Subgroups — #36442111

Order 22784.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018326	C₂	1	2	yes	maximal
2	C₈₉×#1397	C₂	1	1	yes	maximal, characteristic
3	#19151351	—	2	2	no	—
4	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
7	C₄⋊#10623494	—	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	complemented
13	C₂×#10623494	—	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	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₈₉	#108	1	1	yes	characteristic
25	#10623494	—	16	2	no	—
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	—	89	1	no	maximal, Sylow
31	C₂×C₈₉	#1372	1	1	yes	characteristic
32	C₂×C₈₉	#1426	1	2	yes	—
33	#1397	D₈₉	1	1	yes	characteristic
34	#1372	—	89	2	no	—
35	C₈₉	#7463	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃₂	Dic₈₉	1	1	yes	characteristic
37	C₂×C₃₂	Dic₈₉	1	1	yes	characteristic
38	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
39	#335	—	178	2	no	—
40	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
41	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
42	C₄⋊C₈	—	356	2	no	—
43	C₄×C₈	D₃₅₆	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	#10623494	1	2	yes	complemented
46	C₂×C₈	—	712	2	no	—
47	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
48	C₂×C₈	Dic₃₅₆	1	1	yes	characteristic
49	C₄²	D₇₁₂	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₂×#10623494	1	1	yes	characteristic
52	C₁₆	#13083558	1	1	yes	characteristic
53	C₂×C₄	#15667946	1	1	yes	characteristic
54	C₂×C₄	#15667957	1	2	yes	—
55	C₈	—	2	1	no	—
56	C₈	—	1424	2	no	—
57	C₈	C₄⋊#10623494	1	1	yes	characteristic
58	C₈	#15667948	1	1	yes	characteristic
59	C₂²	#19151426	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#19151351	1	1	yes	characteristic
62	C₄	#19151428	1	1	yes	characteristic
63	C₂	#26018326	1	1	yes	characteristic
64	C₂	#26018458	1	2	yes	—

	subgroup	G/H
center	C₂²	#19151426
commutator	C₁₆×C₈₉	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₈₉×#1397	C₂
socle, prime radical	C₂²×C₈₉	#367
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	#1397	D₈₉
hypocenter	C₈₉	#7463
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₈₉	C₈	semidirect
35. C₈₉	#7463	semidirect
45. C₃₂	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26018326	2
C₈₉×#1397	2
#7463	89