Subgroups — #36442330

Order 22784.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018452	C₂	1	2	yes	maximal
2	C₂×C₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151424	—	2	2	no	—
4	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
5	C₆₄×C₈₉	C₄	1	2	yes	complemented
6	#15667946	—	4	2	no	—
7	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
8	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
9	C₃₂×C₈₉	Q₈	1	1	yes	characteristic
10	C₄⋊Dic₈₉	—	8	2	no	—
11	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
12	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
13	C₁₆×C₈₉	Q₁₆	1	1	yes	characteristic
14	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
15	C₂×Dic₈₉	—	16	2	no	—
16	C₈×C₈₉	#107	1	1	yes	characteristic
17	C₈×C₈₉	Q₃₂	1	1	yes	characteristic
18	C₂²×C₈₉	D₃₂	1	1	yes	characteristic
19	C₄×C₈₉	#366	1	1	yes	characteristic
20	C₄×C₈₉	Q₆₄	1	1	yes	characteristic
21	Dic₈₉	—	32	2	no	—
22	#7546	—	89	1	no	maximal, Sylow
23	C₂×C₈₉	#1423	1	1	yes	characteristic
24	C₂×C₈₉	SD₁₂₈	1	2	yes	—
25	C₂×C₆₄	D₈₉	1	1	yes	characteristic
26	#1423	—	89	2	no	—
27	C₈₉	#7546	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
29	#366	—	178	2	no	—
30	C₆₄	Dic₈₉	1	2	yes	complemented
31	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
32	#107	—	356	2	no	—
33	C₃₂	Dic₁₇₈	1	1	yes	characteristic
34	C₃₂	C₂×Dic₈₉	1	1	yes	characteristic
35	C₂×C₈	D₇₁₂	1	1	yes	characteristic
36	C₄⋊C₄	—	712	2	no	—
37	C₁₆	C₄⋊Dic₈₉	1	1	yes	characteristic
38	C₁₆	Dic₃₅₆	1	1	yes	characteristic
39	C₂×C₄	—	1424	2	no	—
40	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
41	C₈	#15667946	1	1	yes	characteristic
42	C₈	Dic₇₁₂	1	1	yes	characteristic
43	C₂²	D₂₈₄₈	1	1	yes	characteristic
44	C₄	—	2848	2	no	—
45	C₄	#19151424	1	1	yes	characteristic
46	C₄	Dic₁₄₂₄	1	1	yes	characteristic
47	C₂	#26018452	1	1	yes	characteristic
48	C₂	#26018468	1	2	yes	—

	subgroup	G/H
center	C₂²	D₂₈₄₈
commutator	C₃₂×C₈₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₂×C₆₄×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	D₃₂
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	C₂×C₆₄	D₈₉
hypocenter	C₈₉	#7546
LCS	G ▷ C₃₂×C₈₉ ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
5. C₆₄×C₈₉	C₄	semidirect
27. C₈₉	#7546	semidirect
30. C₆₄	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26018452	2
C₂×C₆₄×C₈₉	2
#7546	89