂	1	1	yes	maximal, characteristic
2	#26018318	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1398	C₂	1	1	yes	maximal, characteristic
4	#19151419	—	2	1	no	—
5	#19151349	—	2	1	no	—
6	M₆₄×C₈₉	C₄	1	1	yes	characteristic
7	M₆₄×C₈₉	C₄	1	1	yes	characteristic
8	#346×C₈₉	C₂²	1	1	yes	characteristic
9	#15667952	—	4	1	no	—
10	C₄⋊#10623494	—	4	1	no	—
11	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
13	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
14	C₃₂×C₈₉	—	2	1	no	—
15	C₃₂×C₈₉	C₈	1	2	yes	complemented
16	C₂×#10623494	—	8	1	no	—
17	#13083549	—	8	1	no	—
18	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
20	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
21	C₁₆×C₈₉	—	2	1	no	—
22	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
26	C₈×C₈₉	—	2	1	no	—
27	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₈₉	#108	1	1	yes	characteristic
29	#10623494	—	16	1	no	—
30	C₂²×C₈₉	#365	1	1	yes	characteristic
31	C₄×C₈₉	—	4	1	no	—
32	C₄×C₈₉	#334	1	1	yes	characteristic
33	C₄×C₈₉	#365	1	1	yes	characteristic
34	#7457	—	89	1	no	maximal, Sylow
35	C₂×C₈₉	—	2	1	no	—
36	C₂×C₈₉	#1368	1	1	yes	characteristic
37	#1398	D₈₉	1	1	yes	characteristic
38	#1369	—	89	1	no	—
39	#1368	—	89	1	no	—
40	C₈₉	#7457	1	1	yes	characteristic, complemented, Sylow
41	M₆₄	Dic₈₉	1	1	yes	characteristic
42	M₆₄	Dic₈₉	1	1	yes	characteristic
43	#364	—	178	1	no	—
44	#346	C₂×D₈₉	1	1	yes	characteristic
45	#334	—	178	1	no	—
46	M₃₂	—	356	1	no	—
47	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
48	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
49	C₄⋊C₈	—	356	1	no	—
50	C₄×C₈	D₃₅₆	1	1	yes	characteristic
51	C₃₂	—	2	1	no	—
52	C₃₂	#10623494	1	2	yes	complemented
53	C₂×C₈	—	712	1	no	—
54	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
55	C₂×C₈	#13083563	1	1	yes	characteristic
56	C₄²	#13083563	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	—	712	1	no	—
59	C₁₆	C₂×#10623494	1	1	yes	characteristic
60	C₁₆	#13083558	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#15667944	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	1424	1	no	—
65	C₈	C₄⋊#10623494	1	1	yes	characteristic
66	C₈	#15667948	1	1	yes	characteristic
67	C₂²	#19151421	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#19151349	1	1	yes	characteristic
70	C₄	#19151421	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#26018318	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26018318
commutator	C₁₆×C₈₉	C₂×C₈
Frattini	#346	C₂×D₈₉
Fitting	C₈₉×#1398	C₂
socle	C₂×C₈₉	#1368
prime radical	C₂²×C₈₉	#365
cosocle	#346×C₈₉	C₂²
hypercenter	#1398	D₈₉
hypocenter	C₈₉	#7457
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #1398

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₈₉	C₈	semidirect
40. C₈₉	#7457	semidirect
52. C₃₂	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26018320	2
#26018318	2
C₈₉×#1398	2
#7457	89